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# MTH 251 Study Guide - Midterm Guide: Limit Comparison Test, Geometric Progression, Monotonic Function

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School
Department
Course
Professor
Miami University
MTH 251
Calculus II
Winter 2018
Term Test 3
Professor: Konstantinos Beros
Exam Guide
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Table of content
- Section 7.7: Approximation Of Integrals
- Section 11.1: Sequence
- Section 11.2: Series
- Section 11.3: The Integral Test
- Section 11.4: Comparison Test
- Section 11.5: Alternating Series
- Section 11.6: Absolute Convergence and ratio and root tests
- Section 11.8: Power Series
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Section 7.7: Approximation of Integrals
Recall:
If f is continuous on [,] then,
()
=lim
()

Where =
and =+
The area under the curve can be estimated using right and left midpoints.
The area under the curve can also be estimated using midpoints,
=(),
 where =
(+)
The are under the curve can also be estimated using the trapezoid rule, which is just the
average of the left and right sums added together,
=1
2()+()


=()+()
2+()


Simpson’s Rule:
The final way we will look at to approximate integrals is called simpsons rule,
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# What is an Antigrammatic Word™?

In the spirit of the What is a Word™/Phrase™ series started by JLee, a special brand of Phrase™ and Word™ puzzles.

If a word conforms to a special rule, I call it an Antigrammatic Word™.
Use the following examples below to find the rule.

However, by spending too much time in The Sphinx Lair, I have got the ""can't post a puzzle unless it has a single word or phrase answer" syndrome". So, once you figure out the rule, you will need to use the Words provided to get an answer that will both annoy you and soothe you at the same time.

Antigrammatic Words™ Not Antigrammatic Words™
KAFIR ONION
SILENT RELAX
SAINTED SUNRISE
PIT SPAR
HAPPIEST BEGONE
HURTLESS ANGEL

And, if you want to analyze, here is a CSV version:

Antigrammatic Words™,Not Antigrammatic Words™
KAFIR,ONION
SILENT,RELAX
SAINTED,SUNRISE
PIT,SPAR
HAPPIEST,BEGONE
HURTLESS,ANGEL

The puzzle satisfies the series' inbuilt assumption, that each word can be tested for whether it is an Antigrammatic Word™ without relying on the other words.
These are not the only examples of Antigrammatic Words™, more, but a very limited number exist.

What is the special rule these words conform to?

• Whatever font that is has uppercase T as slightly shorter than the rest and I find it unreasonably aggravating. Nov 8 '17 at 20:50
• @EngineerToast completely reasonable, it's abhorrent Nov 8 '17 at 22:12
• @EngineerToast On what platform and browser? They're exactly the same for me on Linux/Firefox, Linux/Chrome, IOS/Safari on iPhone and iPad, and the SE mobile client for IOS.
– Rubio
Nov 9 '17 at 1:50
• @Rubio In a Google Chrome Browser on Windows 8.1 Pro at 1366 X 766 resolution, stem (the top horizontal line of 'T') has two unequal portions. Interestingly at other resolutions it is fine! Nov 9 '17 at 7:02
• @Rubio I note the issue in Win7 Chrome and IE, 1920x1200 and 1440x900. It looks fine on the Android SE app. Based on mine, yours, and Mea Culpa Nay's data, it may be an issue on Windows with \text in MathJax. Even a simple post such as $\text{SILENT}$ has the issue for me. Nov 9 '17 at 14:24

I think an Antigrammatic Word is one that has

an antigram: that is, an anagram with (approximately) opposite meaning.

Thus:

a KAFIR is an infidel, an enemy of Islam; a FAKIR is a Muslim ascetic.
if all is SILENT then there is nothing to LISTEN to.
if two things are UNITED then that can be reversed when they are UNTIED.
if you are SAINTED then presumably your soul is not STAINED.
in a PIT the ground is lowered, whereas in a TIP it is raised.
the HAPPIEST writings are usually not EPITAPHS.
it is hard to be HURTLESS if you are also RUTHLESS.

And

the initial letters of the antigrams of the given antigrammatic words are F,L,U,S,T,E,R; and FLUSTER/RESTFUL is an antigrammatic pair with approximately the meanings requested.

Full disclosure:

I saw that only after some nudging from the OP, though I had noticed that the letters of FLUSTER appear, in the right order, in the given words. D'oh!

• This is indeed correct. Just ponder a bit more on your obscure thoughts and you will see that the answer that you think is indeed the expected one.
– Sid
Nov 8 '17 at 17:03
• @Rob I randomly chose the second column such that they are not antigrams. I didn't bother to check for anagrams of each of them. (Bonus, if you can figure out from which puzzle I copied the words of the second column from.)
– Sid
Nov 8 '17 at 19:19
• Nov 9 '17 at 0:45
• I thought those looked familiar... ;)
– Rubio
Nov 9 '17 at 1:07

format: plain


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# Climate Science for Everyone: How much heat can the air and ocean store?

Let’s look at how much energy the oceans can store compared to the energy storage of the atmosphere.

One way to describe the amount of energy that something can store is called “specific heat.” This is essentially the amount of energy required to heat up a mass of a material by a certain temperature. In our case, we’ll use 1 kg heated by by 1 degree Celsius (1.8° F) because those are the international standards.

The specific heat of air is about 1158 J/(kg*C) while the specific heat of seawater is about 3850 J/(kg*C), where a Joule is a standard measurement of energy. We can see that air has a specific heat a little more than 3x smaller than that of water. But we know from our day-to-day experience that water is a lot denser than air is, and that will matter a great deal to our calculations. (For reference, one Joule is about the amount of energy you need to expend to lift one pound 9 inches.)

While we could go through a huge amount of geometry to estimate how much air and seawater there is on the Earth, but there’s an easier way – use the measurements of experts. for example, this paper calculated that the total mass of the atmosphere is about 5.14 x 1018 kg, while the National Oceanic and Atmospheric Administration (NOAA) has calculated that the total volume of the world’s oceans is about 1.34 x 10^18 m3. In order to get the total mass of the world’s oceans we need an estimate of the density of seawater, which I found at this MIT link – 1027 kg/m3 (other sources have similar values).

Using this, we can multiply the mass of the atmosphere times the specific heat of the air to calculate what the total heat capacity of the atmosphere is:

$5.14\times 10^{18} kg\cdot 1158\frac{J}{kg*C} = 5.95\times 10^{21}\frac{J}{C}$ (Eqn. 1)

In other words, it takes about 5.95 x 1021 Joules to raise the temperature of the atmosphere one degree Celsius.

For ocean we need to add one step – multiplying the volume of the water by its density to get the total mass of the ocean

$1.3410^{18} m^3\cdot 1027\frac{kg}{m^3}\cdot 3850\frac{J}{kg*C} = 5.30\times 10^{24}\frac{J}{C}$ (Eqn. 2)

This shows that the heat capacity of the oceans is about 1000x larger than the heat capacity of the Earth’s atmosphere.

So why do we care? First, it helps to explain why we care about El Nino and La Nina cycles in the Pacific Ocean. If you’re unfamiliar with the terms, La Nina is a massive upwelling of cold water in the Pacific that, because ocean water has a much higher heat capacity than air, cools off the entire planet and affects weather patterns. El Nino is a massive pool of hot water in the Pacific that does the opposite – it dumps heat stored in the ocean back into the atmosphere, warming the globe and affecting weather patterns. Nearly all the energy absorbed by the Pacific Ocean during La Nina periods will eventually be emitted back into the atmosphere during El Nino periods.

Second, the heat capacity of the world’s oceans helps to explain why scientists are so interested in how much energy has been stored in the ocean. Since total ocean heat capacity is about 1000x greater than total atmosphere, it means that a barely measurable temperature increase in the ocean (1/1000th of a degree C) could drive a massive spike in global air temperature (1 degree C).

The difference between measured global surface temperature from various sources and the temperatures adjusted to remove the influence of El Nino, volcanoes, and the solar cycle. Note that the massive 1997/1998 El Nino spike is nearly completely the result of ocean El Nino dumping stored energy into the atmosphere. (Image Credit: Skeptical Science)

Lastly, we care because it demonstrates just why the average global temperature hasn’t been warming as fast over the last several years. We’ve had more La Nina cycles since 1998 than we’ve had El Nino cycles, and that means the Pacific ocean is storing more energy.

El Nino Southern Oscillation index.

The problem with this, however, is that it means that energy is going to come back OUT of the ocean again eventually. And when (not if) that happens next, the average global temperature will spike.

## 5 comments on “Climate Science for Everyone: How much heat can the air and ocean store?”

1. Ryan

Well fuck me, I’m going to go have a drink now…and maybe watch a marathon of Independence Day, The Day After Tomorrow and 2012. But good info… :-)

2. Dr. Denny

This is depressing, Brian.

• Why depressing, Denny? “Worrying” I get, but you’ve got me confused.

3. Fnay

Well ain’t that a kick in the family Joules.

But thank you for the post. This is one (of many actually) subject that keeps me pretty confused. You have just countered most of the anti-climate change arguments I have been seeing lately that cite the last 15 years of temperatures to say it ain’t so.

Onward through the fog.

4. Thanks for the in depth explanation! True, that the average global temperature isnt’ rising as fast as many would think; however, there are still things that we need to do globally to help Mother Nature out!


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# Right triangle eq2

Find the lengths of the sides and the angles in the right triangle. Given area S = 210 and perimeter o = 70.

Correct result:

a =  21
b =  20
c =  29
A =  46.3972 °
B =  43.6028 °
C =  90 °

#### Solution:

Our quadratic equation calculator calculates it.

Try calculation via our triangle calculator.

$c=\sqrt{{a}^{2}+{b}^{2}}=\sqrt{2{1}^{2}+2{0}^{2}}=29$
$A=\frac{18{0}^{\circ }}{\pi }\cdot \mathrm{arcsin}\left(a\mathrm{/}c\right)=\frac{18{0}^{\circ }}{\pi }\cdot \mathrm{arcsin}\left(21\mathrm{/}29\right)=46.397{2}^{\circ }=4{6}^{\circ }2{3}^{\mathrm{\prime }}50\mathrm{"}$
$B=\frac{18{0}^{\circ }}{\pi }\cdot \mathrm{arcsin}\left(b\mathrm{/}c\right)=\frac{18{0}^{\circ }}{\pi }\cdot \mathrm{arcsin}\left(20\mathrm{/}29\right)=43.602{8}^{\circ }=4{3}^{\circ }3{6}^{\mathrm{\prime }}10\mathrm{"}$
$C=90=9{0}^{\circ }$

We would be pleased if you find an error in the word problem, spelling mistakes, or inaccuracies and send it to us. Thank you!

Tips to related online calculators
Looking for help with calculating roots of a quadratic equation?
Do you have a linear equation or system of equations and looking for its solution? Or do you have quadratic equation?
Pythagorean theorem is the base for the right triangle calculator.

#### You need to know the following knowledge to solve this word math problem:

We encourage you to watch this tutorial video on this math problem:

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text:
Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫  π  -¹ ² ³ °

You are not logged in.

## #1 2014-01-14 11:53:11

ninjaman
Member
Registered: 2013-10-15
Posts: 51

### differentiation

hello
I have a question im stuck with

y = (7x-4) 3/2         (3/2 is supposed to be 3 over 2, I don't know how to do that on here)

I got this,

u = 7x-4

du/dx = 7

y = u 3/2

dy/du = 3/2u -3/2  (the -3/2 is power, supposed to be above right of u)

du/dx * dy/du = 3/2u-3/2 * 7 = 21/2 u -3/2

since u = 7x-4,      21/2 (7x-4) -3/2

im thinking most of it is correct except for the -3/2, the fraction part has me concerned.

thanks

simon:)

Offline

## #2 2014-01-14 15:20:38

bobbym
From: Bumpkinland
Registered: 2009-04-12
Posts: 95,897

### Re: differentiation

Hi;

ninjaman wrote:

dy/du = 3/2u -3/2  (the -3/2 is power, supposed to be above right of u)

You made a mistake in the above step. You did not apply the power rule correctly. Here are the steps for the whole problem.

You can use latex for all your math posting. I use this when I am not doing it by hand,

http://latex.codecogs.com/editor.php

In mathematics, you don't understand things. You just get used to them.

If it ain't broke, fix it until it is.

Online

## #3 2014-01-15 05:48:06

ninjaman
Member
Registered: 2013-10-15
Posts: 51

### Re: differentiation

thanks a lot, I looked at the bit where you put in a radical(I think that's what its called, the sqrt sign) that threw me.
thanks also for the latex thing, that will help a lot when writing out maths stuff!

all the best
simon

Offline

## #4 2014-01-15 05:58:11

bobbym
From: Bumpkinland
Registered: 2009-04-12
Posts: 95,897

### Re: differentiation

Hi;

It makes perfect latex so everything looks nice and neat.

In mathematics, you don't understand things. You just get used to them.

If it ain't broke, fix it until it is.

Online


text:
Warning: count(): Parameter must be an array or an object that implements Countable in /home/customer/www/thegatebook.in/public_html/qa/qa-include/qa-theme-base.php on line 177

# DSA - Grand Test -Q22

Warning: count(): Parameter must be an array or an object that implements Countable in /home/customer/www/thegatebook.in/public_html/qa/qa-include/qa-theme-base.php on line 177
+1 vote

Consider the following code fragment.

`y = 0;`

`for ( i = 0; i <= n; i++)`

`{`

`  y += 2^i;`

`}`

Which one of the following conditions is TRUE before every iteration of the loop (where i is the iteration number and ^ is an exponentiation operator)?

(A) $\dpi{100} y_i = 2i - 1$

(B) $\dpi{100} y_i = 2^i - 1$

(C) $\dpi{100} y_i = 2^{i-1}-1$

(D) $\dpi{100} y_i = 2^{i+1}-1$

reshown Sep 10, 2020

When loop runs 3 times,

y = 1 +2 + 4 =7

i.e. $2^{3}-1$

Hence

when loop runs n times ($\leqslant n$)

THen

$2^{n}-1$

And for every iteration of the loop

It is

$2^{i}-1$

answered Sep 26, 2020 by (11,240 points)


text:
# zbMATH — the first resource for mathematics

Arithmetic of Catalan’s constant and its relatives. (English) Zbl 1451.11078
Let $$\beta(s)=\sum_{k=0}^\infty \frac {(-1)^k}{(2k+1)^s}$$ be the Dirichlet beta function. The author proves that at least one of the numbers $$\beta(2)$$-Catalan constant, $$\beta(4)$$, $$\beta(6)$$, $$\beta(8)$$, $$\beta(10)$$, $$\beta(12)$$ is irrational. The proof is in the spirit of Hermite, namely, it evaluates the integral which is from one point of view a small number below one in absolute value and from the second point of view a non-zero integer.
##### MSC:
 11J72 Irrationality; linear independence over a field 11Y60 Evaluation of number-theoretic constants 33C20 Generalized hypergeometric series, $${}_pF_q$$
Full Text:
##### References:
 [1] Fischler, S.: Irrationality of values of $$L$$-functions of Dirichlet characters. Preprint arXiv:1904.02402 [math.NT] (2019) [2] Fischler, S.; Sprang, J.; Zudilin, W., Many odd zeta values are irrational, Compos. Math., 155, 938-952, (2019) · Zbl 1430.11097 [3] Krattenthaler, C.; Zudilin, W., Hypergeometry inspired by irrationality questions, Kyushu J. Math., 73, 189-203, (2019) [4] Rivoal, T.; Zudilin, W., Diophantine properties of numbers related to Catalan’s constant, Math. Ann., 326, 705-721, (2003) · Zbl 1028.11046 [5] Rivoal, T., Zudilin, W.: A note on odd zeta values. Preprint arXiv:1803.03160 [math.NT] (2018) [6] Sprang, J.: Infinitely many odd zeta values are irrational. By elementary means. Preprint arXiv:1802.09410 [math.NT] (2018) [7] Zudilin, W., Arithmetic of linear forms involving odd zeta values, J. Théor. Nombres Bordeaux, 16, 251-291, (2004) · Zbl 1156.11327 [8] Zudilin, W., One of the odd zeta values from $$\zeta (5)$$ to $$\zeta (25)$$ is irrational. By elementary means, SIGMA, 14, 028, (2018) · Zbl 1445.11063
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


text:
## Encyclopedia > Flux

Article Content

# Flux

Flux is the flow that occurs as a result of a potential difference. It can be described as a through variable[?], where potential difference is the across variable[?]. The product of the flux and the potential difference is the power, which is the rate of change of the conserved quantity, e.g energy.

There are many types of flux:

Flux is a quantity proportional to the surface integral of the normal (perpendicular) force field intensity over a given surface.

$\mathrm{Flux} = K \int_s F_N\,dS$

Where FN is the normal component of a field (eg, gravitational field, magnetic field, electric field) and K is the constant of proportionality between the field and the flux density (permittivity, permeability, etc.).

For electromagnetic radiation, flux signifies the energy per unit time (or power) passing through a surface.

The term is also used to denote the volume or mass of fluid or particles transferred across a given area perpendicular to the direction of flow in a given time. For photons or particles, flux is the number passing through a surface per unit time. In nuclear physics, flux commonly means the product n×v, where n is the number of particles per unit volume and v is their mean velocity.

Flux is an aid to melting, a material which by its chemical action facilitates soldering or brazing[?] of metals. Such a flux applied to a metallic surface cleans it and renders it receptive to amalgamation[?] with the solder or brazing metal. Some fluxes are rosin[?], for soldering tin; muriatic acid, for galvanized iron and other zinc surfaces; and borax, for brazing.

A related use of the term flux is to designate the material added to the contents of a smelting furnace or a cupola for the purpose of purging the metal of impurities, and of rendering the slag more liquid. The flux most commonly used in iron and steel furnaces is limestone, which is charged in the proper proportions with the iron and fuel. The slag is a liquid mixture of ash, flux, and other impurities.

Flux is a book by Stephen Baxter.

All Wikipedia text is available under the terms of the GNU Free Documentation License

Search Encyclopedia
 Search over one million articles, find something about almost anything!

Featured Article
 1265 ...     Contents 1265 Centuries: 12th century - 13th century - 14th century Decades: 1210s 1220s 1230s 1240s 1250s - 1260s - 1270s 1280s 1290s 1300s ...


text:
## Tuesday, 19 June 2012

### Bayesian hierarchical glaucoma

Last year (in fact I did some of this while travelling to go to my friend Lorenzo's stag do $-$ he's the one in white, but with no veil), I worked on a clinical paper discussing the prevalence of glaucoma with specific focus on the European population. The objective was relatively straightforward, except for the fact that the studies used to derive the estimations were quite heterogeneous and thus we could not pool them altogether.

So we used a nice (I think) Bayesian hierarchical model where different studies contributed to different parts of the estimation procedure. I built a model in which the overall prevalence was estimated using separate (but connected) modules $-$ basically age groups. So we first estimate a set of "level-1" parameters $\theta_1,\theta_2,\theta_3,\theta_4$ (effectively the age-group specific prevalences) using the observed data from the available studies. Some of these are assumed to be conditionally exchangeable, so that for example $\theta_2,\theta_3,\theta_4$ are used to inform the distribution of the parameter $\theta_5$, representing the prevalence among the over 50s. Again assuming conditionally exchangeability, $\theta_1$ and $\theta_5$ are used to inform the overall prevalence among the over 40s.

We have found sensible (or so I'm told by the clinicians!) results. It wasn't the place to brag about the use of a Bayesian approach, so the paper does not give much detail on the actual model. The observed data were counts of subjects with glaucoma in the study populations.

But it was cool that I persuaded them to report the results in graphical fashion and with the credible intervals.

Some of them do not even know that the model is Bayesian, but they were extremely happy with the results (or if they weren't, they were extremely nice to me anyway).

The paper is out now.


text:
// Home / Documentation / Function Reference / NTLOGNORMINV

# NTLOGNORMINV

Returns the inverse of the cumulative distribution function for a specified log normal distribution.

## Syntax

NTLOGNORMINV(

• prob,
• M,
• S

)

#### Parameters

• prob is a probability corresponding to the normal distribution (0 < prob < 1).
• M is a parameter of the distribution $M$.
• S is a parameter of the distribution $S\;(>0)$.

## Example

• The example may be easier to understand if you copy it to a blank worksheet
How to copy an example
1. Create a blank workbook or worksheet.
2. Select the example in the Help topic.

Note  Do not select the row or column headers.

Selecting an example from Help

3. Press CTRL+C.
4. In the worksheet, select cell A1, and press CTRL+V.
5. To switch between viewing the results and viewing the formulas that return the results, press CTRL+` (grave accent), or on the Tools menu, point to Formula Auditing, and then click Formula Auditing Mode.

1
2
3
4
5
6

A B
Data Description
0.5 Probability associated with the beta distribution
13.5 Value of parameter M
31 Value of parameter B
Formula Description (Result)
=NTLOGNORMINV(A2,A3,A4) Inverse of the cumulative distribution function for the terms above


text:
Games like Karateka (which uses UDK) seem to be able to asynchronously load GL assets as well as display animation.

The splash is incredibly short, then presumably their glView shows up.

I can think of 2 ways to simultaneously load textures, models and assets into OpenGL:

1) Create 2 contexts: the "real" one (which is doing the loading) and a "fake" one (which will be on a separate viewcontroller.. on a separate thread.) I'm not sure if this will work however as I don't know if you can create a 2nd GL context on not the main thread.

2) Cut the load sequence into small, small jobs. Each iteration of your draw loop, load one tiny asset (like a single texture or a single model).
The effect of this is probably a much longer total loading time!

Are either of these 2 right or are they both wrong? Is there another way to do it?

• Where is the question? Apr 5 '13 at 21:53
• Are either of these 2 right or are they both wrong? I have no idea. Apr 5 '13 at 21:53

After some searching, I came across the concept of EAGL Sharegroups.

It seems as though iOS designers really thought things through, because EAGLShareGroup is exactly what I need here.

Docs say EAGL Sharegroups are useful "When you want your application to be able to create new OpenGL ES objects on a thread other than the main thread for the renderer. In this case, a second context runs on a separate thread and is devoted to fetching data and creating resources. After the resource is loaded, the first context can bind to the object and use it immediately."

Creating a new context with the same sharegroup is easy:

EAGLContext* secondContext = [[EAGLContext alloc] initWithAPI:[firstContext API]
sharegroup: [firstContext sharegroup]];

It's ok to read the same OpenGL object on multiple contexts with the same sharegroup provided the object is not being modified.

While a GL object is being modified, you may not read that object on another context. So you can't try and render the texture you are loading on one context with the other context until you are 100% sure the first context is done loading first.

After an object has been modified, you have to re-bind it on all contexts to see the changes.

So much for cutting my code up into little bits!!


text:
# Tag Info

Local quasiparticle excitations and topological quasiparticle excitations To understand and classify anyonic quasiparticles in topologically ordered states, such as FQH states, it is important to understand the notions of local quasiparticle excitations and topological quasiparticle excitations. First let us define the notion of particle-like'' ...


text:
# Difference between revisions of "2012 AMC 12B Problems/Problem 25"

## Problem 25

Let $S=\{(x,y) : x\in \{0,1,2,3,4\}, y\in \{0,1,2,3,4,5\},\text{ and } (x,y)\ne (0,0)\}$. Let $T$ be the set of all right triangles whose vertices are in $S$. For every right triangle $t=\triangle{ABC}$ with vertices $A$, $B$, and $C$ in counter-clockwise order and right angle at $A$, let $f(t)=\tan(\angle{CBA})$. What is $$\prod_{t\in T} f(t)?$$

$\textbf{(A)}\ 1\qquad\textbf{(B)}\ \frac{625}{144}\qquad\textbf{(C)}\ \frac{125}{24}\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ \frac{625}{24}$

## Solution 1

Four points in a rectangular arrangement allow 4 possible right angle triangles. These four congruent triangles will form two pairs of different vertice labelling - two ABC and two ACB. These pairs will multiple to equal 1 due to the fact that $\tan x \tan(90^{\circ}-x) = \tan x \cot x = 1.$

Due to the missing point (0,0) in the grid then in the rectangular arrangement (0,0), (x,0), (x,y), (0,y) there is only one triangle which will not cancel out to zero with a reflected version.

So we need to consider all triangles of the form A(x,y), B(0,y), C(x,0). For these triangle tanB = y/x. Multiplying them all together gives: 1/1 * 1/2 * 1/3 * 1/4 * 2/1 * 2/2 * 2/3 * 2/4 * 3/1 * 3/2 * 3/3 * 3/4 * 4/1 * 4/2 * 4/3 * 4/4 * 5/1 * 5/2 * 5/3 * 5/4 = 625/24

## Solution 2

Consider reflection. For any right triangle $ABC$ with the right labeling described in the problem, any reflection $A'B'C'$ labeled that way will give us $\tan CBA \cdot \tan C'B'A' = 1$. First we consider the reflection about the line $y=2.5$. Then only those triangle $\subseteq S$ that has one vertex at $(0,5)$ does not $\subseteq S$. Within those triangles, consider a reflection about the line $y=5-x$. Then only those triangle $\subseteq S$ that has one vertex on the line $y=0$. There are three cases:

Case 1: $A=(0,5)$. Then $B=(*,0)$ is impossible.

Case 2: $B=(0,5)$. Then we look for $A=(x,y)$ such that $\angle BAC=90^{\circ}$ and that $C=(*,0)$. They are: $(A=(x,5), C=(x,0))$, $(A=(2,4), C=(1,0))$ and $(A=(4,1), C=(3,0)$. The product of their tangents of $\angle CBA$ is $5/1 \cdot 5/2 \cdot 5/3 \cdot 5/4 \cdot 1/4 \cdot 2/3 = 625/144$.

Case 3: $C=(0,5)$. Then $A=(*,0)$ is impossible.

Therefore $\framebox{B}$ is the answer.


text:
# Singular Matrix & Non Singular Matrix | Properties & Examples

Singular matrix is a square matrix whose determinant is zero. It is also known as non invertible matrix or degenerate matrix. A square matrix whose determinant is not zero is known as non singular matrix. It is also known as invertible matrix or non degenerate matrix.

Contents

A square matrix P is said to be singular matrix if |P| = 0. A square matrix Q is said to be non singular matrix if |Q| ≠ 0.

## Singular matrix examples

The example of a singular matrix of order 3 (or matrix size is 3 x 3) is given below.

The example of a singular matrix of order 2 (or matrix size is 2 x 2) is given below.

## Non Singular matrix examples

The example of non singular matrix of order 3 (or matrix size is 3 x 3) is given below.

The example of non singular matrix of order 2 (or matrix size is 2 x 2) is given below.

## Singular matrix properties

1. The determinant of a singular matrix (P) is zero i.e. |P| = 0.

2. The inverse of a singular matrix does not exist. Hence it is also known as non-invertible matrix.

3. This matrix is always a square matrix because determinant is always calculated for a square matrix.

4. There is no multiplicative inverse exist for this matrix.

## Non Singular matrix properties

1. The determinant of a non singular matrix (Q) is not zero i.e. |Q| ≠ 0.

2. The inverse of a non singular matrix does exist. Hence it is also known as invertible matrix.

3. Such matrix is always a square matrix because determinant is always calculated for a square matrix.

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## Sublinear singular elliptic problems with two parameters.(English)Zbl 1039.35042

Let $$\Omega$$ be a smooth bounded domain in $$\mathbb R^N$$ $$(N\geq 2)$$. The authors study the existence or nonexistence of solutions to the following boundary value problem $\begin{cases} -\Delta u+ K(x)g(u)=\lambda f(x,u)+\mu h(x)\quad &\text{in }\Omega,\\ u> 0\quad &\text{in }\Omega,\\ u=0\quad &\text{on }\partial\Omega,\end{cases}\tag{P$$_{\lambda,\mu}$$}$ where $$\lambda$$ and $$\mu$$ are positive parameters. Under suitable assumptions on $$K$$, $$g$$, $$f$$ and $$h$$ the authors using the maximum principle for elliptic equations establish several existence and nonexistence results for $$(\text{P}_{\lambda,\mu})$$.

### MSC:

 35J65 Nonlinear boundary value problems for linear elliptic equations 35B50 Maximum principles in context of PDEs 58J55 Bifurcation theory for PDEs on manifolds 35J60 Nonlinear elliptic equations
Full Text:

### References:

 [1] Aris, R., The mathematical theory of diffusion and reaction in permeable catalysts, (1975), Clarendon Press Oxford · Zbl 0315.76051 [2] Bénilan, P.; Brezis, H.; Crandall, M., A semilinear equation in $$L\^{}\{1\}( R\^{}\{N\})$$, Ann. scuola norm. sup. cl. sci. Pisa, 4, 523-555, (1975) · Zbl 0314.35077 [3] Callegari, A.; Nashman, A., Some singular nonlinear equations arising in boundary layer theory, J. math. anal. appl., 64, 96-105, (1978) · Zbl 0386.34026 [4] Callegari, A.; Nashman, A., A nonlinear singular boundary value problem in the theory of pseudoplastic fluids, SIAM J. appl. math., 38, 275-281, (1980) · Zbl 0453.76002 [5] Chen, H., On a singular nonlinear elliptic equation, Nonlinear anal. T.M.A., 29, 337-345, (1997) · Zbl 0882.35050 [6] Choi, Y.S.; Lazer, A.C.; McKenna, P.J., Some remarks on a singular elliptic boundary value problem, Nonlinear anal. T.M.A., 3, 305-314, (1998) · Zbl 0940.35089 [7] Coclite, M.M.; Palmieri, G., On a singular nonlinear Dirichlet problem, Comm. partial differential equations, 14, 1315-1327, (1989) · Zbl 0692.35047 [8] Cohen, D.S.; Keller, H.B., Some positive problems suggested by nonlinear heat generators, J. math. mech., 16, 1361-1376, (1967) · Zbl 0152.10401 [9] Crandall, M.G.; Rabinowitz, P.H.; Tartar, L., On a Dirichlet problem with a singular nonlinearity, Comm. partial differential equations, 2, 193-222, (1997) · Zbl 0362.35031 [10] Diaz, J.I.; Morel, J.M.; Oswald, L., An elliptic equation with singular nonlinearity, Comm. partial differential equations, 12, 1333-1344, (1987) · Zbl 0634.35031 [11] Gomes, S.M., On a singular nonlinear elliptic problem, SIAM J. math. anal., 17, 1359-1369, (1986) · Zbl 0614.35037 [12] Gilbarg, D.; Trudinger, N.S., Elliptic partial differential equations of second order, (1983), Springer Berlin · Zbl 0691.35001 [13] Lazer, A.C.; McKenna, P.J., On a singular nonlinear elliptic boundary value problem, Proc. amer. math. soc., 3, 720-730, (1991) · Zbl 0727.35057 [14] Luning, C.D.; Perry, W.L., An interactive method for solution of a boundary value problem in non-Newtonian fluid flow, J. non-Newtonian fluid mech., 15, 145-154, (1984) · Zbl 0564.76013 [15] T. Ouyang, J. Shi, M. Yao, Exact multiplicity of positive solutions for a singular equation in unit ball, preprint, 2002. [16] Perry, W.L., A monotone iterative technique for solution of pth order (p<0) reaction – diffusion problems in permeable catalysis, J. comput. chem., 5, 353-357, (1984) [17] del Pino, M., A global estimate for the gradient in a singular elliptic boundary value problem, Proc. roy. soc. Edinburgh sect. A, 122, 341-352, (1992) · Zbl 0791.35046 [18] Shi, J.; Yao, M., On a singular nonlinear semilinear elliptic problem, Proc. roy. soc. Edinburgh sect. A, 128, 1389-1401, (1998) · Zbl 0919.35044 [19] Wong, J., On the generalized emden – fowler equation, SIAM rev., 17, 339-360, (1975) · Zbl 0295.34026 [20] Zhang, Z., On a Dirichlet problem with a singular nonlinearity, J. math. anal. appl., 194, 103-113, (1995) · Zbl 0834.35054
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# x2 +7x +10 x+5 define f(x) at -5 so that it becomes continuous at -5 34. The function f(x)- is defined everywhere except at x--5. If possible,

Question

sketch the graph and find the x-values (if any) at which f is not
continuous. Classify the discontinuities as being removable or essential. If the latter, state
whether it is a jump discontinuity, an infinite discontinuity, or neither.


text:
# American Institute of Mathematical Sciences

November  2013, 12(6): 3027-3046. doi: 10.3934/cpaa.2013.12.3027

## Global dynamics and diffusion limit of a one-dimensional repulsive chemotaxis model

 1 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong 2 Department of Mathematics, Tulane University, New Orleans, LA 70118

Received  April 2012 Revised  November 2013 Published  May 2013

In the first part of this paper, we investigate the qualitative behavior of classical solutions for a one-dimensional parabolic system derived from a repulsive chemotaxis model on bounded domains. It is shown that classical solutions to the initial-boundary value problem exist globally in time for large data and converge to constant equilibrium states exponentially in time. The results indicate that repulsive chemotaxis exhibits a strong tendency against pattern formation. In the second part, we study diffusion limit and convergence rate of the model toward a non-diffusive problem studied in [11]. It is shown that when the chemical diffusion coefficient $\varepsilon$ tends to zero, the solution is convergent in $L^{\infty}$-norm with respect to $\varepsilon$ at order $O(\varepsilon)$.
Citation: Zhi-An Wang, Kun Zhao. Global dynamics and diffusion limit of a one-dimensional repulsive chemotaxis model. Communications on Pure & Applied Analysis, 2013, 12 (6) : 3027-3046. doi: 10.3934/cpaa.2013.12.3027
##### References:
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show all references

##### References:
 [1] W. Alt and D. Lauffenburger, Transient behavior of a chemotaxis system modelling certain types of tissue inflammation,, J. Math. Biol., 24 (1987), 691.  doi: 10.1007/BF00275511.  Google Scholar [2] D. Balding and D. McElwain, A mathematical model of tumour-induced capillary growth,, J. Theor. Biol., 114 (1985), 53.  doi: 10.1016/S0022-5193(85)80255-1.  Google Scholar [3] S. Childress, Chemotactic collapse in two dimensions,, Lect. Notes in Biomath., 55 (1984), 61.  doi: 10.1007/978-3-642-45589-6_6.  Google Scholar [4] T. Cieślak, P. Laurencot and C. Morales-Rodrigo, Global existence and convergence to steady states in a chemorepulsion system,, Banach Center Publ., 81 (2008), 105.  doi: 10.4064/bc81-0-7.  Google Scholar [5] J. Guo, J. Xiao, H. Zhao and C. Zhu, Global solutions to a hyperbolic-parabolic coupled system with large initial data,, Acta Math. Sci. Ser. B Engl. Ed, 29 (2009), 629.  doi: 10.1016/S0252-9602(09)60059-X.  Google Scholar [6] D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its con- sequences I,, Jahresberichte der DMV, 105 (2003), 103.   Google Scholar [7] E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar [8] E. Keller and L. Segel, Traveling bands of chemotactic bacteria: a theoretical analysis,, J. Theor. Biol., 26 (1971), 235.  doi: 10.1016/0022-5193(71)90051-8.  Google Scholar [9] H. Levine and B. Sleeman, A system of reaction diffusion equations arising in the theory of reinforced random walks,, SIAM J. Appl. Math., 57 (1997), 683.  doi: 10.1137/S0036139995291106.  Google Scholar [10] D. Li, T. Li and K. Zhao, On a hyperbolic-parabolic system modeling chemotaxis,, Math. Models Methods Appl. Sci, 21 (2011), 1631.  doi: 10.1142/S0218202511005519.  Google Scholar [11] T. Li, R. Pan and K. Zhao, Global dynamics of a chemotaxis model on bounded domains with large data,, SIAM J. Appl. Math., 72 (2012), 417.  doi: 10.1137/110829453.  Google Scholar [12] T. Li and Z. Wang, Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis,, SIAM J. Appl. Math., 70 (2009), 1522.  doi: 10.1137/09075161X.  Google Scholar [13] T. Li and Z. Wang, Nonlinear stability of large amplitude viscous shock waves of a hyperbolic-parabolic system arising in chemotaxis,, Math. Models Methods Appl. Sci., 20 (2010), 1967.   Google Scholar [14] T. Li and Z. Wang, Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis,, J. Differential Equations, 250 (2011), 1310.  doi: 10.1016/j.jde.2010.09.020.  Google Scholar [15] T. Li and Z. Wang, Steadily propagating waves of a chemotaxis model,, Math. Biosci., 240 (2012), 161.  doi: 10.1016/j.mbs.2012.07.003.  Google Scholar [16] C. Lin, W. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system,, J. Differential Equations, 72 (1988), 1.  doi: 10.1016/0022-0396(88)90147-7.  Google Scholar [17] J. Murray, "Mathematical Biology I: An Introduction,", 3$^{rd}$ edition, (2002).   Google Scholar [18] H. Othmer and A. Stevens, Aggregation, blowup and collapse: The ABC's of taxis in reinforced random walks,, SIAM J. Appl. Math., 57 (1997), 1044.  doi: 10.1137/S0036139995288976.  Google Scholar [19] L. Segel, A theoretical study of receptor mechanisms in bacterial chemotaxis,, SIAM J. Appl. Math., 32 (1977), 653.  doi: 10.1137/0132054.  Google Scholar [20] J. Sherratt, E. Sage and J. Murray, Chemical control of eukaryotic cell movement: a new model,, J. Theor. Biol., 162 (1993), 23.  doi: 10.1006/jtbi.1993.1074.  Google Scholar [21] Y. Tao, L. Wang and Z. Wang, Large-time behavior of a parabolic-parabolic chemotaxis model with logarithmic sensitivity in one dimension,, Discrete Contin. Dyn. Syst - Series B., 18 (2013), 821.   Google Scholar [22] Y. Tao and Z. Wang, Competing effects of attraction vs. repulsion in chemotaxis,, Math. Models. Methods Appli. Sci., 23 (2013), 1.  doi: 10.1142/S0218202512500443.  Google Scholar [23] Z. Wang and T. Hillen, Shock formation in a chemotaxis model,, Math. Methods. Appl. Sci., 31 (2008), 45.  doi: 10.1002/mma.898.  Google Scholar [24] M. Winkler, Global solutions in a fully parabolic chemotaxis system with singular sensitivity,, Math. Methods Appl. Sci., 34 (2011), 176.  doi: 10.1002/mma.1346.  Google Scholar [25] Y. Yang, H. Chen and W. Liu, On existence of global solutions and blow-up to a system of the reaction-diffusion equations modelling chemotaxis,, SIAM J. Math. Anal., 33 (2001), 763.  doi: 10.1137/S0036141000337796.  Google Scholar [26] M. Zhang and C. Zhu, Global existence of solutions to a hyperbolic-parabolic system,, Proc. Amer. Math. Soc., 135 (2006), 1017.  doi: 10.1090/S0002-9939-06-08773-9.  Google Scholar
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Environ Eng Res > Volume 26(4); 2021 > Article
Sahu, Bhan, and Singh: Removal of fluoride from an aqueous solution by batch and column process using activated carbon derived from iron infused Pisum sativum peel: characterization, Isotherm, kinetics study

### Abstract

The present study investigated the adsorption efficiency of magnetic activated carbon was synthesized by waste biomass of Pisum sativum (peel) and pyrolysis at 500°C temperature (MPPAC-500). Derived activated carbon was applied for removal of fluoride from aqueous solution. The MPPAC-500 was characterized by Fourier Transform Infrared Spectroscopy (FTIR), Scanning Electron Microscopy (SEM), zeta potential, X-ray Diffraction (XRD) and Particle Size Analyser. The fluoride sequestration study was performed in both batch and column systems. The batch adsorption study was focused on parameter like, adsorbent dose, contact time, pH and initial fluoride concentrations. The maximum capacity of fluoride removal was qo = 4.71 (mg/g). Freundlich isotherm model (R2-0.995) obeyed better than Langmuir (R2-0.979) model. The RL values observed between 0–1 (RL-0.057) inferred the favourable adsorption. Pseudo-second-order model favoured well than pseudo-first-order in the whole experimental data. In case of column study was performed at two different bed height 5 cm and 10 cm having flow rate of 5 mL/min as well as 10 mL/min. The breakthrough curve and column data were interpreted by Thomas, Adams-Bohart, Yoon-Nelson and Clark model. These finding showed that MPPAC-500 has potential adsorptive capacity for fluoride removal from aqueous solutions in batch and column systems.

### 1. Introduction

Excessive and dearth presence of Fluoride (F) in groundwater exposed more than two hundred million people across the world [1]. Many countries like China, India, Nigeria, Kenya, Pakistan, South America, Africa and Sri Lanka are facing severe problem of F contamination in groundwater [2, 3]. According to WHO [4] 1.5 mg/L is maximum concentration of fluoride in water for drinking purpose. Fluoride act as double edged sword like element in water as lower concentration of F is essential for human health, but its concentration more than 1.0 mg/L leads fluorosis related dental and skeletal in which teeth enamel and bone tissues get harmed because of its strong attraction with calcium found in teeth and bones [5].
Fluoride is naturally present in earth crust and rocks and it reached to the groundwater by slow leaching of fluoride containing parent rocks (viz. granite, basalt, synginte etc.) and minerals such as biotite, fluorite topaz [6]. Therefore, the natural geological sources are eventually responsible for reaching of fluoride to groundwater. Besides natural sources pharmaceuticals, cosmetics, effluents from fertilizers, metal processing units, semiconductors, and glass manufacturing factories are also contributing in reaching of fluoride to the surface water and groundwater [7]. Globally about more than 20 developed and developing nations like USA, Africa and Asia, where fluorosis became endemic, wherever China, Sri Lanka, India and Rift Valley countries of Africa are badly affected. In India, it has been reported that about 17 states are prevalently affected by fluorosis specially Andhra Pradesh, Gujarat, Rajasthan, Haryana, Assam, Tamil Nadu and Uttar Pradesh [8]. At present about more than 1 million people of India are thought to be affected by endemic fluorosis [9]. Therefore, today defluoridation of water became a necessity for vital human health by feasible technologies. Defluorination of water has been conducted by applying different conventional technologies such as membrane filtration, reverse osmosis, ion exchange, electrodialysis, flotation and adsorption. However, among these only few technologies can be applied at large scale because these technologies require high initial operating investment and maintenance. Therefore, there is a need to focus on cost effective and efficient technologies for the treatment of fluoride containing water. Adsorption is one of centre of focus of many researchers now days because of its cost-effectiveness, ease in operation, less chemical consumption, and better efficiency [10].
From the literature study, it is found that number of materials have been used for fluoride adsorption like oxides of Fe-Al [11], Al-impregnated carbon [12], activated charcoal, Mn-Ce oxide [13], composites of biopolymer [14], agricultural waste [15], algal biomass, leaf biomass, biochar [16]. Activated and modified carbon is well known adsorbent for removal of pollutants from the water and wastewater due to its large surface area, good porosity, stable chemical characters [17]. Impregnation of Fe(III) with activated carbon enhances the adsorption ability ion aqueous solutions. The Fe(III) containing material bound with carbonyl groups, amine, carboxylic and hydroxyl groups through complexation reaction [18]. The different adsorbents having their adsorption capacity were reported for fluoride adsorption such as Cerium impregnated activated carbon (4.6 mg/g) [1], activated carbon (6.5 mg/g) [19], Zirconium doped activated carbon (5 mg/g) at 4 pH [20], SnO2 doped activated carbon (4.6 mg/g) [21] and CaCl2 modified activated carbon (2.1 mg/g) [22].
In this study, authors attempt to prepare iron based activated carbon from the kitchen waste pea (pea peel) for the removal of fluoride by batch as well as column process that is the novelty of this research work.

### 2.1. Chemicals and Reagents

In this study magnetic adsorbent was prepared from the peel of pea (Pisum sativum) collected from the local vegetable market and grains separated from peel. The analytical grade chemicals and reagents were used in present study (sodium fluoride, solution of Ionic strength adjuster (ISA) solution, hydrochloric acid (HCl), ferric chloride (FeCl3), sodium hydroxide (NaOH),) were acquired from sigma Aldrich and thermo fisher scientific.

### 2.2. Synthesis of Activated Carbon

Activated carbon prepared from the pea peel according to the method explained by Lung et al. [23]. The collected pea peel was washed several times to eliminate the dust particles followed by drying at 80°C for 24 h, after that it was ground and sieved in fraction of 0.25 mm mesh. The material was prepared in 1:10 ratio, 1 part is pea peel and 10 part is ferric chloride (FeCl3) solution, then the mixture was stirrer for 240 mins and kept overnight for the mixing of FeCl3 with pea peel powder. The excess iron solution was separated by vacuum filtration and the obtained solid material dried for 24 h at 80°C. The material was then filled in the quartz crucible with 100ml capacity and capped with fitted lid then pyrolyzed by using muffle furnace at 500°C temperature for 2 h. The obtain Activated Carbon was washed number of times with distilled water and then oven dried for 12 h. The dried activated carbon homogenized and used for the removal of Fluoride. In this study synthesized pea peel magnetic activated carbon was termed as PPMAC-500.

### 2.3. Characterization of the Bioadsorbent

SEM (Scanning electron microscope), (JSM-6490 LV, JEOL, JAPAN) equipped with EDS used to determine surface analysis of the synthesized adsorbent (PPMAC-500). Surface functional group of the adsorbent observed by using Fourier-transform-infrared-spectroscopy (FTIR), (NICOLET 6700, Thermo Fisher Scientific, USA). X-Ray diffractometer (PW 3040/60 Pan Analytical Nether Land) was used for the analysis of crystalline structure of the activated carbon. The activated carbon scanned in the range 10° to 90° with 2° min−1 speed rate. Zeta nano-sizer (model no-Nano-ZS90, Malvern Instrument) used for observation of stability and particles size distribution by after dissolving activated carbon in ethanol. Point zero charge on the surface of adsorbent was also analysed.

### 2.4. Experimental Batch Adsorption Design

All batch experiment performed by adding of decided MPPAC-500 doses (1 to 4 g/L) having 50 mL solution of fluoride with varying concentration (5 to 20 mg/L) in a 250 mL of Erlenmeyer flask. The flasks were shaken at constant speed (100 rpm) in an orbital shaker at ambient temperature. After adsorption, samples were collected at regular interval differences and then filtered, residual concentration of fluoride analysed by using fluorimeter (EUTECH ION 6+ THERMO SCIENTIFIC, U.S.A. Whole batch experiment was done at ambient temperature except (from temperature values 25°C to 55°C and pH study conducted with different pH (2 to 10 pH). A pH of the fluoride solution was maintained by 0.1 M hydrochloric acid and 0.1 M sodium hydroxide solution. All batch adsorption studies were conducted in triplicates to minimize the error of the data. The removal percentage (Re) and adsorption capacity (qt) of fluoride were determined by Eq. (1) and (2), respectively.
##### (1)
$Re=(Co-CeCo)×100$
##### (2)
$Qt=(Co-Cew)×V$
Where, Co; Initial concentration, Ce; final concentration, w; adsorbent mass, V; solution volume.

### 2.5. Experimental Column Design and Procedure

The continuous column adsorption performed for the removal of on MPPAC-500 borosilicate glass column at room temperature and neutral pH. Vertical glass column was used for fixed bed column study with 1.5 cm diameter and 38.3 cm length is shown in Fig. 1. Column were performed in downflow mode. Two different quantities of MPPAC-500 were packed at 5 cm and 10 cm bed height with used doses 4.8 g for 5 cm and 9.6 g for 10 cm bed depth.
The MPPAC-500 was packed by the 1cm layer of glass wool and sieved with pore size 0.5 mm were placed at both side (up and down) of the column, which are used for the uniform flow. solution having 10 mg/L concentration was passed out from reservoir knob with 5 mL/min and 10 mL/min flow rate. The treated samples (5 mL) were collected at regular time interval. The main focus of column adsorption study to determined, how much volume of the effluent can treat at a time.

### 2.5.1.1. Breakthrough curve

Breakthrough curve is used to check adsorption performance of the fixed-bed-column. The breakthrough point was taken until the outlet fluoride concentration was reaches 99.5% of inlet fluoride concentration. The breakthrough curve was expressed by plotting Cin/Cout against contact time [24]. Where Cin is the starting fluoride concentration and Cout is the final fluoride concentration, respectively.
Total outlet volume, Veff can be determined by the following Eq. (3).
##### (3)
$Veff=Q×ttotal$
Where Q (mL/min); flow rate in volume, ttotal; total flow time (min).
Total adsorbed amount of fluoride concentration, qtotal (mg) can be observed from the Eq. (4).
##### (4)
$qtotal=Q1000∫t=0t=totalCaddt$
Adsorption capacity denotes by qe (mg/g) can be found by the formula as given below.
##### (5)
$qe=qtotalm$
where m denotes dry amount of the adsorbent (g) filled in a column.
Total fluoride ions (Mtotal) delivered to the column is calculated from Eq. (6).
##### (6)
$Mtotal=Co×Q×ttotal$
Total removal percentage of fluoride ions can be calculated from Eq. (7).
##### (7)
$Total fluoride removal %=madmtotal×100$
Where mad is the dry amount of adsorbent sent to the column.

### 3.1. Characterization of MPPAC-500

#### 3.1.1. SEM and EDX

The SEM and EDS analysis were conducted for morphological investigation of prepared adsorbent. Image shows that surface of iron modified biochar is rough and partially porous in nature as presented in Fig. 2(a). Surface roughness and porosity of adsorbent may determine the maximum adsorption sites that facilitate high adsorption of fluoride. After adsorption of fluoride the structure of adsorbent particle altered that is shown Fig. 2(b). The EDS analysis revealed that C, O, Cl, and Fe are present in adsorbent in different proportion as shown in Fig. 5(c) and (d).

#### 3.1.2. FTIR

The FTIR spectra of MPPAC-500 before and after uptake of fluoride, are shown in Fig. 2(c). The spectra of materials were observed from scanning of wavenumber ranged from 400–4,000 cm−1. The strong band observed 3,013. 7 cm−1 in MPPAC-500 before fluoride adsorption and 3,046.8 cm−1 in MPPAC-5000 after adsorption represent the –CH stretching vibration due to presence of methyl group. The peak ranged between 1,513.1 to 1,578.7 cm−1 are due to C = C aromatic group [25]. The adsorption bands observed at 1,408.1 to 1,238.9 cm−1 could be assigned to –OH (hydroxyl) group vibration mode. The shifting of wavenumber from 1,108.7–1,059.9 cm−1 represent the > C = O (ketone) group functioning during adsorption of fluoride. The adsorption band between 400 cm−1 to 500 cm−1 attributed to the Fe-O vibrations. The presence of Fe-O shows that the iron was present on the synthetic adsorbent [26].

#### 3.1.3. XRD

The XRD pattern of MPPAC-500 before adsorption is shown in Fig. 2(d). The XRD pattern observed in the 2θ range of 5–80º. The fine and sharp peaks of MPPAC-500 were found at 2θ range of 20–40º. Fig. 2(d) illustrates the XRD diffractogram pattern of iron coated adsorbent (MPPAC-500). The major and sharp peaks 23.58º, 31.04º and 35.7º, 40.48°, 49.06º and 53.72º are resembling the presence of Fe3O4 [27, 23]. Similar peaks i.e. 30.2º, 35.5º and 53.6º were reported for Fe3O4 [28]. A significant hump in the range of 20º–40º is depicted a crystalline graphitic structure [29].

#### 3.1.4. Zeta Potential

Zeta potential of MPPAC-500 was studied at different pH (2–10) by using the (Zeta Nano Zs90, Malvern Instrument). The value of Zeta potential indicates the stability or instability of the material in dispersed medium [30]. Zeta potential was found to be +56 mV at pH-4 that is shown in Fig. 2(e). This value indicates that adsorbent is highly stable in acidic pH while at higher pH, Zeta potential value was observed + 10 mV. Therefore, it is clear that the adsorbent MPPAC-500 is stable for the application of fluoride.

#### 3.1.5. PSA

Particle size distribution of adsorbent is measured by several techniques such as based on light, ultrasound, electric field and centrifugation. In this study we have used the light-based measurement particle size distribution (Zeta Nano ZS90, Malvern). Fig. 2(f) shows the average particle size range ~0.11 to 0.30 μm, for first peak 20% particle in range ~0.11 μm and second peak 80 % of particles in ~ 0.30 μm. Adsorbent MPPAC-500 has small average particle size distribution its mean synthesized adsorbent has high surface area that facilitate highest adsorption of adsorbate.

#### 3.1.6. pHZPC

The pHZPC value of synthesized MPPAC-500 was measured according to a described method by Singh et al. [31]. pHZPC value was used to determine the surface charge of the adsorbent. Fig. S1 shows that pHZPC of MPPAC-500 was found to be 2.8 and this pHzpc value indicates that surface of adsorbent is dominated with cationic groups. It means the adsorbent is acidic in nature. When pH of the solution does below the pHZPC, surface of the adsorbent contains positive charge while above the pHZPC it contains negative charge.

### 3.2. Batch Study

#### 3.2.1. Effect of adsorbent dose

Quantity of adsorbent plays crucial role in adsorption process. As given in Fig. 3(a) four varying doses 1, 2, 3, 4 g/L of MPPAC-500 were selected for the dose study for the removal of fluoride ions, while other parameters like starting concentration-10 mg/L, contact time- 470 min, at neutral pH in an ambient temperature were maintained. It was observed that with increasing the dosage of the adsorbent from 1 g/L to 4 g/L, the removal of fluoride increased from 78% to 99.5%. The highest adsorption capacity was achieved to be 2.48 mg/g, so that adsorbent dose 4 g/L was selected for the further study. The results revealed that percentage removal of fluoride was increased with increasing the amount of adsorbent because of more active sites are available on MPPAC-500 for the removal of fluoride ions.

#### 3.2.3. Effect of initial fluoride concentration with contact time on adsorption

The effect of fluoride concentration was assessed with varying the starting concentration, taken from 5 mg/L to 20 mg/L at the neutral pH and at ambient temperature. As shown in Fig. 3(c). It was obtained that adsorption rate was declined from 99% to 87% with enhancing the concentration of fluoride from 5 to 20 mg/L. The highest adsorption capacity was observed to be 4.35 mg/g. Further increasing the initial concentration of fluoride ions, removal percentage was decreased this may occur due to active sites on adsorbent surface were decreases after a saturation point [34].
To evaluate the effect of interaction time between fluoride ions and MPPAC-500, batch experiment was performed at several time intervals. Fig. 5(c) illustrates, the removal of fluoride increase with increasing contact time, after contact time of 300 min, there is no further increment in fluoride adsorption so adsorption equilibrium was achieved at 300 min at which removal rate of fluoride was constant because adsorption of fluoride ions in aqueous solution on the surface of MPPAC-500 is equal to the desorption rate.

#### 3.2.4. Effect of Temperature

In an adsorption phenomenon temperature is a very important factor, which influence the reaction rate of it. The effect of the temperature was studied at different temperature range (25°C, 35°C, 45°C, and 55°C) for the adsorption of fluoride on MPPAC-500, while all other parameters were kept at optimum conditions such as dose-4 g/L, concentration-10 mg/L. pH-7, temperature- 25°C. As shown in Fig. 3(d) the removal percentage of the adsorbate was increased with increasing temperature, which proves that adsorption process is endothermic. The adsorption of fluoride was slightly increased from 95% to 99% with enhancing temperature from 25°C to 55°C, this happen might be due to various adsorbent required high temperature for activation of vacant sites on adsorbent at high temperature, many vacant sites are present on the surface of adsorbent, that need sometimes high or sometimes low temperature for activation [35].

### 3.3. Isotherm

An adsorption isotherm was applied to analysed the equilibrium relationship between adsorbate and adsorbent surface. Langmuir, Freundlich, Temkin were widely adapted for validity of adsorption phenomenon. The summarization of isotherm parameters is listed in Table 1.

#### 3.3.1. Langmuir isotherm

Langmuir model illustrate the monolayer adsorption between the adsorbent and adsorbate due to presence of active sites on the surface of adsorbent. Langmuir isotherm presumes that adsorption force and chemical interaction are similar. Langmuir isotherm express by the given Eq. (8).
##### (8)
$Ce/qe=1Qob+(1Qo)$
Where, qe denotes the quantity of fluoride adsorbed per unit gram of the adsorbent (mg/g) and Ce is the fluoride ion concentration at equilibrium (mg/L). Langmuir constant (Qo) and highest adsorption capacity of MPPAC-500 and adsorption rate are represented by b. The value Qo and b can be obtained from plot of Ce/qe and Ce [36] shown in the supplementary Fig. S2. Table 1 show the values of Langmuir isotherm parameter. RL equilibrium parameter known as dimensionless constant that is significant characteristic of the Langmuir isotherm, expressed as following Eq. (9).
##### (9)
$RL=11+bCo$
Where, Co and b are stand for the starting fluoride concentration and Langmuir equilibrium constant, respectively. The RL value express the Langmuir isotherm type could be favourable (0 < RL < 10, (RL = 1) linear, (RL > 1) unfavourable, (RL = 0) irreversible. The value of RL was found to be 0.057 that is calculated by given Eq. (9).

#### 3.3.2. Freundlich isotherm

This isotherm represents the heterogeneous layer formation on the surface of adsorbent due to adsorption. Freundlich model equation is expressed by the given Eq. (10)
##### (10)
$lnqe=lnKF+(1n)CelnCe$
Where, at equilibrium quantity of fluoride (mg/g) and adsorbate concentration (mg/L) are denoted by qe and Ce. The n value illustrates the Freundlich isotherm favouring or not favouring, whereas KF represents the capacity (mg/g(L/mg)1/n) of MPPAC-500 to adsorption fluoride. The heterogenous nature of adsorbent surface and adsorption intensity were calculated through the slope value between 0–1 was observed from the graph plotted lnqe against lnCe. When the slope value is near about 0, indicate that the surface of adsorbent is highly heterogenous. Plotted graph is illustrated in the Fig. 3(e) and obtained values are listed in Table 1.
The Freundlich constant KF value was 3.206 (mg/g(L/mg1/n), which shows the biosorption capacity of adsorbent. The value of ‘n’ was also lying between 1 to 10 (n-4.92) which also favours the favourable adsorption process [37]

#### 3.3.3. Temkin isotherm

This isotherm model gives information about interaction of adsorbent-adsorbate. This model assumes that adsorption heat of whole molecule will not remain same. Interaction between sorbate and sorbent decreases the heat during adsorption process due to uniform dispersal of binding energy [38]. Temkin isotherm represented in the Eq. (11).
##### (11)
$qe=B ln (AT)+B ln Ce$
Where, qe represents the quantity of fluoride adsorbed on MPPAC-500 (g/L) and Ce is the concentration of fluoride. BT is constant for Temkin isotherm related to the adsorption heat (L/g). AT is Temkin isotherm equilibrium binding constant, which is dimensionless. Values of AT and B presented in Table 1 and supplementary Fig. S3.
After comparing of these isotherm models, it is revealed that Freundlich model fitted well (R2 = 0.995) than Langmuir model (R2 = 0.979). Table 1 shows that Qo value of MPPAC-500 was 4.717 mg/g which is quite higher than data available in literature, vitex negundo plant carbon 1.150 mg/g [30], treated banana peel 0.395 mg/g and treated coffee husk 0.416 mg/g [39]. The RL values was ranged between 0–1 (RL-0.22) indicating the favourable fluoride adsorption onto MPPAC-500 [37].

Adsorption kinetic quantifies the rate of reaction and adsorption mechanism involved in the uptake of fluoride onto MPPAC-500 [40]. Both kinetic models (pseudo-first-order and pseudo-second-order) and their parameters were applied for present experimental data. The rate constant of Fluoride adsorption is analysed from the reaction of Pseudo-first-order (PFO) given by Lagergren (1898) that is expressed by Eq. (12).
##### (12)
$log (qe-qt)=log qe-(K12.3030).t$
Where, qe denotes amount of fluoride adsorbed on MPPAC-500 at a point of equilibrium, qt is the amount of fluoride adsorbed at a time (min). K1 is the fluoride adsorption constant rate. The K1 value was obtained from the plot of log (qe–qt) and time (min) was given in supplementary Fig. S4.
The kinetics data of fluoride were also analysed through pseudo- second-order [31] and the equation for pseudo-second-order is given below.
##### (13)
$t/qe=1/(k2qe2)+(1/qe)t$
Where, k2 is stand for the pseudo-second-order constant (g/(mg min) and qe is the fluoride adsorption capacity. There values were found from the plot of t/qt against t as represented in Fig. 3(f).
After comparing data of both kinetic models, it is confirmed that pseudo-second-order favoured better with adsorption data than pseudo-first-order. Values of both kinetics parameters are given in the supplementary Table S1.

### 3.5. Intraparticle Diffusion Model

Adsorption is considered as multi-step phenomenon because it includes transportation of solute from aqueous medium to the adsorbent surface and also solute diffusion through pores. Through the Intraparticle model, large amount of adsorbate present in solution are transferred to the adsorbent surface. The equation for Intraparticle diffusion model is expressed in the below.
##### (14)
$qt=kit1/2+C$
Where, Ki (mg/g min1/2) represents to the rate constant and C is the constant, which related to thickness of boundary layer. Maximum value of constant indicates the maximum effect of boundary layer. The value of Ki and C can be determined from the m and C of the plot of qt verses t1/2. It was investigated from the Fig. 4(a) and Table 2, at each concentration linear plot of fluoride does not go through the origin, hence the diffusion rate was not only responsible for adsorption of fluoride. That means other mechanism may also affect the fluoride removal on MPPAC-500.

### 3.6. Thermodynamics

Thermodynamic study describes the role of temperature in adsorption process. Parameters related thermodynamic such as change in entropy (ΔS), change in enthalpy (ΔS) and change in Gibb’s free energy (ΔG) for the adsorption of fluoride ions on adsorbent were determine by using following equation.
##### (15)
$Kc=qeCe$
##### (16)
$ln Kc=ΔSR-ΔHRT$
##### (17)
$ΔG=ΔH-TΔS$
Where, K is the thermodynamic distribution coefficient constant, While T and R denote temperature (K) and universal gas constant, respectively in the reaction. Ce and qe are fluoride ions in aqueous phase (mg/L) and adsorbed quantity of fluoride (mg/g) on MPPAC-500 at equilibrium. The changed entropy and changed enthalpy were obtained through the value of intercept and slope of the plot between lnKc and 1/T (Fig. 4(b)). From the Table 2, it is evidenced that the reaction was spontaneous in nature due to negative value of Gibb’s free energy. The positive value of enthalpy (ΔH) was 90.78 KJ/mol, it also stated that the fluoride uptake onto MPPAC-500 was endothermic in nature.

### 3.7. Column Experiment

#### 3.7.1. Effect of packed column bed length

The breakthrough curve was obtained at two different bed depth 5 cm and 10 cm at a fixed flow rate 5 mL/min and starting concentration 10 mg/L. The breakthrough curve and other parameter are depicted in Fig. 4(c) and interpreted values of breakthrough parameter are presented in Table 3. When the bed depth was increased from 5cm to 10cm, saturation time from 345 min to 420 min and treated effluent volume from 1,800 mL to 2,250 mL and the breakthrough time from 135 min to 285 min were increased, which due to available adsorption sites of the adsorbent were increased for uptake of fluoride ions.

#### 3.7.2. Effect of flow rate

To evaluate the effect of volumetric flow rate of inlet Fluoride ions on the capacity adsorption and breakthrough curve for MPPAC-500. The operational experiments were implemented at two different flow rates (5 mL/min and 10 mL/min), although experimental parameters were placed on concentration 10 mg/L and 10 cm bed height are constant. Obtained breakthrough curve shown in Fig. 4(d) and other breakthrough curve parameters such as breakthrough time, saturation time were appraised and treated volume was epitomized in Table 3. It is clear from the Fig. 4(d) that increasing the feed fluoride ions flow rate decreases the breakthrough time and exhaustion time, experimental results indicate when the flow rate increased which decreases breakthrough time and saturation time, because at high flow rate adsorbate have not enough time to contact with adsorbent bed in the column. The result show that higher breakthrough time, saturation time, treated volume was found at lower flow rate because at low flow rate solute has more residence time to pores of the adsorbent.

#### 3.7.3. Thomas model

Thomas model employed to predict column performance and breakthrough curves [41]. The model relied on with Langmuir kinetic of adsorption and desorption process [42, 43]. This model also describes constant separation factor which applied either favourable or unfavourable isotherms [41]. The linear Eq. (18) of Thomas model is given as below.
##### (18)
$ln(CoCt-1)=kTHqexQ-kTHCot$
Where, kTH (mL/min.mg) is the rate constant for Thomas; Q (mL/min) is flow rate of the solution; qe (mg/g) is the adsorption capacity and x (g) is the mass of adsorbent.
Fig. 4(e) shows the linear plot of ln[(Ci/Ct)-1] vs ‘t’. The values obtained from Thomas models were shown in Table 3. It can be stated that as increasing in flow rates from 5–10 mL min−1 at bed height of 5 cm, the values of KTH increased and qo decreased shown in Fig. 4(f). The reason of increased values of KTH and decreased value of qo is due to the driving force between adsorbate and adsorbent. Similar results were also observed at 10 cm bed height for Fluoride adsorption in the solution [44]. These results revealed that lower the flow rates with higher bed height leads to maximum fluoride uptake on to adsorbent.

This model assume that adsorption rate is proportional to the concentration of the adsorbate (fluoride) and residual capacity of the adsorbent. The equation of Adams-Bohart model in linear form is given in Eq. (19).
##### (19)
$ln(CoCt-1)=kABNozμo-kABCot$
Where, No is the maximum uptake capacity of the adsorbent per unit volume in the column; kAB is the rate constant for Adams-Bohart and μo is the linear flow rate of the influent solution (cm/min).

#### 3.7.5. Yoon-Nelson model

Yoon- Nelson model stipulates the adsorption kinetic of a fixed bed. This model assumes that the rate decrease in the probability of adsorption for particular sorbate molecule is proportional to the probability of sorbate breakthrough and the probability of the sorbate (fluoride) adsorption. The linearized equation is presented as below.
##### (20)
$ln(CtCo-Ct)=kYNt-KYNτ$
Where kYN (1/min) is the rate constant for Yoon-Nelson and τ(min) is taken for 50% adsorbate breakthrough.

#### 3.7.6. Clark model

This model introduced by Clark for fixed bed column adsorption. The mass transfer can be evaluated with the help of Freundlich parameter (n). Clark equation is presented as below.
##### (21)
$ln [(CoCt)n-1-1]=ln A-rt$
Where, A and r (1/h) is the Clark constant, and n is the parameter of Freundlich. The value of these constant can obtain from the intercept and slope of the graph plotted ln[C0/Ct)n−1-1] vs time (min).
The parameters of Adams-Bohart, Yoon-Nelson and Clark models are given in supplementary data (Table S2, S3 and S4, respectively) and plot of these model is shown in Fig. S5, S6 and S7, respectively. Adams-Bohart model revealed that the rate constant increased as the bed depth of adsorbent increases. While, adsorption capacity (No) decreased. The rate constant (KYN) of Yoon-Nelson and (time needed for 50% adsorbate breakthrough decreased with increasing flow rate, while increased with increasing adsorbent dose. The Clark parameter (r) increased with increasing flow rate.

### 5. Conclusions

This work explored the applicability of waste pea peel as novel adsorbent for the elimination of fluoride ions from aqueous medium. The pea peel was modified with iron (FeCl3) for enhancing its adsorption efficiency. Surface characterization explored the existence of heterogeneous structure, showing small and large pore, provides exchangeable sites for adsorption. Infrared Spectroscopy results demonstrated the role of hydroxyl, methyl and ketone groups were involved in the removal of fluoride. In batch study, the maximum fluoride removal 99% was found at 5 mgL−1 initial fluoride concentration at neutral pH, 4 gm/L dose, and 420 min contact time. Adsorption study favoured Freundlich isotherm model. Fluoride uptake was well described with pseudo-second-order kinetic model. Thermodynamic parameters indicated that the adsorption process was followed endothermic and spontaneous in nature. The column study results were fitted well with Adams-Bohart model followed by Thomas model. The maximum removal capacity at column system was found 3.75 mg/g. The present adsorbent shows the efficient adsorptive capacity for uptake of fluoride in batch and column systems, which may suggest for remediation of fluoride from groundwater and contaminated surface water by biomass based biosorbent is highly recommended.

### Acknowledgment

The research work was financially supported by Department of Science and Technology (DST) Science and Engineering Research Board (SERB), India. (Reference no. ECR/2016/001924).

### Notes

Author Contributions

Ms. N.S. (Project Assistant), has conducted all the experiments and writing work, Mr. C.B. (Ph.D) has drafted the manuscript and Dr. J.S. (Assistant Professor) has supervised the work and corrected the manuscript.

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##### Fig. 1
Experimental fixed bed column setup.
##### Fig. 2
SEM and EDS of MPPAC500 (a) before and (b) after adsorption, (c) FTIR spectra of MPPAC500, (d) XRD analysis, (e) Zeta Potential and (f) Particle Size Analysis of MPPAC500.
##### Fig. 3
(a) Effect of adsorbent dose, (b) effect of pH on adsorption, (c) effect of initial concentration of fluoride, (d) effect of temperature on adsorption, (e) Plot of Freundlich adsorption isotherm for MPPAC500 and (f) Plots of pseudo second order kinetic.
##### Fig. 4
(a) Plot of Intraparticle Diffusion Model, (b) Plot of Thermodynamic, (c) Breakthrough curve for effect of bed height on MPPAC-500, (d) effect of flow rate on MPPAC-500, (e) Thomas model of MPPAC-500 at 5 cm bed height, (f) Thomas model of MPPAC-500 at 10 cm bed height.
##### Fig. 5
Possible adsorption mechanism of fluoride on MPPAC-500.
##### Table 1
Parameters of Langmuir, Freundlich and Temkin Isotherm for the Fluoride Ions Removal on to AC-500
 Langmuir Isotherm Qo(mg/g) 4.717 B(L/mg) 3.355 RL 0.057 R2 0.979 Freundlich Isotherm KF (mg/g(L/mg)1/n) 3.206 N 3.345 R2 0.995 Temkin Isotherm KT 29.254 B 0.715 R2 0.945
##### Table 2
Values of Intraparticle Diffusion Model and Thermodynamic
Intraparticle diffusion model Thermodynamic

C0 (mg/L) Ki (mg/g.min1/2) C(mg−1) R2 parameters Temperature (K)
5 0.013 1.004 0.76 ΔG°(KJ/mol)
10 0.032 1.818 0.71 298 933338.5
15 0.054 2.415 0.955 308 −96473.7
20 0.104 2.399 0.965 318 −99608.9
328 −102744
ΔH (J/mol) 90.7889

ΔS (J/mol K) 313.5209
##### Table 3
Values of different fixed bed and Thomas model parameters
Column parameters Thomas parameters

Co (mg L−1) Z (cm) Q (mL min−1) ttotal (min) mtotal (mg) qtotal (mg) qeq (mg/g) Veff (mL) KTH (mL min−1 mg−1) Qo (mg g−1) R2
10 5 5 360 18 17.9982 3.7497 1800 0.00354 3083.21 0.9588
10 5 10 105 10.5 10.4998 2.1878 1050 0.01168 1931.37 0.951
10 10 5 450 22.5 22.4998 2.3438 2250 0.0033 2327.7 0.9043
10 10 10 195 19.5 19.4998 2.0313 1950 0.0079 1829.38 0.8224
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