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# Derivation Of Nernst Equation Pdf

Derivation of unstirred-layer transport number equations. This is the Nernst equation. And, is a kinetic definition of the standard potential, related to And, is a kinetic definition of the standard potential, related to microscopic reaction rates., 13/12/2017В В· ELECTROCHEMISTRY most important topic (NERNST equation) derivation and some of their formulas are explained in this video..Detailed information about Gibbs free вЂ¦.

### Derivation of unstirred-layer transport number equations

Butler-Volmer equation with concentration dependence?. The electrode potential difference between the two half-cells can be calculated using the Nernst equation. Concentration cells are galvanic (or voltaic) cells made of two half-cells, each of which containing the same electrodes, but different concentrations., Set equation=0 to get Nernst equation (no current) 0 = -рќњ‡рќ‘§рќ‘…рќ‘‡рќњ•[рќђ¶]рќњ•рќ‘Ґ - рќњ‡рќђ№рќ‘§2 [рќђ¶]рќњ•рќ‘‰рќњ•рќ‘Ґ Factor out- рќњ‡рќ‘§ 0= - рќњ‡рќ‘§(рќ‘…рќ‘‡рќњ•[рќђ¶]рќњ•рќ‘Ґ +рќђ№рќ‘§ [рќђ¶]рќњ•рќ‘‰рќњ•рќ‘Ґ) (We see that -рќњ‡рќ‘§ is a solution, so we can get rid of it.) Now we have the variables we want: 0=рќ‘…рќ‘‡рќњ•[рќђ¶]рќњ•рќ‘Ґ +рќђ№рќ‘§ [рќђ¶]рќњ•рќ‘‰рќњ•рќ‘Ґ Move the.

The Nernst Equation and Pourbaix Diagrams Click here for actual (non-printable) TLP pages. Note: DoITPoMS Teaching and Learning Packages are intended to be used interactively at a computer! The Nernst-Planck equation is a physical reformulation of FickвЂ™s macroscale diffusion law, extended to the motion of charged particles. The equation was established and experimentally underpinned by Nernst, and theoretically elaborated by Planck, and acquired growing significance when it was

Procedure to verify the Nernst equation, the standard potential of a redox system, and ion-ion interaction theory. Equation (2.53) is a general form of the Nernst equation. It expresses the emf involved in an electrochemical reaction, such as equation (2.46) in terms of R, T, F, n activities of products and activities of reactants.

Nernst Equation Table of Contents 1. Introduction 2. Biological Application of the Nerst Equation 3. Action Potential of a Neuron Cell 4. References 5. Outside Links The Nernst Equation enables one to determine electromotive forces (emf) of many processes, for instance the resting potential of cell membranes. We can then deduce the biological standard potentials which are important in studying The Nernst Equation and Pourbaix Diagrams Click here for actual (non-printable) TLP pages. Note: DoITPoMS Teaching and Learning Packages are intended to be used interactively at a computer!

Appendix A Derivation of the Nernst equation Consider the general reaction [1, 2] m AAГѕm BBГѕГѕ e Г°metalГћ m CCГѕm DDГѕ ; Г°A:1Гћ where m i are the stoichiometric coefп¬Ѓcients of вЂ¦ The electrode potential difference between the two half-cells can be calculated using the Nernst equation. Concentration cells are galvanic (or voltaic) cells made of two half-cells, each of which containing the same electrodes, but different concentrations.

So this is the Nernst equation, alright, we'll talk about why the Nernst equation is so important, we'll talk more about that at the end of the video. Right now, let's go ahead and derive another form of the Nernst equation, I should say, the form when you're talking about a certain temperature. So at 25 degrees C, right, most of our reactions take place at 25 degrees C, well, temperature Note that for this simple reaction, the Nernst equation shows that the equilibrium potential, E e is independent of the pH of the solution. Many half-cell reactions contain H + ions and their Nernst equations therefore depend on the pH .

Electrochemical Cells and the Nernst Equation Objective: To observe the effect of concentration on electrochemical potential; to use these results to verify the Nernst equation 2.1 Biophysics of neurons - Overview 2.2 Reversal potential - Nernst equation 2.3 Hodgin-Huxley Model 2.4 Threshold in the Hodgkin-Huxley Model

Derivation of the Steady-State Nernst-Planck Equation Nernst (1888)startedfromvanвЂ™tHoffвЂ™s(1887)insight thatsolute particles generate, inasolvent,an osmotic pressure that obeys the gas laws, and from the observation that they move, under the Simulation of Dynamic Electrochemical Processes. William Breen and John Cassidy, School of Chemistry, Dublin Institute of Technology, Kevin Street, Dublin 8, IRELAND. Abstract. This work is designed to introduce electrochemists in a tutorial manner to the basics of modeling of electrochemical systems based primarily on diffusion equations. There is an introduction to analytical and numerical

according to the Cottrell equation. This was explained in terms of the concentration profile extending from the electrode surface into the bulk of solution. The thickness of this diffusion layer increases with time, decreasing the slope of the concentration gradient, and thus reducing the observed current. 0 вЂ™, with the ratio of C. Red. to C. Ox (at 25 . o. C) given by the Nernst equation 29/01/2016В В· Derivation of Nernst Equation explained step-by-step for 1st semester VTU chemistry students. Detailed Notes: http://www.vturesource.com/post/1558/...

3-2 S = 0 T 0 в€† в†’ lim (3-1) In 1912 Nernst offered a proof that the unattainability of zero absolute temperature was dictated by the second law of thermodynamics and was able to 2.1 Biophysics of neurons - Overview 2.2 Reversal potential - Nernst equation 2.3 Hodgin-Huxley Model 2.4 Threshold in the Hodgkin-Huxley Model

Concentration cell (video) Khan Academy. Equation (2.53) is a general form of the Nernst equation. It expresses the emf involved in an electrochemical reaction, such as equation (2.46) in terms of R, T, F, n activities of products and activities of reactants., The Nernst-Planck equation is a physical reformulation of FickвЂ™s macroscale diffusion law, extended to the motion of charged particles. The equation was established and experimentally underpinned by Nernst, and theoretically elaborated by Planck, and acquired growing significance when it was.

### What is the derivation for the Nernst equation? Quora

Butler-Volmer equation with concentration dependence?. 2.1 Biophysics of neurons - Overview 2.2 Reversal potential - Nernst equation 2.3 Hodgin-Huxley Model 2.4 Threshold in the Hodgkin-Huxley Model, This potential is determined using the concentration of the ion both inside and outside the cell:. The expression of time dependence has been established by Karaoglanoff..

A simple derivation of the Nernst Equation. Nernst equation its importance and all topics about it Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website., 2.1 Biophysics of neurons - Overview 2.2 Reversal potential - Nernst equation 2.3 Hodgin-Huxley Model 2.4 Threshold in the Hodgkin-Huxley Model.

### On the NernstвЂ“Planck equation IOS Press

The Nernst Equation Derivation from Previously. The General Nernst Equation The general Nernst equation correlates the Gibb's Free Energy 'G and the EMF of a chemical system known as the galvanic cell. Equation (2.53) is a general form of the Nernst equation. It expresses the emf involved in an electrochemical reaction, such as equation (2.46) in terms of R, T, F, n activities of products and activities of reactants..

• Nernst heat theorem Wikipedia

• Equation (1) is referred to as the Nernst Equilibrium Potential. вЂ“ p. 9. Membrane currents вЂ“ p. 10. Accumulation around the membrane The membrane has properties similar to a capacitor Consists of two conducting medias (intra- and extra cellular space) These are separated by an insulating material (the membrane) The potential over a capacitor is proportional to the separated charge (q): v 29/01/2016В В· Derivation of Nernst Equation explained step-by-step for 1st semester VTU chemistry students. Detailed Notes: http://www.vturesource.com/post/1558/...

Nernst equation its importance and all topics about it Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. Confusion exists concerning the form of the Nernst-Planck equation for ideal gas mixtures and the Nernst-Planck equation for ideal liquid solutions. This was made clear in an earlier paper (del

Derivation of Poisson and Nernst-Planck equations in a bath and channel from a molecular model Z. Schuss* and B. NadlerвЂ  Department of Applied Mathematics, Tel-Aviv University, Ramat-Aviv, 69978 Tel вЂ¦ The Nernst Equation is derived from the emf and the Gibbs energy under non-standard conditions. Eo=Eoreductionв€’Eooxidation. When Eo is positive, the reaction is spontaneous.

This Nernst equation calculator presents the fundamental formula of electrochemistry - the Nernst equation, also known as the cell potential equation. It is a formula that allows you to calculate the reduction potential of a half-cell or full cell reaction. If you don't know what is the reduction Any electrode kinetics theory must reduce to the Nernst equation when equilibrium to the Nernst equation when equilibrium conditions are placed upon it 2. Any electrode kinetic theory must also explain Tafel behaviour 3. Rate equations must reduce to the Nernst and Tafel equations when the appropriate conditions are introduced. 19 Kinetics of Electrode Reactions. 20 Rates of forward and

Note that for this simple reaction, the Nernst equation shows that the equilibrium potential, E e is independent of the pH of the solution. Many half-cell reactions contain H + ions and their Nernst equations therefore depend on the pH . The electrode potential difference between the two half-cells can be calculated using the Nernst equation. Concentration cells are galvanic (or voltaic) cells made of two half-cells, each of which containing the same electrodes, but different concentrations.

So this is the Nernst equation, alright, we'll talk about why the Nernst equation is so important, we'll talk more about that at the end of the video. Right now, let's go ahead and derive another form of the Nernst equation, I should say, the form when you're talking about a certain temperature. So at 25 degrees C, right, most of our reactions take place at 25 degrees C, well, temperature Equation (2.53) is a general form of the Nernst equation. It expresses the emf involved in an electrochemical reaction, such as equation (2.46) in terms of R, T, F, n activities of products and activities of reactants.

Note that for this simple reaction, the Nernst equation shows that the equilibrium potential, E e is independent of the pH of the solution. Many half-cell reactions contain H + ions and their Nernst equations therefore depend on the pH . вЂў apply Nernst equation for calculating the emf of galvanic cell and define standard potential of the cell; вЂў derive relation between standard potential of the cell, Gibbs energy of cell reaction and its equilibrium constant; вЂў define resistivity (ПЃ), conductivity (Оє) and molar conductivity ( m) of ionic solutions; вЂў differentiate between ionic (electrolytic) and electronic

Derivation of the Steady-State Nernst-Planck Equation Nernst (1888)startedfromvanвЂ™tHoffвЂ™s(1887)insight thatsolute particles generate, inasolvent,an osmotic pressure that obeys the gas laws, and from the observation that they move, under the For a simple elementary reaction on electrode $$O+e^-\rightleftharpoons R$$ We can derive the Butler-Volmer equation. But вЂ¦

A simple derivation of the Nernst Equation The goal of this handout is to help you avoid taking notes during the lecture. I hope this derivation Derivation of Poisson and Nernst-Planck equations in a bath and channel from a molecular model Z. Schuss* and B. NadlerвЂ  Department of Applied Mathematics, Tel-Aviv University, Ramat-Aviv, 69978 Tel вЂ¦

## Einstein relation (kinetic theory) Wikipedia

Nernst equation IPFS. Set equation=0 to get Nernst equation (no current) 0 = -рќњ‡рќ‘§рќ‘…рќ‘‡рќњ•[рќђ¶]рќњ•рќ‘Ґ - рќњ‡рќђ№рќ‘§2 [рќђ¶]рќњ•рќ‘‰рќњ•рќ‘Ґ Factor out- рќњ‡рќ‘§ 0= - рќњ‡рќ‘§(рќ‘…рќ‘‡рќњ•[рќђ¶]рќњ•рќ‘Ґ +рќђ№рќ‘§ [рќђ¶]рќњ•рќ‘‰рќњ•рќ‘Ґ) (We see that -рќњ‡рќ‘§ is a solution, so we can get rid of it.) Now we have the variables we want: 0=рќ‘…рќ‘‡рќњ•[рќђ¶]рќњ•рќ‘Ґ +рќђ№рќ‘§ [рќђ¶]рќњ•рќ‘‰рќњ•рќ‘Ґ Move the, Derivation of the Nernst Equation appendix A The Nernst equation is used extensively in the discussion of resting membrane potential and action potentials in this book..

### Nernst Equation Calculator Omni

Concentration cell (video) Khan Academy. THE NERNST EQUATION AND ION FLOW The Nernst equation is V = In RT zF C o C i zFV вЂ“ RT In = 0 C o C i RT zF V = In C o C i 2.3 = 58 mV at 20RT oC and 61.5 mV at 37 C, Equation (2.53) is a general form of the Nernst equation. It expresses the emf involved in an electrochemical reaction, such as equation (2.46) in terms of R, T, F, n activities of products and activities of reactants..

Set equation=0 to get Nernst equation (no current) 0 = -рќњ‡рќ‘§рќ‘…рќ‘‡рќњ•[рќђ¶]рќњ•рќ‘Ґ - рќњ‡рќђ№рќ‘§2 [рќђ¶]рќњ•рќ‘‰рќњ•рќ‘Ґ Factor out- рќњ‡рќ‘§ 0= - рќњ‡рќ‘§(рќ‘…рќ‘‡рќњ•[рќђ¶]рќњ•рќ‘Ґ +рќђ№рќ‘§ [рќђ¶]рќњ•рќ‘‰рќњ•рќ‘Ґ) (We see that -рќњ‡рќ‘§ is a solution, so we can get rid of it.) Now we have the variables we want: 0=рќ‘…рќ‘‡рќњ•[рќђ¶]рќњ•рќ‘Ґ +рќђ№рќ‘§ [рќђ¶]рќњ•рќ‘‰рќњ•рќ‘Ґ Move the Equation (1) is referred to as the Nernst Equilibrium Potential. вЂ“ p. 9. Membrane currents вЂ“ p. 10. Accumulation around the membrane The membrane has properties similar to a capacitor Consists of two conducting medias (intra- and extra cellular space) These are separated by an insulating material (the membrane) The potential over a capacitor is proportional to the separated charge (q): v

From the full set of equations (the Nernst equations Eqs and and the continuity equation Eq. ( 14 ), both being combined into Eqs ( 15 ) and ( 16 ), along with the Poisson equation Eq. ( 17 )), Planck is able to calculate the speed at which the diffusion potential is generated between two solutions of different ionic composition. Dr. SobersвЂ™ Lecture Notes (Last Updated 03 December 2012) The Nernst Equation: Derivation from Previously Encountered Equations Some Previously encountered Equations:

The Nernst equation for a given ion determines the difference of potential at both sides of the membrane at which this ion is at equilibrium between inward and outward flux (zero net current). The Nernst Equation is derived from the emf and the Gibbs energy under non-standard conditions. Eo=Eoreductionв€’Eooxidation. When Eo is positive, the reaction is spontaneous.

4 The Nernst equation 21 Concentration cells:::::23 Analytical applications Electrochemistry is the study of reactions in which charged particles (ions or electrons) cross the interface between two phases of matter, typically a metallic phase (the electrode) and a conductive solution, or electrolyte. A process of this kind is known generally as an electrode process. Electrode processes Confusion exists concerning the form of the Nernst-Planck equation for ideal gas mixtures and the Nernst-Planck equation for ideal liquid solutions. This was made clear in an earlier paper (del

The Nernst equation can therefore be used to determine the direction in which a redox reaction will proceed spontaneously (eq. 6 ). Reaction ( 1 ) will proceed forward signi cantly ( K Equation (1) is referred to as the Nernst Equilibrium Potential. вЂ“ p. 9. Membrane currents вЂ“ p. 10. Accumulation around the membrane The membrane has properties similar to a capacitor Consists of two conducting medias (intra- and extra cellular space) These are separated by an insulating material (the membrane) The potential over a capacitor is proportional to the separated charge (q): v

The Nernst equation can therefore be used to determine the direction in which a redox reaction will proceed spontaneously (eq. 6 ). Reaction ( 1 ) will proceed forward signi cantly ( K Equation (1) is referred to as the Nernst Equilibrium Potential. вЂ“ p. 9. Membrane currents вЂ“ p. 10. Accumulation around the membrane The membrane has properties similar to a capacitor Consists of two conducting medias (intra- and extra cellular space) These are separated by an insulating material (the membrane) The potential over a capacitor is proportional to the separated charge (q): v

RMP > Theory Nernst Equation : The value of the equilibrium potential for any ion depends upon the concentration gradient for that ion across the membrane. If the concentrations on the two sides were equal, the force of the concentration gradient would be zero, and the equilibrium potential would also be zero. The larger the concentration gradient, the larger is the equilibrium potential. The 29/01/2016В В· Derivation of Nernst Equation explained step-by-step for 1st semester VTU chemistry students. Detailed Notes: http://www.vturesource.com/post/1558/...

3-2 S = 0 T 0 в€† в†’ lim (3-1) In 1912 Nernst offered a proof that the unattainability of zero absolute temperature was dictated by the second law of thermodynamics and was able to Dr. SobersвЂ™ Lecture Notes (Last Updated 03 December 2012) The Nernst Equation: Derivation from Previously Encountered Equations Some Previously encountered Equations:

Derivation of Nernst Equation explained step-by-step for 1st semester VTU chemistry students. Detailed Notes: http://www.vturesource.com/post/1558/News/derivation-of Equation (2.53) is a general form of the Nernst equation. It expresses the emf involved in an electrochemical reaction, such as equation (2.46) in terms of R, T, F, n activities of products and activities of reactants.

Equation (2.53) is a general form of the Nernst equation. It expresses the emf involved in an electrochemical reaction, such as equation (2.46) in terms of R, T, F, n activities of products and activities of reactants. THE NERNST EQUATION AND ION FLOW The Nernst equation is V = In RT zF C o C i zFV вЂ“ RT In = 0 C o C i RT zF V = In C o C i 2.3 = 58 mV at 20RT oC and 61.5 mV at 37 C

The Nernst distribution law, which is used by many authors in geology as the theoretical basis of geothermometry and trace-component distribution, is derived using excess functions' It is demonstrated that the free-energy term in the Nernst equation, as it is applied in many of The General Nernst Equation The general Nernst equation correlates the Gibb's Free Energy 'G and the EMF of a chemical system known as the galvanic cell.

Nernst equation its importance and all topics about it Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. 13/12/2017В В· ELECTROCHEMISTRY most important topic (NERNST equation) derivation and some of their formulas are explained in this video..Detailed information about Gibbs free вЂ¦

RMP > Theory Nernst Equation : The value of the equilibrium potential for any ion depends upon the concentration gradient for that ion across the membrane. If the concentrations on the two sides were equal, the force of the concentration gradient would be zero, and the equilibrium potential would also be zero. The larger the concentration gradient, the larger is the equilibrium potential. The This Nernst equation calculator presents the fundamental formula of electrochemistry - the Nernst equation, also known as the cell potential equation. It is a formula that allows you to calculate the reduction potential of a half-cell or full cell reaction. If you don't know what is the reduction

Derivation of Poisson and Nernst-Planck equations in a bath and channel from a molecular model Z. Schuss* and B. NadlerвЂ  Department of Applied Mathematics, Tel-Aviv University, Ramat-Aviv, 69978 Tel вЂ¦ Procedure to verify the Nernst equation, the standard potential of a redox system, and ion-ion interaction theory.

4 The Nernst equation 21 Concentration cells:::::23 Analytical applications Electrochemistry is the study of reactions in which charged particles (ions or electrons) cross the interface between two phases of matter, typically a metallic phase (the electrode) and a conductive solution, or electrolyte. A process of this kind is known generally as an electrode process. Electrode processes L12 : Nernst Equation Derivation - Electrochemistry, Chemistry, Class 12 video for Class 12 is made by best teachers who have written some of the best books of Class 12.

Nernst equation is a general equation that relates the Gibbs free energy and cell potential in electrochemistry. It is very helpful in determining cell potential, equilibrium constant etc. It takes into account the values of standard electrode potentials, temperature, activity and the reaction quotient for the calculation of cell potential. For any cell reaction, Gibbs free energy can be Equation (2.53) is a general form of the Nernst equation. It expresses the emf involved in an electrochemical reaction, such as equation (2.46) in terms of R, T, F, n activities of products and activities of reactants.

Recently, the validity of these equations has been questioned. This paper has demonstrated that, by going back to the Nernst-Planck flux equations, exactly the same resultant equations can be derived and therefore that the equations derived directly from the transport number definitions and standard diffusion equations are indeed valid. 4 The Nernst equation 21 Concentration cells:::::23 Analytical applications Electrochemistry is the study of reactions in which charged particles (ions or electrons) cross the interface between two phases of matter, typically a metallic phase (the electrode) and a conductive solution, or electrolyte. A process of this kind is known generally as an electrode process. Electrode processes

Nernst equation its importance and all topics about it Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. Procedure to verify the Nernst equation, the standard potential of a redox system, and ion-ion interaction theory.

In this page I have given the concept of Nernst potential and derivation of Nernst equation. Nernst Potential and Equation. German chemist Walther Nernstt was the scientist who gave an equation to calculated the potential difference between inside and outside a nerve cell. But first we have to know the concepts of nernst potential. By definition "Nernst potential is amount of electric 2.1 Biophysics of neurons - Overview 2.2 Reversal potential - Nernst equation 2.3 Hodgin-Huxley Model 2.4 Threshold in the Hodgkin-Huxley Model

Nernst Equation tayloredge.com. This is the Nernst equation. The 2.303 is log e(10). It assumes equal activity coefп¬Ѓcients on the two sides of the membrane. For univalent ions the slope, 2.303RT/zF, of the potential-concentration relationship is about 60 mV per decade (10-fold) change in ionic concentration; for divalent ions it is 30 mV/decade. Values for RT/zF depend only on temperature: with R =8.3145 J molв€’1 Kв€’1, So this is the Nernst equation, alright, we'll talk about why the Nernst equation is so important, we'll talk more about that at the end of the video. Right now, let's go ahead and derive another form of the Nernst equation, I should say, the form when you're talking about a certain temperature. So at 25 degrees C, right, most of our reactions take place at 25 degrees C, well, temperature.

### Nernst equation (video) Khan Academy

Concentration cell (video) Khan Academy. 29/01/2016В В· Derivation of Nernst Equation explained step-by-step for 1st semester VTU chemistry students. Detailed Notes: http://www.vturesource.com/post/1558/..., вЂў apply Nernst equation for calculating the emf of galvanic cell and define standard potential of the cell; вЂў derive relation between standard potential of the cell, Gibbs energy of cell reaction and its equilibrium constant; вЂў define resistivity (ПЃ), conductivity (Оє) and molar conductivity ( m) of ionic solutions; вЂў differentiate between ionic (electrolytic) and electronic.

### Derivation of Poisson and Nernst-Planck equations in a

Nernst Equation SlideShare. Derivation of the Steady-State Nernst-Planck Equation Nernst (1888)startedfromvanвЂ™tHoffвЂ™s(1887)insight thatsolute particles generate, inasolvent,an osmotic pressure that obeys the gas laws, and from the observation that they move, under the 29/01/2016В В· Derivation of Nernst Equation explained step-by-step for 1st semester VTU chemistry students. Detailed Notes: http://www.vturesource.com/post/1558/....

• What is the importance of Nernst Equilibrium Potential
• Nernst-Planck Equation Springer
• What is the importance of Nernst Equilibrium Potential

• Nernst equation its importance and all topics about it Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. Physical Chemistry: Concepts and Theory provides a comprehensive overview of physical and theoretical chemistry while focusing on the basic principles that unite the sub-disciplines of the field.

2.1 Biophysics of neurons - Overview 2.2 Reversal potential - Nernst equation 2.3 Hodgin-Huxley Model 2.4 Threshold in the Hodgkin-Huxley Model THE NERNST EQUATION AND ION FLOW The Nernst equation is V = In RT zF C o C i zFV вЂ“ RT In = 0 C o C i RT zF V = In C o C i 2.3 = 58 mV at 20RT oC and 61.5 mV at 37 C

Electrochemical Cells and the Nernst Equation Objective: To observe the effect of concentration on electrochemical potential; to use these results to verify the Nernst equation Set equation=0 to get Nernst equation (no current) 0 = -рќњ‡рќ‘§рќ‘…рќ‘‡рќњ•[рќђ¶]рќњ•рќ‘Ґ - рќњ‡рќђ№рќ‘§2 [рќђ¶]рќњ•рќ‘‰рќњ•рќ‘Ґ Factor out- рќњ‡рќ‘§ 0= - рќњ‡рќ‘§(рќ‘…рќ‘‡рќњ•[рќђ¶]рќњ•рќ‘Ґ +рќђ№рќ‘§ [рќђ¶]рќњ•рќ‘‰рќњ•рќ‘Ґ) (We see that -рќњ‡рќ‘§ is a solution, so we can get rid of it.) Now we have the variables we want: 0=рќ‘…рќ‘‡рќњ•[рќђ¶]рќњ•рќ‘Ґ +рќђ№рќ‘§ [рќђ¶]рќњ•рќ‘‰рќњ•рќ‘Ґ Move the

The Nernst equation can therefore be used to determine the direction in which a redox reaction will proceed spontaneously (eq. 6 ). Reaction ( 1 ) will proceed forward signi cantly ( K Appendix A Derivation of the Nernst equation Consider the general reaction [1, 2] m AAГѕm BBГѕГѕ e Г°metalГћ m CCГѕm DDГѕ ; Г°A:1Гћ where m i are the stoichiometric coefп¬Ѓcients of вЂ¦

Simulation of Dynamic Electrochemical Processes. William Breen and John Cassidy, School of Chemistry, Dublin Institute of Technology, Kevin Street, Dublin 8, IRELAND. Abstract. This work is designed to introduce electrochemists in a tutorial manner to the basics of modeling of electrochemical systems based primarily on diffusion equations. There is an introduction to analytical and numerical The Nernst equation can therefore be used to determine the direction in which a redox reaction will proceed spontaneously (eq. 6 ). Reaction ( 1 ) will proceed forward signi cantly ( K

For a simple elementary reaction on electrode $$O+e^-\rightleftharpoons R$$ We can derive the Butler-Volmer equation. But вЂ¦ Derivation of the equation Nernst-Aibassov in a magnetic field 220 of Organometallic ChemistryвЂќ, Saint-Petersburg, Russia. [9] Aibassov, E. Z., and Aibassova, S. M

Any electrode kinetics theory must reduce to the Nernst equation when equilibrium to the Nernst equation when equilibrium conditions are placed upon it 2. Any electrode kinetic theory must also explain Tafel behaviour 3. Rate equations must reduce to the Nernst and Tafel equations when the appropriate conditions are introduced. 19 Kinetics of Electrode Reactions. 20 Rates of forward and THE NERNST EQUATION AND ION FLOW The Nernst equation is V = In RT zF C o C i zFV вЂ“ RT In = 0 C o C i RT zF V = In C o C i 2.3 = 58 mV at 20RT oC and 61.5 mV at 37 C

This is the Nernst equation. The 2.303 is log e(10). It assumes equal activity coefп¬Ѓcients on the two sides of the membrane. For univalent ions the slope, 2.303RT/zF, of the potential-concentration relationship is about 60 mV per decade (10-fold) change in ionic concentration; for divalent ions it is 30 mV/decade. Values for RT/zF depend only on temperature: with R =8.3145 J molв€’1 Kв€’1 The Nernst Equation is derived from the emf and the Gibbs energy under non-standard conditions. Eo=Eoreductionв€’Eooxidation. When Eo is positive, the reaction is spontaneous.

Note that for this simple reaction, the Nernst equation shows that the equilibrium potential, E e is independent of the pH of the solution. Many half-cell reactions contain H + ions and their Nernst equations therefore depend on the pH . The General Nernst Equation The general Nernst equation correlates the Gibb's Free Energy 'G and the EMF of a chemical system known as the galvanic cell.

The Nernst-Planck equation is a physical reformulation of FickвЂ™s macroscale diffusion law, extended to the motion of charged particles. The equation was established and experimentally underpinned by Nernst, and theoretically elaborated by Planck, and acquired growing significance when it was The Nernst equation can therefore be used to determine the direction in which a redox reaction will proceed spontaneously (eq. 6 ). Reaction ( 1 ) will proceed forward signi cantly ( K

For a simple elementary reaction on electrode $$O+e^-\rightleftharpoons R$$ We can derive the Butler-Volmer equation. But вЂ¦ This is the Nernst equation. And, is a kinetic definition of the standard potential, related to And, is a kinetic definition of the standard potential, related to microscopic reaction rates.

L12 : Nernst Equation Derivation - Electrochemistry, Chemistry, Class 12 video for Class 12 is made by best teachers who have written some of the best books of Class 12. Luckily for us, Nernst derived an equation that allows us to determine at what point the two forces (chemical and electrical gradients) balance each other - in other words, at what point we have an

L12 : Nernst Equation Derivation - Electrochemistry, Chemistry, Class 12 video for Class 12 is made by best teachers who have written some of the best books of Class 12. 4 The Nernst equation 21 Concentration cells:::::23 Analytical applications Electrochemistry is the study of reactions in which charged particles (ions or electrons) cross the interface between two phases of matter, typically a metallic phase (the electrode) and a conductive solution, or electrolyte. A process of this kind is known generally as an electrode process. Electrode processes

13/12/2017В В· ELECTROCHEMISTRY most important topic (NERNST equation) derivation and some of their formulas are explained in this video..Detailed information about Gibbs free вЂ¦ Cellular Physiology of Nerve and Muscle, Fourth Edition Published Online: 25 JUL 2013

Confusion exists concerning the form of the Nernst-Planck equation for ideal gas mixtures and the Nernst-Planck equation for ideal liquid solutions. This was made clear in an earlier paper (del From the full set of equations (the Nernst equations Eqs and and the continuity equation Eq. ( 14 ), both being combined into Eqs ( 15 ) and ( 16 ), along with the Poisson equation Eq. ( 17 )), Planck is able to calculate the speed at which the diffusion potential is generated between two solutions of different ionic composition.

Derivation of the Steady-State Nernst-Planck Equation Nernst (1888)startedfromvanвЂ™tHoffвЂ™s(1887)insight thatsolute particles generate, inasolvent,an osmotic pressure that obeys the gas laws, and from the observation that they move, under the Luckily for us, Nernst derived an equation that allows us to determine at what point the two forces (chemical and electrical gradients) balance each other - in other words, at what point we have an

Equation (2.53) is a general form of the Nernst equation. It expresses the emf involved in an electrochemical reaction, such as equation (2.46) in terms of R, T, F, n activities of products and activities of reactants. Nernst equation is a general equation that relates the Gibbs free energy and cell potential in electrochemistry. It is very helpful in determining cell potential, equilibrium constant etc. It takes into account the values of standard electrode potentials, temperature, activity and the reaction quotient for the calculation of cell potential. For any cell reaction, Gibbs free energy can be

13/12/2017В В· ELECTROCHEMISTRY most important topic (NERNST equation) derivation and some of their formulas are explained in this video..Detailed information about Gibbs free вЂ¦ The Nernst equation for a given ion determines the difference of potential at both sides of the membrane at which this ion is at equilibrium between inward and outward flux (zero net current).


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### Communication Systems

goals
The number of telephone conversations by a fibre of band width 40MHz, with much less intensity loss, are:
In sky wave communication which of the frequency will be reflected back by the ionospheric layer having electron density $4 \times 10^6/m^3$ ?
The power of a  AM transmitter is 100 W. If the modulation index is 0.5 and the transmission is  having single side band, the percentage of useful power is:
In a communication system, noise is most likely to affect the signal
Cellular Mobile works in the frequency range of :

format: plain


text: Prime factorization of 24 Get the answers you need, now! Mathematics. Example of Prime Factorization The number 24 can be written as a product of prime numbers 24 = 2 x 2 x 2 x 3 It is called the prime factorization of 24. Factors of 18. Example. Natural numbers bigger than 1 that are not prime numbers are called composite numbers. They are integers and prime numbers of 24, they are also called composite number. Creating a factor tree involves breaking up the composite number into factors of the composite number, until all of the numbers are prime. Prime numbers or primes are natural numbers greater than 1 that are only divisible by 1 and with itself. algebra. 24 divided by 2, 2, 2, 3, gives no remainder. To find the prime factors, you start by dividing the number by the first prime number, which is 2. Prime factorization is a process of factoring a number in terms of prime numbers i.e. Also, despite being closely related, the prime factors of 24 and the prime factorization of 24 â¦ 1. They are integers and prime numbers of 24, they are also called composite number. There are different methods which can be utilized to find the prime factorization of a number. If it is not correct, give the correct prime factorization â¦ Why is the prime factorization of 24 written as 2 3 x 3 1? Prime factorization of 24: By prime factorization of 24 we follow 5 simple steps: 1. 3 times 25, 25 is 5 times 5. If you multiply all the prime factors of 24 you will get 24. The factors of 24 are 1,2,3,4,6,8,12,24. 147 ÷ 2 = 73½. Math. Thus, the Prime Factors of 24 are: 2, 2, 2, 3. The factorization of a number into its prime factors and expression of the number as a product of its prime factors is known as the prime factorization of that number. Let's divide by 2 to get 24 = 2 x 12. That worked, now we try factoring 49. Equcation for number 24 factorization is: 2 * 2 * 2 * 3; It is determined that the prime factors of number 24 are: 2, 3; Prime Factorization Of 23; Prime Factorization Of 25 List of positive integer factors of â¦ Answer. No it can't. Middle School. what is the prime factorization of 120 . So this is a prime factorization, but they want us to write our answer using exponential notation. This is the prime factorization of 24. A number is written with the following factorization: 23 × 4 × 5. Prime factorization of 24 in exponential form is: 24 = 2 3 × 3 1. Join now. Number 6 has four factors 1, 2, 3 and 6 itself. Log in. And we're done with our prime factorization because now we have all prime numbers here. For finding other factors you will start to divide the number starting from 2 and keep on going with dividers increasing until reaching the number that was divided by 2 in the beginning. Prime factors of 24 are 2, 3. Now this isn't a hard question. 5 points kawaiidino11 Asked 09.05.2019. Let's try the next prime number, 3: 147 ÷ 3 = 49. Can we divide 147 exactly by 2? Prime factorization of 18 2.2.2.2 Prime factorzation of 24 2.2.2.3 Using the p eaconations what is the â¦ Prime Factorization of 24. It's not dividable by 2 evenly that's why we skip it(Remembe 4,5 so you know when to stop later). Prime factorization or prime factor decomposition is the process of finding which prime numbers can be multiplied together to make the original number.. Finding the prime factors of 24. When we count the number of prime numbers above, we find that 24 has a total of 4 Prime Factors. iihhuggug iihhuggug 11/09/2016 Mathematics Middle School Prime factorization of 24 See answer cheetahdoglover1 cheetahdoglover1 3 x 8 â¦ Let's create an example for factorization with the number nine. 25 is 5 times 5. Related Calculators. Prime factorization of 24 in exponential form is: 24 = 2 3 × 3 1. 3â¢2â¢2â¢2 is the prime factorization of 24, since the numbers multiply to 24, and are all prime numbers. The aim of factoring is usually to reduce something to basic building blocks, such as numbers to prime numbers, or polynomials to irreducible polynomials. Join now. Example: If you don't know any factors of 24, try dividing it by small prime numbers. Another common way to conduct prime factorization is referred to as prime decomposition, and can involve the use of a factor tree. It is also known as prime decomposition. Factor Trees. Prime factorization of 96 (by division): 96 ÷ 2 = 48 48 ÷ 2 = 24 24 ÷ 2 = 12 12 ÷ 2 = 6 6 ÷ 2 = 3 3 ÷ 3 = 1 96 = 2 * 2 * 2 * 2 * 2 * 3 24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43. The prime factors of 24 are the prime numbers that can be divided into 24 exactly, with no remainder. Drawing factor trees is a good way of doing prime factorization. Simple example showing how to find the prime factorization of 24. When you multiply all the Prime Factors of 24 together it will result in â¦ What is the prime factorization of 24? The prime factorization of 24 is the list of prime factors of 24. We write number 24 above a 2-column table 2. The Prime Factors of 24: 23 â¢ 3. View this answer The prime factorization of 24 using exponents is 23 â3 2 3 â 3. One good strategy is to try dividing the number by 2, then 3, then 5, working your way up through the prime numbers until you find one that divides evenly. Product of Prime Factors of 24 The Prime Factors of 24 are unique to 24. Again, all the prime numbers you used to divide above are the Prime Factors of 24. The prime factors are 2 and 3. So we can write that 75 is 3 times 5 times 5. "From MathWorld--A Wolfram Web Resource.. Weisstein, Eric W. "Direct Search Factorization. USING OUR SERVICES YOU AGREE TO OUR USE OF. So, $\color{blue}{60 = 6 \cdot 10}$. Is this factorization a prime factorization? Once you get to the primes in your "tree", they are the "leaves", and you stop factoring in that âbranchâ. "From MathWorld--A Wolfram Web Resource.. Weisstein, Eric W. "Prime Factorization â¦ 24 is not a prime number . The prime factors of 24 are 2 and 3. Math is Fun: Prime Factorization. Explain why or why not. In mathematics, factorization (also factorisation in some forms of British English) or factoring is the decomposition of an object (for example, a number, a polynomial, or a matrix) into a product of other objects, or factors, which when multiplied together give the original. Study the tree to see the step by step division. It is simple to factor numbers in a natural numbers set. And don't think that i don't know this cuz i learned it 2 weeks ago. Hence, the HCF of 24 and 15 is 3. For example 7 has two factors 1 and 7. It is determined that the prime factors of number 24 are: 2, 3. To get the number that you are factoring just multiply whatever number in the set of whole numbers with another in the same set. What are the factors of prime factors of 24? 8 and 5 do not have a common prime factor. This article is a stub. So 75 is equal to 3 times 5 times 5. Finding prime factorization and factor tree. The next primeâ¦ The answer should be a whole number, and 73½ is not. A factor tree shows the prime factors of a composite number in a "tree-like" form. List of positive integer factors of â¦ How do you calculate natural number factors? Division of 24 and 15 by 3 will leave 8 and 5 as their remainders respectively. The number of primes is infinite. It is simple 6th grade math. I want to see how many adults can get the question right. Step 1: Start with any number that divides 60, in this we will use 10. Because all numbers have a minimum of two factors(one and itself). Factors of 20. Weisstein, Eric W. "Prime Number. Work your way up until you arrive to 5 (9 divided by 2, rounded up). You can help Math Wiki by expanding it. Next to What is the prime factorization of 249? Factor tree or prime decomposition for 24 As 24 is a composite number, we can draw its factor tree: You can also write this with exponents as 2 3 × 3. We write down on the left side of the table the prime factor and next number to factorize on the ride side 4. List the prime factors of each number: 18 = 2 × 3 × 3 24 = 2 × 2 × 2 × 3 For each prime number listed, underline the most repeated occurrence of this number in any prime factorization. The prime factorizations of 16 and 24 are shown below. Log in. The factors of 24 and the prime factors of 24 differ because twenty-four is a composite number. Stay tuned with BYJUâS â The Learning App to learn more about division method, Prime Factorization method and much more. References and Further Reading. We can say it's 3 times 25. Prime Factorization of 24. Prime factors of 24 are 2, 3. 0. Here, all the concepts of prime factors and prime factorization methods have been explained which will help the students understand how to find the prime factors of a number easily.. How many Prime Factors of 24? Nine can be divided by 3, now add 3 to your factors. Back to What is the prime factorization of 247? The prime factorization of a number includes ONLY the prime factors, not any products of those prime factors. We divide 24 by the smallest possible prime factor 3. The prime factorization of 24 = 2 3 â¢3. For example, the number 15 factors into primes as 3 x 5, and the polynomial x2 - 4 factors as (x - 2)(x + 2). What is prime factorization? The prime factorization of the number 24 is 2 x 2 x 2 x 3. Example: Find prime factorization of 60. ... Prime factorization of 24: 2 3 × 3. In the end you have 1, 3 and 9 as a list of factors. the factors will be prime numbers. Example 2: What is the prime factorization of 147 ? Use the form below to do your conversion, separate numbers by comma and prime factorization calculator. Find the prime factors of 24. what is the prime factorization of 132 . The prime factorization of the number 24 is 2 × 2 × 2 × 3. One way is to repeatedly divide by prime numbers: EXAMPLE 1. Ultimate Math Solver (Free) Free Algebra Solver ... type anything in there! All numbers without remainders are factors including the divider itself. The Factor Tree of 24 above shows the level of divisions carried out to get the factor numbers. Prime number are numbers that can divide without remainder, This means that 24 is divisible by 2, 2, 2, 3, numbers. Find an answer to your question What is the prime factorization of 24? 1. So 24 = 2 × 2 × 2 × 3. In all cases, a product of simpler objects is obtained. For a list of the first 1000 prime numbers see our See 1000 Prime Numbers Table. Have fun! Solution. 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# Calculating the surface area of sphere above a plane

How do I calculate the surface area of the unit sphere above the plane $z=\frac12$?

EDIT: I have been attempting things and I am thinking about parameterizing this... While I know that surface area is given by the double integral of the cross products of partial derivatives of the new parameters, I don't know what to set them to.. (sorry I'm not good with the fancy notation)

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If you just want a formula, Wikipedia has it. MathWorld has a derivation as well. –  Rahul Jun 3 '12 at 22:15

The circumference of an infinitesimal ring of the unit sphere between $z$ and $z+\mathrm dz$ is $2\pi\sqrt{1-z^2}$, and its width is $\mathrm dz/\sqrt{1-z^2}$. Thus its surface area is $2\pi\,\mathrm dz$. That is, the surface area of a slab of the unit sphere between two $z$ coordinates (or in fact between any two parallel planes) is simply $2\pi$ times the difference of the $z$ coordinates (or, generally, the distance between the two planes). Thus the surface area of the slab of the unit sphere between $z=1/2$ and $z=1$ is $2\pi\cdot(1-1/2)=\pi$.

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So if this is your paramterization $$X\left(u,v\right)=\left(\begin{array}{c} r\sin u\cos v\\ r\sin u\sin v\\ r\cos u \end{array}\right)$$ these are the elements of tangent space (partial derivatives wrt $u$ and $v$ respectively): $$X_{u}=\left(\begin{array}{c} r\cos u\cos v\\ r\cos u\sin v\\ -r\sin u \end{array}\right)$$ $$X_{v}=\left(\begin{array}{c} -r\sin u\sin v\\ r\sin u\cos v\\ 0 \end{array}\right)$$ Then by direct calculation: $$\left|X_{u}\times X_{v}\right|=\left|\begin{array}{ccc} i & j & k\\ r\cos u\cos v & r\cos u\sin v & -r\sin u\\ -r\sin u\sin v & r\sin u\cos v & 0 \end{array}\right|=\left|\left(r^{2}\sin^{2}u\cos v\right)i+\left(-r^{2}\sin^{2}u\sin v\right)j+\left(r^{2}\sin u\cos u\right)k\right|=r^{2}\sin u$$ The area of half a sphere is found as follows: $$A=r^2\int_0^{\pi}\int_0^{\pi}\sin ududv=2\pi r^2$$

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so your approach is similar to ananda's below, but you parameterized everything into polar coordinates first right? –  Mike Jun 3 '12 at 22:30
also, what exactly (intuitively) is the cross product doing? and why are the limits of the integral just 0 to pi and not to 2pi? –  Mike Jun 3 '12 at 22:32
aha found that it represents the area of a parallelogram! –  Mike Jun 3 '12 at 22:39
yes, exactly. also if you need a slice above $z=1/2$, then $u$ ranges from 0 to $\arccos{\frac{1}{2}}$. I wrongly assumed you were looking for the surface of half a sphere –  Valentin Jun 3 '12 at 22:45

Surface area is given by

$$\iint_R \left| \vec r_u \times \vec r_v \right| \ dA$$

where $\vec r(u,v)$ is the parametrization of the surface. We can rewrite this as (derivation shown here: http://tutorial.math.lamar.edu/Classes/CalcIII/SurfaceIntegrals.aspx):

$$\iint_D \sqrt{ \left(\frac{\partial z}{\partial x}\right)^2 + \left(\frac{\partial z}{\partial y}\right)^2 + 1} \ dA$$

for a function $z = f(x,y)$ where $D$ is the projection of the surface onto the xy-plane.

Since we are only concerned with the portion of the unit sphere above $z = 0$, we can write it as

$$z = \sqrt{1-x^2-y^2}$$

Computing the partial derivatives with respect to $x$ and $y$,

$$\frac{\partial z}{\partial x} = \frac{-x}{\sqrt{1-x^2-y^2}} \rightarrow \left(\frac{\partial z}{\partial x}\right)^2 = \frac{x^2}{1-x^2-y^2}$$

$$\frac{\partial z}{\partial y} = \frac{-y}{\sqrt{1-x^2-y^2}} \rightarrow \left(\frac{\partial z}{\partial y}\right)^2 = \frac{y^2}{1-x^2-y^2}$$

Substituting these into our expression for surface area,

$$\iint_D \sqrt{ \frac{x^2}{1-x^2-y^2} + \frac{y^2}{1-x^2-y^2} + 1} \ dA$$

which simplifies to (omitting a bit of algebra)

$$\iint_D \frac{1}{\sqrt{1-x^2-y^2}} \ dA$$

Observe that $D$ (the projection of our surface into the xy-plane) is given by

$$z = \sqrt{1-x^2-y^2}$$

$$\frac{1}{2} = \sqrt{1-x^2-y^2}$$

$$\frac{1}{4} = 1-x^2-y^2$$

$$x^2+y^2 = \frac{3}{4}$$

which is a circle of radius $\frac{\sqrt{3}}{2}$. The integral over $D$ is easiest done in polar coordinates. I'll assume you know how to do that and omit the computation.

$$\int_{0}^{2\pi} \int_{0}^{\frac{\sqrt{3}}{2}} \frac{1}{\sqrt{1-r^2}} \ r \ dr \ d\theta$$

$$= \pi$$

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We will basically project the part of the unit sphere above $z=\frac1 2$ onto $xy$ plane. I will assume that $\int \int_s||\frac {\partial r } {\partial x }\times \frac {\partial r } {\partial y }|| dy dx$ Now $r= f(x,y,z) = f(x,y,z(x,y))$. So $\frac {\partial r } {\partial x }=f(1,0,\frac {\partial z } {\partial x})$ and $\frac {\partial r } {\partial y }=f(0,1,\frac {\partial z } {\partial y})$. so $||\frac {\partial r } {\partial x }\times \frac {\partial r } {\partial y }||=$ $({\frac {\partial z } {\partial x}}^2+{\frac {\partial z } {\partial y}}^2+1)^{1/2}$

so now you have just find the derivatives and plug in . and the limit of the integral will be around the circle $x^2+y^2=3/4$. you can use polar co-ordinates . let me know if u have doubts , i think the answer will be $3/2$ times $\pi$.

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So, z=sqrt(1-x^2-y^2), right and I need to take the partial w.r.t. x and y, respectively? what do you mean by the limit of the integral though? –  Mike Jun 3 '12 at 22:12
yes , after that u have to evaluate double integral for which u need the limits . Can you see what the region $S$ looks like ? its just a circle . –  Theorem Jun 3 '12 at 22:15
ok I get a very messy formula for the cross product: sqrt($(x+y+1-x^2-y^2)/(1-x^2-y^2)$)... I take that this is right and I set x^2+y^2=r^2? the limits would then be r=0 to r=sqrt(3/4) and \$\theta from 0 to 2pi? how did you "guess" the answer so quickly when this is so complicated thought? –  Mike Jun 3 '12 at 22:28


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## Physics for Scientists and Engineers: A Strategic Approach with Modern Physics (3rd Edition)

Published by Pearson

# Chapter 16 - A Macroscopic Description of Matter - Exercises and Problems - Page 464: 12

#### Answer

The numerical value in $^{\circ}F$ matches numerical value in $^{\circ}C$ when the temperature is $-40^{\circ}F$ (which is equal to $-40^{\circ}C$)

#### Work Step by Step

We can use the equation to convert from Fahrenheit to Celsius to solve for F when C = F; $C = \frac{5}{9}(F-32)$ $F = \frac{5}{9}(F-32)$ $9F = 5F-160$ $4F = -160$ $F = -40$ The numerical value in $^{\circ}F$ matches numerical value in $^{\circ}C$ when the temperature is $-40^{\circ}F$ (which is equal to $-40^{\circ}C$).

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


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Consider the character table for the C2v group shown at the You In a similar approximation, the frequency of vibration between two atoms is unique and varies depending on the strength of the bond (k) and the size of the atoms (m). J Which transitions are allowed is based on the Hydrogen-like atom. observed. the basis of the selection rules for infrared ... C 3 V E 2 C 3 3 σ v IR Activity Raman. 2 If {\displaystyle y(x)=-y(-x)} or B2 symmetry then it will be both IR and Raman active. Selection Rules for Vibrational Spectroscopy. [15] In rovibronic transitions, the excited states involve three wave functions. Fateley, W. G., Neil T. McDevitt, and Freeman F. Bentley. Displacements from the ideal structure can result in relaxation of the selection rules and appearance of these unexpected phonon modes in the spectra. Yahoo fa parte del gruppo Verizon Media. In Raman spectroscopy, vibrational ) Two of the Raman lines are totally symmetric (A1 symmetry) If you know the point group of the molecule and the symmetry labels for the In surface vibrational spectroscopy, the surface selection rule is applied to identify the peaks observed in vibrational spectra. J Informazioni su dispositivo e connessione Internet, incluso l'indirizzo IP, Attività di navigazione e di ricerca durante l'utilizzo dei siti web e delle app di Verizon Media. must be bent and symmetrical (C2v symmetry). Fortunately this information is also found in the character tables. A vibration is IR active if there is a change in dipole moment during the vibration. is the Spin quantum number, and The frequency oscillation is proportional to $${\displaystyle {\sqrt {\frac {k}{m}}}}$$ ,where k is the spring constant and m is the mass of the object. This page was last updated on March 9, 2005 If the polarization of the scattered beam is the The sulfur dioxide molecule has C2v symmetry. Watch the recordings here on Youtube! [9], In the harmonic approximation, it can be shown that overtones are forbidden in both infrared and Raman spectra. [3] p orbitals also have u symmetry, so the symmetry of the transition moment function is given by the triple product u×u×u, which has u symmetry. The selection rules may differ according to the technique used to observe the transition. + For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. Raman and mid-IR spectroscopy are complementary techniques and usually both are required to completely measure the vibrational modes of a molecule. (whisnantdm@wofford.edu). <>>> The PCOL community that partial support for this work was provided by the National In the following, mainly atomic and molecular transitions are considered. Vibrations which occur with a change in dipole moment have the same symmetry properties as translations - corresponding to the x, y or z in the molecular properties column. a polarizing filter, first parallel and the perpendicular to the polarization is the total angular momentum of the electron(s). An allowed transition has a high probability of occurring and will result in a strong band. = expect to be observed in the IR and Raman for each structure. In the spring-mass analogy the moving system can be approximated by a simple harmonic oscillator. are, respectively, the initial and final angular momenta of the atom. vibration has a fixed frequency. We usually do not expect more strong bands to exist than are predicted A vibration is IR active if there is a change in, Raman spectroscopy involves inelastic light scattering and the selection rules are different from those of IR. .

.

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# Puzzle - where did the extra dollar come from? [duplicate]

Possible Duplicate:
Riddle (simple arithmetic problem/illusion)

I have this small story (sorry about my bad English) : One man (M) took 25 dollar from man (A), and another 25 dollar from man (B), so now man (M) has 50 dollar. Man (M) spent 45 dollar from the 50 he had. The reminder he has now is 5 dollar. He used this reminder to give man(A) 1 dollar, and also give man(B) 1 dollar. That means as though that man (A) gave man(M) 24 dollar from the beginning, the same is for man(B), it seems he gave man(M) 24 dollar. Now man (M) has 3 dollar. Now, if we calculate the total: 24 + 24 + 3 =51, so we have 1 more dollar! From where it came? ?

And how can I show the trick by mathematical equations ?

-

## marked as duplicate by Guess who it is., Byron Schmuland, Henning Makholm, Zev ChonolesJan 3 '12 at 16:23

Same problem as in this question. Shall we close this as a duplicate? –  Zev Chonoles Jan 3 '12 at 16:10
@Zev Chonoles: I am asking here about showing it by equations .. not the same question. –  Adban Jan 3 '12 at 16:12
It is really the same problem, you're just not seeing it. For the sake of the question, here's why you get that weird $51$ ; you're not adding the right things.. if you compute the total, man $M$ took $24$ from man $A$ and man $B$, and he has spent $45$ dollars over his things, so he's left with $3$ dollars, fine. But the total would be $1+1+3 + 45 = 50$, which has nothing to do with $24+24+3$. (One dollar in A/B's pockets, $3$ in M's pockets and 45 dollars in candy.) –  Patrick Da Silva Jan 3 '12 at 16:17

$(25 + 25) - 2 = 48$
$48 - 45 = 3$


text:
# ATLAS Conference Notes

უკანასკნელი დამატებები:
2014-03-04
17:45
 A general search for new phenomena with the ATLAS detector in pp collisions at $\sqrt{s}=8$ TeV This note presents a model-independent general search for new phenomena in proton-proton collisions at a centre-of-mass energy of 8 TeV with the ATLAS detector at the LHC. [...] ATLAS-CONF-2014-006. - 2014. Original Communication (restricted to ATLAS) - Full text

2014-03-04
16:03
 A search for resonant Higgs-pair production in the $b\bar{b}b\bar{b}$ final state in $pp$ collisions at $\sqrt{s}=8$ TeV A search for TeV-scale resonances decaying via a pair of Higgs bosons to the $b\bar{b}b\bar{b}$ final state is performed with 19.5 fb$^{-1}$ of proton-proton collision data at $\sqrt{s}=8$ TeV recorded by ATLAS in 2012. [...] ATLAS-CONF-2014-005. - 2014. - 19 p. Original Communication (restricted to ATLAS) - Full text

2014-02-25
19:43
 Calibration of $b$-tagging using dileptonic top pair events in a combinatorial likelihood approach with the ATLAS experiment This note presents a new method to derive the $b$-jet tagging efficiency from data using a combinatorial likelihood approach applied to dileptonic $t\bar{t}$ events. [...] ATLAS-CONF-2014-004. - 2014. - 20 p. Original Communication (restricted to ATLAS) - Full text

2014-02-17
21:57
 Performance of shower deconstruction in ATLAS This note describes the performance of the shower deconstruction algorithm, a jet tagging algorithm, using proton-proton collisions at a centre of mass energy of $\sqrt{s}= 8$ $\mathrm{\ Te\kern -0.1em V}$. [...] ATLAS-CONF-2014-003. - 2014. - 11 p. Original Communication (restricted to ATLAS) - Full text

2014-02-11
23:13
 The ATLAS transverse-momentum trigger performance at the LHC in 2011 The transverse momentum triggers of the ATLAS experiment at the CERN Large Hadron Collider are designed to select events with non-interacting particles passing through the detector and events with a large amount of outgoing momentum transverse to the beam axis. [...] ATLAS-CONF-2014-002. - 2014. - 29 p. Original Communication (restricted to ATLAS) - Full text

2014-01-07
09:01
 Search for Diphoton Events with Large Missing Transverse Momentum in 8 TeV pp Collision Data with the ATLAS Detector A search for diphoton events with large missing transverse momentum has been performed using 20.3 fb$^{-1}$ of proton-proton collision data at a centre-of-mass energy of 8 TeV recorded with the ATLAS detector. [...] ATLAS-CONF-2014-001. - 2014. - 25 p. Original Communication (restricted to ATLAS) - Full text

2013-12-26
23:26
 Calibration of the $b$-tagging efficiency for $c$ jets with the ATLAS detector using events with a $W$ boson produced in association with a single $c$ quark A novel method is used to measure the $b$-jet tagging efficiency of $c$ jets of the standard MV1 tagging algorithm in proton-proton collisions at $\sqrt{s}=7$ TeV using data collected by the ATLAS experiment at the LHC in the year 2011, corresponding to an integrated luminosity of about 4.6 fb$^{−1}$. [...] ATLAS-CONF-2013-109. - 2013. Original Communication (restricted to ATLAS) - Full text

2013-11-28
20:44
 Evidence for Higgs Boson Decays to the $\tau^+\tau^-$ Final State with the ATLAS Detector A search for the Higgs boson with a mass of about $125$ GeV decaying into a pair of $\tau$ leptons is performed with a data sample of proton-proton collisions, corresponding to an integrated luminosity of $\mathcal{L}=20.3$ fb$^{-1}$, collected with the ATLAS detector at the LHC at a centre-of-mass energy of $\sqrt{s}=8$ TeV. [...] ATLAS-CONF-2013-108. - 2013. Original Communication (restricted to ATLAS) - Full text

2013-11-05
10:48
 Transverse momentum, rapidity, and centrality dependence of Charged Particle Production in $p$+Pb $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV collisions measured by ATLAS experiment at the LHC ATLAS has measured transverse momentum ($p_{\mathrm{T}}$) and pseudorapidity ($\eta$) distributions of charged particles produced in $\sqrt{s_{\mathrm{NN}}}=5.02$ GeV $p$+Pb collisions. [...] ATLAS-CONF-2013-107. - 2013. Original Communication (restricted to ATLAS) - Full text

2013-11-04
12:51
 Measurement of the $W\rightarrow\mu\nu$ charge asymmetry and centrality dependence in Pb+Pb collisions at $\sqrt{s_{NN}} = 2.76 ~TeV$ with the ATLAS detector A study of $W$ production via the muon decay channel in lead-lead collisions at $\sqrt{s_{NN}} = 2.76 ~TeV$ is presented using data corresponding to an integrated luminosity of 0.14 $nb^{-1}$ collected with the ATLAS detector at the Large Hadron Collider in 2011. [...] ATLAS-CONF-2013-106. - 2013. - 19 p. Original Communication (restricted to ATLAS) - Full text


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# Nice plot of f(x)

A user has this nice graph of f(x) on his profile and I want to learn how to do it in matlab or Gnu octave.

f(x) = -(x*x*x +x + 1)
(0,0) = center
color(x,y) = arctan2(x, f(x))
lum(x,y) = sin(hypot(x, f(x))/alpha)

But when I try in matlab I fail:

>> y= -(x.*x.*x+x+1)

>> x = linspace(0,2*pi,100);

>> plot(x,y)

The matlab code does not make a plot that resembles the nice visual graph. I understand that I must also accomplish the effects, but can you help me on the way? Is there a program that is more or less suitable to make a plot like the above?

• Why don't you ask the user from whose site you copied the picture? – Wolfgang Bangerth Jun 3 '16 at 4:01
• I may misunderstood but your three MATLAB code lines are completely off topic. You need at least a contour command to reproduce the isocontour lines you have displayed, not a plot command. Only then you can focus on the colormap used for obtaining such a graph but it seems to have been postprocessed with another software. – Coriolis Jun 3 '16 at 7:43

import matplotlib

Easy peasy.... See here for the documentation on mpmath.cplot


text:
# Jacobi elliptic function sn,cn,dn (chart) Calculator

## Calculates a table of the Jacobi elliptic function sn(u,k), cn(u,k) and dn(u,k) and draws the chart.

 k -1≦k≦1 [ initial value u increment repetition ]
 $\normal Jacobi\ elliptic\ functions\\\hspace{120} sn(u,k),\ cn(u,k),\ dn(u,k)\\[10](1)\ sn(u,k)=sin(am(u,k))\\[10](2)\ am(u,k)=\phi=F^{-1}(u,k)\\\hspace{25}u=F(\phi,k)={\large\int_{\small 0}^{\hspace{25pt}\phi}\frac{d\theta}{\sqrt{1-k^2sin^2\theta}}}\\[10](3)\ cn(u,k)=\sqrt{1-sn^2(u,k)}\\\hspace{25}dn(u,k)=\sqrt{1-k^2sn^2(u,k)}\\[10](4)\ sn^2(u,k)+cn^2(u,k)=1\\\hspace{25}k^2sn^2(u,k)+dn^2(u,k)=1\\$

Jacobi elliptic function sn,cn,dn (chart)
 [1-6] /6 Disp-Num5103050100200
[1]  2017/04/28 00:03   - / 50 years old level / A teacher / A researcher / Very /
Purpose of use
See what the solution of an ODE from quantum theory looks like
Comment/Request
The period of sn does not seem to be very accurate for m = 0.999999
[2]  2014/12/23 23:07   Male / 60 years old level or over / A teacher / A researcher / Very /
Comment/Request
I should appreciate it very much if it is extended to sc, nc, and dc (imaginary argument).
[3]  2014/04/15 03:16   Male / 20 years old level / High-school/ University/ Grad student / Very /
Purpose of use
Investigating properties of the Jacobi elliptic functions.
[4]  2013/06/06 08:33   Male / 20 years old level / Others / Very /
Purpose of use
MONOGRAFIAS
[5]  2013/02/27 06:44   Male / 30 years old level / A teacher / A researcher / Very /
Purpose of use
Wanted to see the behavior of the functions in a plot.
[6]  2011/08/28 08:17   Male / 60 years old level / Others / Very /
Purpose of use
Comment/Request
Excellent !

Sending completion

To improve this 'Jacobi elliptic function sn,cn,dn (chart) Calculator', please fill in questionnaire.
Male or Female ?
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text:
# Thread: Impulse with wall and change of direction

1. ## Impulse with wall and change of direction

Got an impulse question here that I wouldn't normally struggle with, however there's something about a change of direction and not sure what to do here?

A ball of mass 0.1 kg hits a wall at 15 m/s and rebounds at a speed of 5 m/s.

If the path of the ball's diverted by $\frac{\pi}{2}$ radians by the impact, find the magnitude of the impulse exerted on the ball by the wall.

$I = m(v_2 - v_1)$

$I = 0.1(-5 - 15) = -20$, so the magnitude of the impulse would be $20 Ns$?

Think that's right anyway. However I'm not sure how I'd solve it with the change of direction involved?

2. If you use vector subtraction, $|m(v_2 - v_1)| = m\sqrt{5^2 + 15^2}$ since the ball bounces at a right angle. This gives the magnitude of the impulse to be $$1.581 N \cdot s.$$ I'm not sure if this is the correct way to do the problem, but it does account for the change in direction.


text:
# Dissipative particle dynamics

Dissipative particle dynamics (DPD) is a stochastic simulation technique for simulating the dynamic and rheological properties of simple and complex fluids. It was initially devised by Hoogerbrugge and Koelman [1][2] to avoid the lattice artifacts of the so-called lattice gas automata and to tackle hydrodynamic time and space scales beyond those available with molecular dynamics (MD). It was subsequently reformulated and slightly modified by P. Español [3] to ensure the proper thermal equilibrium state. A series of new DPD algorithms with reduced computational complexity and better control of transport properties are presented.[4] The algorithms presented in this article choose randomly a pair particle for applying DPD thermostating thus reducing the computational complexity.

DPD is an off-lattice mesoscopic simulation technique which involves a set of particles moving in continuous space and discrete time. Particles represent whole molecules or fluid regions, rather than single atoms, and atomistic details are not considered relevant to the processes addressed. The particles’ internal degrees of freedom are integrated out and replaced by simplified pairwise dissipative and random forces, so as to conserve momentum locally and ensure correct hydrodynamic behaviour. The main advantage of this method is that it gives access to longer time and length scales than are possible using conventional MD simulations. Simulations of polymeric fluids in volumes up to 100 nm in linear dimension for tens of microseconds are now common.

## Equations

The total non-bonded force acting on a DPD particle i is given by a sum over all particles j that lie within a fixed cut-off distance, of three pairwise-additive forces:

$f_i =\sum_{j \ne i}(F^C_{ij} + F^D_{ij} + F^R_{ij})$

where the first term in the above equation is a conservative force, the second a dissipative force and the third a random force. The conservative force acts to give beads a chemical identity, while the dissipative and random forces together form a thermostat that keeps the mean temperature of the system constant. A key property of all of the non-bonded forces is that they conserve momentum locally, so that hydrodynamic modes of the fluid emerge even for small particle numbers. Local momentum conservation requires that the random force between two interacting beads be antisymmetric. Each pair of interacting particles therefore requires only a single random force calculation. This distinguishes DPD from Brownian dynamics in which each particle experiences a random force independently of all other particles. Beads can be connected into ‘molecules’ by tying them together with soft (often Hookean) springs. The most common applications of DPD keep the particle number, volume and temperature constant, and so take place in the NVT ensemble. Alternatively, the pressure instead of the volume is held constant, so that the simulation is in the NPT ensemble.

## Parallelization

In principle, simulations of very large systems, approaching a cubic micron for milliseconds, are possible using a parallel implementation of DPD running on multiple processors in a Beowulf-style cluster. Because the non-bonded forces are short-ranged in DPD, it is possible to parallelize a DPD code very efficiently using a spatial domain decomposition technique. In this scheme, the total simulation space is divided into a number of cuboidal regions each of which is assigned to a distinct processor in the cluster. Each processor is responsible for integrating the equations of motion of all beads whose centres of mass lie within its region of space. Only beads lying near the boundaries of each processor's space require communication between processors. In order to ensure that the simulation is efficient, the crucial requirement is that the number of particle-particle interactions that require inter-processor communication be much smaller than the number of particle-particle interactions within the bulk of each processor's region of space. Roughly speaking, this means that the volume of space assigned to each processor should be sufficiently large that its surface area (multiplied by a distance comparable to the force cut-off distance) is much less than its volume.

## Applications

A wide variety of complex hydrodynamic phenomena have been simulated using DPD, the list here is necessarily incomplete. The goal of these simulations often is to relate the macroscopic non-Newtonian flow properties of the fluid to its microscopic structure. Such DPD applications range from modelling the rheological properties of concrete[5] to simulating liposome formation in biophysics.[6] Other recent three-phase phenomena such as dynamic wetting.[7]

The full trace of the developments of various important aspects of the DPD methodology since it was first proposed in the early 1990s can be found in "Dissipative Particle Dynamics: Introduction, Methodology and Complex Fluid Applications - A Review"[8]

The state-of-the-art in DPD was captured in a CECAM workshop in 2008.[9] Innovations to the technique presented there include DPD with energy conservation; non-central frictional forces that allow the fluid viscosity to be tuned; an algorithm for preventing bond crossing between polymers; and the automated calibration of DPD interaction parameters from atomistic molecular dynamics.

## References

1. ^ P. J. Hoogerbrugge and J. M. V. A. Koelman. Simulating microscopic hydrodynamic phenomena with dissipative particle dynamics. Europhysics Letters, 19(3):155–160, JUN 1 1992
2. ^ J. M. V. A. Koelman and P. J. Hoogerbrugge. Dynamic simulations of hard-sphere suspensions under steady shear. Europhysics Letters, 21(3):363–368, JAN 20 1993
3. ^ P. Español and P. B. Warren. Statistical-mechanics of dissipative particle dynamics. Europhysics Letters, 30(4):191–196, MAY 1 1995
4. ^ N. Goga, A.J. Rzepiela, A.H. de Vries, S.J. Marrink, H.J.C. Berendsen: Efficient algorithms for Langevin and DPD dynamics, J. Chem. Th. Comp., 2012, DOI:10.1021/ct3000876
5. ^ James S. Sims and Nicos S. Martys: Modelling the Rheological Properties of Concrete
6. ^ Petri Nikunen, Mikko Karttunen, and Ilpo Vattulainen: Modelling Liposome formation in biophysics
7. ^ B. Henrich, C. Cupelli, M. Moseler, and M. Santer": An adhesive DPD wall model for dynamic wetting, Europhysics Letters 80 (2007) 60004, p.1
8. ^ Moeendarbary et al. (2009). "Dissipative Particle Dynamics: Introduction, Methodology and Complex Fluid Applications - A Review" (Subscription required). International Journal of Applied Mechanics (World Scientific Journals) 1 (4): 737–763. Bibcode:2009IJAM...01..737M. doi:10.1142/S1758825109000381.
9. ^ Dissipative Particle Dynamics: Addressing deficiencies and establishing new frontiers, CECAM workshop, July 16–18, 2008, Lausanne, Switzerland.

## Available packages

Some available simulation packages that can (also) perform DPD simulations are:

• CULGI: The Chemistry Unified Language Interface, Culgi B.V., The Netherlands
• DL_MESO: Open-source mesoscale simulation software.
• DPDmacs
• ESPResSo
• Fluidix: The Fluidix simulation suite available from OneZero Software.
• GPIUTMD: Graphical processors for Many-Particle Dynamics
• Gromacs-DPD: A modified version of Gromacs including DPD.
• HOOMD-blue: Highly Optimized Object-oriented Many-particle Dynamics—Blue Edition
• LAMMPS
• LAMMPS interface in the MAPS suite of Scienomics
• Materials Studio: Materials Studio - Modeling and simulation for studying chemicals and materials, Accelrys Software Inc.
• SYMPLER: A freeware SYMbolic ParticLE simulatoR from the University of Freiburg.


text:
# Tag Info

## Hot answers tagged ancova

9 votes
Accepted

### How do I understand ANCOVA in basic layman's terms?

To answer your question, I would like to invite you to think of a broader picture for, then, take you back to your original question. First, I would like to introduce a comparison between ANOVA and ...
• 1,483
8 votes

### Between-subjects effect becomes non-signicant after centering covariate. Should I center or not?

Neither, or both, are correct. You often cannot interpret the main effect when you have an interaction in there. The main effect is the difference between the groups when the covariate is equal to ...
• 15.1k
8 votes

### Student's t-test with a covariate?

One common means of controlling for some other covariate would be via regression. Put the X and Y values into the response (DV), and a Y-group indicator (0 if in X, 1 if in Y) as a DV, along with your ...
• 264k
8 votes
Accepted

### Two-way ANOVA vs ANCOVA in R

Order matters whenever the predictors aren't independent. They're correlated in your first example, as they're continuous measurements on each of the diamonds, but not in the second, as those are ...
7 votes

### Difference between two-way ANOVA, factorial ANOVA and ANCOVA: as analogies of linear regression

Compared to one way ANOVA: Two way ANOVA adds one more categorical independent variable to the regression (and possibly the interaction between the two IVs). Factorial ANOVA adds any number of ...
• 95.7k
6 votes
Accepted

### Good data example needed with covariate affected by treatments

You may want to check out the mediation R package. It does include experimental data like jobs and ...
• 1,722
6 votes

### Should covariates that are not statistically significant be 'kept in' when creating a model?

We really need more information about your goals to answer this question. Regressions are used for two main purposes: Prediction Inference Prediction is when your goal is to be able to guess at ...
• 160
6 votes
Accepted

• 4,772
4 votes

### Do I have a justified reason to exclude a non-significant covariate from my ANCOVA? How interesting is unequal variance?

First, ask yourself why IQ is always included in such models. There is probably some reason. It might be that IQ is a mediator (see below) Second, from what you say, it sounds like IQ is a type of ...
• 95.7k
4 votes
Accepted

### Do I have a justified reason to exclude a non-significant covariate from my ANCOVA? How interesting is unequal variance?

First, let me assure you that - as mentioned by @amoeba - you are on the right path to land in "research hell", that is the place where researchers (should) go when they let their p-value to decide ...
• 644
4 votes
Accepted

### Can ANCOVA be used with a dichotomous dependent variable? If not, what have these authors done?

The authors are talking about odds, so I'd guess they used logistic regression with multiple covariables ("multivariable regression"). The terms "ANOVA" and "ANCOVA" do not make much sense in the ...
• 10.9k
4 votes

### Fat tail? Short tail? Long tail? Where do I go from here?

Your Q-Q plot doesn't look like it has a fat tail. I'll show you how a fat tail looks like: Your tail is like a Victoria Secret's model compared to the above. I wish some of my model residuals had ...
• 57.1k
4 votes

### Two-Way ANCOVA: It is necessary to include a non-significant interaction term in the model?

There is nothing in statistical theory or practice which requires you to include any interaction, or any main effect for that matter. You include in your model the variables which your scientific ...
• 16.8k
4 votes

### Two-Way ANCOVA: It is necessary to include a non-significant interaction term in the model?

There are two different definitions or understandings of the term ANCOVA. The first and a broader one is "Any linear model containing continuous/scale predictors besides factors (categorical ...
• 53.3k
4 votes

### Questions concerning visualizing model results with the R-package visreg

The paper by Patrick Breheny and Woodrow Burchett (the authors of visreg) discuss all the issues raised in your question. Before I go into the specific questions, ...
• 26.4k
4 votes
Accepted

### GPower: Difference in Sample Size for ANCOVA vs. Repeated Measures ANOVA in clinical trials

This is a topic for seriously misunderstanding G*Power! Thank you for bringing this up. We thought the following was an explanation for the matter: "When calculating the sample size for "...
4 votes

### ANCOVA vs linear mixed model in R, using different functions gives me different results

To summarise: You've discovered that different packages & functions may implement different methods, so you shouldn't expect to get the same results when you throw your data at a sundry collection ...
• 5,787
3 votes
Accepted

### When to create a control group with paired T test

I think what you are stating is directionally right. But, to clarify you can conduct a paired t test with your one single group and measure how they fare before and after the treatment. You can also ...
• 7,042

Only top scored, non community-wiki answers of a minimum length are eligible


text:
2015
04-14

# Harry Potter and the Present

Hogwarts is welcoming the new term! After Voldemort and the Death Eater were wiped out by Harry Potter and his friends, Harry becomes the glory of the Wizarding World, magician treat him as model, hero, king… and much more than that. So on the ceremony held to signal the beginning of a new school semester, they decide to invite Harry Potter and his three friends: Ginny, Ron and Hermione.
Back to their Alma Mater, Harry and his three friends find a store to purchase some present. They can use at most M units of money, what’s more, they don’t have enough energy to take the presents to Hogwarts if their total weight is exceeding W or the total number is exceeding S. In the store, there are N kinds of present, each of them has price Pi, weight Wi, and value Vi, and as it’s a magical world, any present’s amount is infinite and can be purchased for many times.
Unfortunately, all of them have different idea to buy the gifts. Ginny wants the total number to be the largest, while Ron thinks the more money they cost, the more sincerity there will be shown, Harry wonders the biggest weight they can take because he always likes to challenge himself, Hermione, of course, is already beginning to calculating the biggest value.
Here you see, your task is very simple, calculating the four numbers.

The first line contains a single integer T, indicating the number of test cases.
Each test case begins with four integers N, S, W, M. Then N lines following, each line contains three integers Pi, Wi, Vi, describing a present.

Technical Specification

1. 1 <= T <= 100
2. 1 <= N <= 1 000
3. 1 <= S <= 1 000 000 000
4. 0 <= Wi, Pi, W, M <= 1 000
5. 0 <= W*M <= 1 000
6. 0 <= Vi <= 1 000 000

The first line contains a single integer T, indicating the number of test cases.
Each test case begins with four integers N, S, W, M. Then N lines following, each line contains three integers Pi, Wi, Vi, describing a present.

Technical Specification

1. 1 <= T <= 100
2. 1 <= N <= 1 000
3. 1 <= S <= 1 000 000 000
4. 0 <= Wi, Pi, W, M <= 1 000
5. 0 <= W*M <= 1 000
6. 0 <= Vi <= 1 000 000

2
2 2 3 4
1 1 2
1 2 3
2 3 3 4
1 1 2
1 2 3

Case 1: 2 2 3 5
Case 2: 3 3 3 6


text:
# Question on definite integral

1. Jul 24, 2013

### yungman

I want to proof
$$\int_{-\pi}^0 e^{jz\cos \theta}\cos(m\theta) d\theta=\int_0^{\pi} e^{jz\cos \theta}\cos(m\theta) d\theta$$
(1) Let $\theta=-\theta\;\Rightarrow\; d\theta=-d\theta,\;\cos(-\theta)=\cos\theta,\;\cos[m(-\theta)]=\cos(m\theta)$
$$\int_{-\pi}^0 e^{jz\cos \theta}\cos(m\theta) d\theta=-\int_{\pi}^0 e^{jz\cos \theta}\cos(m\theta) d\theta=\int_0^{\pi}e^{jz\cos \theta}\cos(m\theta) d\theta$$

BUT

(2) Let $\theta=\pi+\theta\;\Rightarrow\; d\theta=d\theta,\;\cos(\pi+\theta)=-\cos\theta$
$$\int_{-\pi}^0 e^{jz\cos \theta}\cos(m\theta) d\theta=\int_0^{\pi} e^{-jz\cos \theta}\cos(m\pi+m\theta) d\theta$$
$$\cos(m\pi+m\theta)=\cos(m\pi)\cos(m\theta)-\sin(m\pi)\sin(m\theta)=(-1)^m \cos(m\theta)$$
$$\Rightarrow\;\int_{-\pi}^0 e^{jz\cos \theta}\cos(m\theta) d\theta=\int_0^{\pi} e^{-jz\cos \theta}(-1)^m \cos(m\theta) d\theta$$

Obviously the two ways give different result. What did I do wrong?

Thanks

2. Jul 24, 2013

### vanhees71

Both of your calculations are correct. Where do you see a contradiction?

3. Jul 24, 2013

### yungman

But
$$\int_0^{\pi}e^{jz\cos \theta}\cos(m\theta) d\theta\neq \int_0^{\pi} e^{-jz\cos \theta}(-1)^m \cos(m\theta) d\theta$$

4. Jul 24, 2013

### yungman

Anyone please, I don't think I did anything wrong.

5. Jul 24, 2013

### Mandelbroth

Could you give a counterexample for me? I'm feeling tired and lazy right now, but both your derivations, at a glance, look okay to me.

6. Jul 24, 2013

### yungman

I don't have counter example, I am confused enough already.

This identity is definitely true given but many books:
$$J_m(z)=\frac{j^{-m}}{2\pi}\int_{-\pi}^{\pi}e^{jz\cos u}\cos(m\theta)d\theta=\frac{j^{-m}}{\pi}\int_0^{\pi} e^{jz\cos \theta}\cos(m\theta) d\theta$$
$$J_m(z)=\frac{j^{-m}}{2\pi}\int_{-\pi}^{\pi}e^{jz\cos u}\cos(m\theta)d\theta=\frac{j^{-m}}{2\pi}\int_0^{\pi} e^{jz\cos \theta}\cos(m\theta) d\theta+\frac{j^{-m}}{2\pi}\int_{-\pi}^{0} e^{jz\cos \theta}\cos(m\theta) d\theta$$

The original question is regarding the second integral.

Thanks

7. Jul 24, 2013

### Mandelbroth

Can you give an $m$ such that the two integrals are not equal? Checking the trivial case $m=0$, we get that the two integrals are equal. You say that the two ways "obviously" give a different result, but I'm pretty sure the two integrals are equal. I want to know why you think they wouldn't be.

8. Jul 24, 2013

### yungman

But for m=1

$$\int_0^{\pi} e^{-jz\cos \theta}(-1) \cos(\theta) d\theta=-\int_0^{\pi} \frac{1}{e^{jz\cos \theta}} \cos(\theta) d\theta$$

$$\int_0^{\pi}e^{jz\cos \theta}\cos(\theta) d\theta\neq -\int_0^{\pi} \frac{1}{e^{jz\cos \theta}} \cos(\theta) d\theta$$

for m=3

$$\int_0^{\pi} e^{-jz\cos \theta}(-1) \cos(3\theta) d\theta=-\int_0^{\pi} \frac{1}{e^{jz\cos \theta}} \cos(3\theta) d\theta$$

$$\int_0^{\pi}e^{jz\cos \theta}\cos(3\theta) d\theta\neq -\int_0^{\pi} \frac{1}{e^{jz\cos \theta}} \cos(3\theta) d\theta$$

Last edited: Jul 24, 2013
9. Jul 24, 2013

### Mandelbroth

This is not true. For $m=1$, they are both equal to $j\pi J_1(z)$, where $J_1$ is a Bessel function of the first kind. The integrals are also equal for $m=3$.

10. Jul 24, 2013

### yungman

Please explain a little more, I don't get what you mean. All I am looking is that part of the equation is not equal and I don't see a reason.

The important thing is not the m. It's the $\;e^{jx\cos\theta}\;$ can never equal to $\;e^{-jx\cos\theta}\;$ except $\;\theta=\pi/2$

11. Jul 24, 2013

### Mandelbroth

You're just looking at the integrands. You CAN'T do that. Different integrands can have the same definite integrals.

12. Jul 24, 2013

### micromass

Staff Emeritus
Let's take a look at the graphs. Here is the graph for the real part for $z=3$ and $m=3$:

As you see, the graphs won't be equal, but the integrals will be. I hope this reassures you that your result is correct.

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13. Jul 24, 2013

### yungman

Sorry I still don't get this.
It is given:
$$J_m(z)=\frac{j^{-m}}{2\pi}\int_{-\pi}^{\pi}e^{jz\cos u}\cos(m\theta)d\theta=\frac{j^{-m}}{\pi}\int_0^{\pi} e^{jz\cos \theta}\cos(m\theta) d\theta$$

From that we have:
$$J_m(z)=\frac{j^{-m}}{2\pi}\int_{-\pi}^{\pi}e^{jz\cos u}\cos(m\theta)d\theta=\frac{j^{-m}}{2\pi}\int_0^{\pi} e^{jz\cos \theta}\cos(m\theta) d\theta+\frac{j^{-m}}{2\pi}\int_{-\pi}^{0} e^{jz\cos \theta}\cos(m\theta) d\theta$$

This mean it has to be:
$$\frac{j^{-m}}{2\pi}\int_0^{\pi} e^{jz\cos \theta}\cos(m\theta) d\theta=\frac{j^{-m}}{2\pi}\int_{-\pi}^{0} e^{jz\cos \theta}\cos(m\theta) d\theta$$

From my original post I use two different change of variables and get two different result, and:
$$\frac{j^{-m}}{2\pi}\int_0^{\pi}e^{jz\cos \theta}\cos(m\theta) d\theta\neq \frac{j^{-m}}{2\pi}\int_0^{\pi} \frac{1}{e^{jz\cos \theta}}(-1)^m \cos(m\theta) d\theta$$

Or
$$J_m(z)=\frac{j^{-m}}{\pi}\int_0^{\pi} e^{jz\cos \theta}\cos(m\theta) d\theta\neq\frac{j^{-m}}{2\pi}\int_0^{\pi} e^{jz\cos \theta}\cos(m\theta) d\theta+\frac{j^{-m}}{2\pi}\int_{0}^{\pi} \frac{1}{e^{jz\cos \theta}}(-1)^m\cos(m\theta) d\theta$$

This is regardless of whether it's a Bessel function or not.

I am confused.

Thanks

14. Jul 24, 2013

### micromass

Staff Emeritus
Why do you think the integrals are not equal? Doesn't my graph show you that they were equal (for a special case, ok)?

Wait, didn't you mean:
$$\frac{j^{-m}}{2\pi}\int_0^{\pi}e^{jz\cos \theta}\cos(m\theta) d\theta\neq \frac{j^{-m}}{2\pi}\int_0^{\pi} e^{-jz\cos \theta}(-1)^m \cos(m\theta) d\theta$$

15. Jul 24, 2013

### yungman

Yes, I am trying to derive the equation:
$$J_m(z)=\frac{j^{-m}}{2\pi}\int_{-\pi}^{\pi}e^{jz\cos u}\cos(m\theta)d\theta=\frac{j^{-m}}{\pi}\int_0^{\pi} e^{jz\cos \theta}\cos(m\theta) d\theta$$
AND I can't get there if I cannot prove
$$\frac{j^{-m}}{2\pi}\int_0^{\pi}e^{jz\cos \theta}\cos(m\theta) d\theta\neq \frac{j^{-m}}{2\pi}\int_0^{\pi} e^{-jz\cos \theta}(-1)^m \cos(m\theta) d\theta$$

My original post is not even a Bessel function question. It's a change of variable problem.....that I use two different change of variable and get two different answer that I cannot prove they are equal.

Last edited: Jul 24, 2013
16. Jul 24, 2013

### micromass

Staff Emeritus
I think your post in your OP actually provides a good proof of the formula.

17. Jul 24, 2013

### yungman

Can you explain more? I am more confused now. How come I have two ways and get two different result and I cannot prove it's the same. This is not even issue with Bessel function. Just a change of variable problem.

Last edited: Jul 24, 2013
18. Jul 24, 2013

### micromass

Staff Emeritus
Well, you started from one integral. You did two different substitutions (namely $\theta\rightarrow - \theta$ and $\theta \rightarrow \pi + \theta$) and you obtained two different integrals equal to the first integral. So these two integrals must equal.

19. Jul 24, 2013

### yungman

That's how it work? Isn't it pushing it? How do you justify the two equations in my OP are equal? Even you use numerical method, it's not going to be equal. I thought math has to have reason and valid justification. What I showed is very basic change of variables only!!!

Forget that it's part of the Bessel function. How do you justify to beginning math students that this basic change of variables give you the same solution......as it absolutely does not. Unless you are going to amend that there are exception to the change of variables when comes to some difficult functions.

No matter how you cut it
From my original post I use two different change of variables and get two different result, and:
$$\int_0^{\pi}e^{jz\cos \theta}\cos(m\theta) d\theta\neq\int_0^{\pi} \frac{1}{e^{jz\cos \theta}}(-1)^m \cos(m\theta) d\theta$$
As you cannot make:
$$e^{jz\cos \theta}\neq \frac{1}{e^{jz\cos \theta}}$$
For all z and $\theta$.

I don't mean to challenge you or others, it just does not make sense.

Thanks

Last edited: Jul 24, 2013
20. Jul 24, 2013

### Mandelbroth

So, by your logic, $\displaystyle \int\limits_{[0,2\pi]}\cos{x}\, dx = \int\limits_{[0,2\pi]}\sin{x}\, dx$ is not true because $\sin{x}\neq\cos{x}$?