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# In each term of a sequence, 9 is added to get the next term. If the

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Magoosh GMAT Instructor
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In each term of a sequence, 9 is added to get the next term. If the [#permalink]

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20 Mar 2013, 14:18
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In each term of a sequence, 9 is added to get the next term. If the first term is 2, what is the eighty-first term?

(A) 632
(B) 695
(C) 713
(D) 722
(E) 731

For more on sequences, as well as a complete explanation of this question, see:
http://magoosh.com/gmat/2012/sequences-on-the-gmat/

Mike
[Reveal] Spoiler: OA

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Mike McGarry
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Kudos [?]: 144 [3] , given: 6

Re: In each term of a sequence, 9 is added to get the next term. If the [#permalink]

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20 Mar 2013, 14:35
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1rst term + 80 terms = 2 + 9+9+9+9+9+9+9+...+9 (80 times)

2 + (9 x 80) = 2 + 720 = 722

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20 Mar 2013, 22:10
mikemcgarry wrote:
In each term of a sequence, 9 is added to get the next term. If the first term is 2, what is the eighty-first term?
(A) 632
(B) 695
(C) 713
(D) 722
(E) 731

For more on sequences, as well as a complete explanation of this question, see:
http://magoosh.com/gmat/2012/sequences-on-the-gmat/

Mike

Quick tip on this type of question: the GMAT is frequently counting on you (pun intended) to miscount and think the answer is off by one iteration in either direction, so the correct answer tends to not be the biggest or smallest you find. This is far from ironclad, but if you find yourself picking the biggest number (in this case E), you've probably fallen for a classic GMAT trap.
The solution outlined by johnwesley above is excellent, but a lot of students find themselves doing 81*9+2 = 731 and falling into the trap.

Hope this helps!
-Ron
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Re: In each term of a sequence, 9 is added to get the next term. If the [#permalink]

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22 Mar 2013, 13:23
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Xo = 2
X1 = Xo + 9 = 2 + 9
X2 = X1 + 9 = (2 + 9) + 9
X3 = X2 + 9 = ((2+ 9) +9) +9

Xn = 2 + 9n
X80 = 2 + 9(80)

X80 = 722
D
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Re: In each term of a sequence, 9 is added to get the next term. If the [#permalink]

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26 Jul 2013, 06:52
There is formula where you can find the nth term.

a + (n-1)d

a = first term = 2
d = common difference = 9

so for 81th term

2 + (81-1)9 = 722
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Re: In each term of a sequence, 9 is added to get the next term. If the [#permalink]

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22 Feb 2015, 19:13
# of terms =[ (last term - first term)/interval] + 1
i.e. [(81-2)/1] + 1 = 80

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Re: In each term of a sequence, 9 is added to get the next term. If the [#permalink]

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28 Feb 2015, 11:49
mikemcgarry wrote:
In each term of a sequence, 9 is added to get the next term. If the first term is 2, what is the eighty-first term?

(A) 632
(B) 695
(C) 713
(D) 722
(E) 731

For more on sequences, as well as a complete explanation of this question, see:
http://magoosh.com/gmat/2012/sequences-on-the-gmat/

Mike

First term = 2
common difference = 9
Nth term = First term + (N-1)(common difference)
So, 81th term = 2 + (81-1)(9)
= 2 + 80*9
= 2 + 720
= 722
Hence option D.

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Re: In each term of a sequence, 9 is added to get the next term. If the [#permalink]

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27 Feb 2017, 05:29
First term is 2
Second term is 2 + 9 = 2 +(2-1)9
Third term is 2 +9 +9 = 2 +(3-1)9
.
.
.
.
81st term is 2 +(81-1)*9 = 722. Option D
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01 Mar 2017, 18:21
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mikemcgarry wrote:
In each term of a sequence, 9 is added to get the next term. If the first term is 2, what is the eighty-first term?

(A) 632
(B) 695
(C) 713
(D) 722
(E) 731

We are given a sequence in which 9 is added to each term and the first term is 2. Let’s list the first few terms:

term 1 = 2 (notice that 2 = 9(0) + 2)

term 2 = 11 (notice that 11 = 9(1) + 2)

term 3 = 20 (notice that 20 = 9(2) + 2)

Thus, term n = 9(n-1) + 2

So the 81st term is 9(80) + 2 = 722

Alternate Solution:

You might recognize this as an arithmetic sequence, with first term 2 and common difference d = 9.
To determine the nth term of an arithmetic sequence, we can use the formula a_n = a_1 + (n – 1) * d. We can thus determine the 81st term as follows:

a_81 = 2 + (81 – 1)(9)

a_81 = 2 + (80)(9)

a_81 = 722

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Re: In each term of a sequence, 9 is added to get the next term. If the   [#permalink] 01 Mar 2017, 18:21
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# American Institute of Mathematical Sciences

October  2021, 26(10): 5337-5354. doi: 10.3934/dcdsb.2020346

## Qualitative analysis of a generalized Nosé-Hoover oscillator

 1 School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou, Henan 450046, China 2 School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China 3 Hubei Key Laboratory of Engineering Modeling and Science Computing, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China

* Corresponding author: Xiao-Song Yang

Received  November 2019 Revised  July 2020 Published  October 2021 Early access  November 2020

Fund Project: The first author is supported by NSFC grant 51979116

In this paper, we analyze the qualitative dynamics of a generalized Nosé-Hoover oscillator with two parameters varying in certain scope. We show that if a solution of this oscillator will not tend to the invariant manifold $\{(x,y,z)\in \mathbb R^3|x = 0,y = 0\}$, it must pass through the plane $z = 0$ infinite times. Especially, every invariant set of this oscillator must have intersection with the plane $z = 0$. In addition, we show that if a solution is quasiperiodic, it must pass through at least five quadrants of $\mathbb R^3$.

Citation: Qianqian Han, Xiao-Song Yang. Qualitative analysis of a generalized Nosé-Hoover oscillator. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5337-5354. doi: 10.3934/dcdsb.2020346
##### References:

show all references

##### References:
The grid is part of $S_{1}$ and the shadow is part of $S_{2}$
The shadow is the projection of the region $I$ on the plane $z = 0$
$A_{1}\rightarrow A_{2}$ means there are solutions from $A_{1}$ to $A_{2}$, $B_{1}\dashrightarrow A_{2}$ means there are solutions from $B_{1}$ to $A_{2}$ and these solutions have intersection with $X$-axis or $Y$-axis
From right to left are $l_{10}$ and $l_{20}$
From right to left are $l_{01}$, $l_{02}$, $l_{03}$ and $l_{04}$
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## Elements having octets

$FC=V-(L+\frac{S}{2})$

Matt Sanruk 2H
Posts: 131
Joined: Wed Sep 18, 2019 12:21 am

### Elements having octets

I am still slightly confused on how some elements can over an octet, while we also automatically assume some elements have octets no matter what. Can someone clarify for me?

Jessica Booth 2F
Posts: 101
Joined: Fri Aug 30, 2019 12:18 am

### Re: Elements having octets

C,N,O,F always form an octect and anything with an atomic number above F has a d-shell, so they can form expanded octects.

Chem_Mod
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Has upvoted: 435 times

### Re: Elements having octets

The most common exception to the octet rule is a molecule or ion with with at least one atom that possesses more than an octet of electrons. These compounds are found for elements of period 3 and beyond. For example, sulfur can accommodate more than eight valence electrons by using one or more d orbitals. Sulfur and oxygen are the most common elements seen to exceed the octet rule and have an expanded valence shell.

Sarah Nichols 4C
Posts: 52
Joined: Sat Aug 24, 2019 12:18 am

### Re: Elements having octets

The octet (8 e-) comes from the full s- and p- subshells; s2p6 totals 8 electrons. Atoms in period 3 or higher have valence shell d-orbitals, so they can accommodate more electrons than the standard 8 for atoms with lower atomic numbers

Matt Sanruk 2H
Posts: 131
Joined: Wed Sep 18, 2019 12:21 am

### Re: Elements having octets

Okay, so CNOF are always octets since they do not have a d orbital, but those that have d orbitals (periods 3+) can have more if their formal charge allows for it?


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# Casella and Berger 2002 Statistical Inference p71

I wonder if this is a mistake on P71 of Casella and Berger (2002), Statistical Inference. When they define the function $g(x,\lambda)$, they have $$g(x,\lambda) = \frac{x^n e^{-x/(\lambda+\delta_{0})}}{(\lambda-\delta_{0})^2} \left(\frac{x}{\lambda-\delta_{0}}+1 \right).$$ But, shouldn't the power of the exponet be $-x/(\lambda-\delta_{0})$ instead of having a plus sign in front of $\delta_{0}$? This function $g$ is designed to be larger than $$\frac{x^ne^{-x/\lambda}}{\lambda^2} \left( \frac{x}{\lambda}+1 \right).$$ For some constant $\delta_{0}$ satifying $0<\delta_{0}<\lambda$ and $x>0$, $$\frac{x}{\lambda}+1\leq \frac{x}{\lambda-\delta_{0}}+1$$ and $$\frac{x^n}{\lambda^2}\leq \frac{x^n }{(\lambda-\delta_{0})^2}$$ but $$e^{-x/\lambda} \geq e^{-x/(\lambda+\delta_{0})}.$$

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2019
Том 71
№ 11

# Vitrychenko I. E.

Articles: 11
Brief Communications (Ukrainian)

### On a special critical case of stability of a nonautonomous essentially nonlinear system

Ukr. Mat. Zh. - 2005. - 57, № 12. - pp. 1711–1718

We obtain sufficient conditions for the Lyapunov stability of the trivial solution of a nonautonomous essentially nonlinear differential system in a special critical case.

Brief Communications (Ukrainian)

### Critical Cases of the π-Stability of a Nonautonomous Quasilinear Equation of the nth Order

Ukr. Mat. Zh. - 2004. - 56, № 2. - pp. 264-270

We establish sufficient conditions for the π-stability of the trivial solution of a quasilinear equation of the nth order.

Article (Ukrainian)

### Global λ-Stability of One Nonautonomous Quasilinear Second-Order Equation

Ukr. Mat. Zh. - 2002. - 54, № 9. - pp. 1172-1189

We establish sufficient conditions for the λ-stability of the trivial solution of one quasilinear differential equation of the second order.

Article (Russian)

### On the functional polystability of certain essentially nonlinear nonautonomous differential systems

Ukr. Mat. Zh. - 2000. - 52, № 2. - pp. 197-207

For essentially nonlinear differential systems with the limit matrix of coefficients of the first-approximation system, we establish sufficient conditions for functional polystability, which generalizes the notion of exponential polystability.

Article (Russian)

### A critical case of stability of one quasilinear difference equation of the second order

Ukr. Mat. Zh. - 1999. - 51, № 12. - pp. 1593–1603

We obtain sufficient conditions for the Perron stability of the trivial solution of a real difference equation of the form $$y_{n + 1} - 2\lambda _n y_n + y_{n - 1} = F(n,y_n ,\Delta y_{n - 1} ), n \in N$$ where $y_n \in \left] { - 1,1} \right[,\left| {F(n,y_n ,\Delta y_{n - 1} )} \right| \le L_n \left( {\left| {y_n \left| + \right|\Delta y_{n - 1} } \right|} \right)^{1 + \alpha } ,L_n \ge 0$ and $\alpha \in \left] {0, + \infty } \right[$ . The resuits obtained are valid for the case where $\left| {\lambda _n } \right| = 1 + o(1), n \to + \infty$ .

Brief Communications (Russian)

### Functional polystability of some nonautonomous quasilinear differential systems

Ukr. Mat. Zh. - 1999. - 51, № 7. - pp. 989–995

For quasilinear differential systems with a boundary matrix of coefficients of the system of the first approximation, we obtain sufficient conditions of functional polystability, which generalizes the notion of exponential polystability.

Brief Communications (Russian)

### On the instability of one nonautonomous essentially nonlinear equation of thenth order

Ukr. Mat. Zh. - 1999. - 51, № 6. - pp. 835–841

We establish sufficient conditions for the Lyapunov instability of the trivial solution of a nonautonomous equation of thenth order in the case where its limit characteristic equation has a multiple zero root. The instability is determined by nonlinear terms.

Brief Communications (Ukrainian)

### Critical cases of stability of one nonautonomous essentially nonlinear equation of the nth order

Ukr. Mat. Zh. - 1997. - 49, № 5. - pp. 720–724

We establish sufficient conditions of the Lyapunov stability of the trivial solution of a nonautonomous ordinary differential equation of the nth order in the case where its characteristic equation has a multiple zero root. The stability is determined by nonlinear terms.

Brief Communications (Russian)

### On stability of annth-order equation in a critical case

Ukr. Mat. Zh. - 1995. - 47, № 8. - pp. 1138–1143

We obtain sufficient conditions for the Lyapunov stability of the trivial solution of a nonautonomousnth-order equation in the case where the root of the boundary characteristic equation is equal to zero and has multiplicity greater than one.

Brief Communications (Russian)

### On stability of the trivial solution of a nonautonomous quasilinear system whose characteristic equation has multiple roots

Ukr. Mat. Zh. - 1994. - 46, № 8. - pp. 1072–1079

For $t \uparrow \omega, \quad \omega \leq +\infty$, we obtain sufficient conditions for Lyapunov stability of the zero solution of a specific nonautonomous quasilinear differential system in the case where the matrix of the first-degree approximation has the Jordan form with triangular blocks. Methods to reduce certain classes of general differential systems to differential systems of special type are given.

Article (Ukrainian)

### On oscillation of solutions of a nonautonomous quasilinear second-order equation

Ukr. Mat. Zh. - 1994. - 46, № 4. - pp. 347–356

Sufficient conditions are obtained for the initial values of nontrivial oscillating (for $t = ω$) solutions of the nonautonomous quasilinear equation $y'' \pm \lambda (t)y = F(t,y,y'),$, where $t ∈ Δ = [a, ω[,-∞ < a < ω ≤ + ∞, λ(t) > 0, λ(t) ∈ C_Δ^{(1)},$ $|F((t,x,y))| ≤ L(t)(|x|+|y|)^{1+α}, L(t) ≥ -0, α ∈ [0,+∞[,$ $F: Δ × R^2 →R, F ∈ C_{Δ × R^2}, R$ is the set of real numbers, and $R^2$ is the two-dimensional real Euclidean space.


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# Category:Search Convenience

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QUESTION

# Evolution and Taxonomy and Innovation and Diversit

Evolution and Taxonomy and Innovation and Diversit

Question 1: Evolution and Taxonomy

Select one species of organism that is interesting to you (try to pick something different from your classmates’ choices—you only have about two million to choose from). Show the hierarchy of classification for the organism you choose from Domain through to binomial name (genus and species). Be sure to write the names in correct format and to spell correctly.

What are the nearest relatives of your chosen species? How do the levels of classification for your species trace the major evolutionary steps the ancestors of your species took over time? Be prepared to discuss similarities and differences in the evolution of diversity in the examples chosen by your classmates.

Question 2: Innovation and Diversity

Over the long period of time that life has existed on Earth, there have been a number of important or significant innovations including (but not limited to) endosymbiosis to create mitochondria and chloroplasts; multicellularity; adaptation to land by plants and animals; development of exoskeletons in arthropods, shells in molluscs, and notochords followed by vertebral columns in chordates and vertebrates; and bipedalism in the ancestry of humans. All of these had to come about by natural selection in response to changing environmental forces.

Pick one of these significant innovations and describe:

How the innovation appears to have happened What environmental challenges were met and overcome by this innovation, and What opportunities were opened for the organism that made this innovation.

Be prepared to discuss the general concept of innovation and opportunity in response to selective challenges in life and how this can greatly increase biodiversity over time.

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Monthly Archives: July 2011

From the Exchange

I speak of Math Stackexchange frequently for two reasons: because it is fantastically interesting and because I waste inordinate amounts of time on it. But I would like to again share some of the more interesting things from the exchange here.

Prime rich and prime poor

A short excursion –

The well-known Euler’s Polynomial $x^2 – x + 41$ generates 40 primes at the first 40 natural numbers. It is sometimes called a prime-rich polynomial. There are many such polynomials, and although Euler’s Polynomial is perhaps the best-known, it is not the best. The best that I have heard of is $(x^5 – 133 c^4 + 6729 x^3 – 158379 x^2 + 1720294x – 6823316)/4$, which generates 57 primes. But this morning, I was reading an article on Ulam’s Spiral when I heard of the opposite – a prime-poor polynomial. The polynomial $x^{12} + 488669$ doesn’t produce a prime until $x = 616980$. Who knew?

And to give them credit, that prime-rich polynomial was first discovered by Jaroslaw Wroblewski & Jean-Charles Meyrignac in one of Al Zimmerman’s Programming Contests (before being found by a few other teams too).

Giving Journals

Firstly, I wanted to note that keeping a frequently-updated blog is hard. It has its own set of challenges that need to be overcome. Bit by bit.

But today, I talk about a sort of funny experience. Suppose for a moment that you had acquired a set of low-level math journals throughout the undergrad days, journals like the College Mathematics Journal, Mathematics Magazine, etc. Presuming that you didn’t want to keep them in graduate school (I don’t – they’re heavy and I have online access), what would you do with them?

Posted in Humor, Mathematics, Story | | 1 Comment

Back and Forum-ing: Beating a dead horse

I’m back! Croatia, Greece, Turkey… all behind me. In the meantime, I’ve fallen even more in love with math.stackexchange and have ended up as a temporary moderator for philsophy.stackexchange (check them out). To announce my return, a little fun:

from Loers Hey everyone thanx for the amazing effort that u provide us with , over here . just gotta a simple questions why cannot we differentiate |x|when x = 0 ? or let’s say |x+2| when x = -2 this is really annoying me I cannot see a proper reason for it thanx again

We strive to develop our humor. The Chaz (quite the internet sensation, if you haven’t run across him) writes: (more…)


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# Number of ways to break $n$ into $\lceil n/k\rceil$ positive integers each at most $k$

Basically we want to write $n$ as the sum of positive integers not greater than $k$, and we want to minimize number of terms. So there are $\lceil n/k\rceil$ terms. How many such sequences are there?

Two sequences are distinct if they contain different numbers at some position. That is we're counting ordered sequences.

• Are you counting, for example, $7=3+2+2$ and $7=2+3+2$ as two distinct ways? – Mark Fischler May 6 '17 at 14:26
• Yes. Added to the question. – Artur May 6 '17 at 14:42

## 2 Answers

Let $\bigl\lceil{n\over k}\bigr\rceil=:r$. Then $r-1<{n\over k}\leq r$, or $$0\leq rk-n<k\ .$$ Let $$x_i=k-p_i\>, \quad 0\leq p_i<k\qquad(1\leq i\leq r)$$ be the sizes of the $r$ parts. Then $$n=\sum_{i=1}^r x_i=rk-\sum_{i=1}^r p_i$$ and therefore $$\sum_{i=1}^r p_i=rk-n\in[0,k[\ .\tag{1}$$ This means that we have to count the number of nonnegative solutions $(p_1,\ldots, p_r)$ of $(1)$ satisfying $p_i<k$ for all $i$. This latter condition will be automatically fulfilled since $rk-n<k$. "Stars and bars" then produces the following number $N$ of solutions: $$N={rk-n+r-1\choose r-1}\ .$$

Adopting @ChristianBlatter's terminology, we define

\begin{align} r &= \lceil {n \over k} \rceil \\ h &= rk - n \\ \end{align}

• $r$ is the number of partitions we divide $n$ into; and
• $h$ is the number of holes (shortfalls) we have to distribute amongst the $r$ partitions.

Using a Stars and Bars argument, We have $r - 1$ divisions to put among the $h$ holes. So the number of arrangements is

$${h + r - 1 \choose h}$$

This result is equivalent to @ChristianBlatter's.


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# An original sample of a radioisotope had a mass of 10 grams. After 2 days, 5 grams of the radioisotope remains unchanged. What is the half life of the radioisotope?

$2$ days
Sample was initially $\text{10 g}$. After $2$ days it’s decreased to $\text{5 g}$. Therefore, half life of sample is $2$ days.


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Environ Eng Res > Volume 23(4); 2018 > Article
Mohanadhas and Govindarajan: Modeling the sensitivity of hydrogeological parameters associated with leaching of uranium transport in an unsaturated porous medium

### Abstract

The uranium ore residues from the legacies of past uranium mining and milling activities that resulted from the less stringent environmental standards along with the uranium residues from the existing nuclear power plants continue to be a cause of concern as the final uranium residues are not made safe from radiological and general safety point of view. The deposition of uranium in ponds increases the risk of groundwater getting contaminated as these residues essentially leach through the upper unsaturated geological formation. In this context, a numerical model has been developed in order to forecast the 238U and its progenies concentration in an unsaturated soil. The developed numerical model is implemented in a hypothetical uranium tailing pond consisting of sandy soil and silty soil types. The numerical results show that the 238U and its progenies are migrating up to the depth of 90 m and 800 m after 10 y in silty and sandy soil, respectively. Essentially, silt may reduce the risk of contamination in the groundwater for longer time span and at the deeper depths. In general, a coupled effect of sorption and hydro-geological parameters (soil type, moisture context and hydraulic conductivity) decides the resultant uranium transport in subsurface environment.

### 1. Introduction

Modeling the fate and transport of radioactive wastes has been known to be a valuable approach in accessing the radiological impact assessment of uranium tailing ponds in sub-soil system. The long-term impacts on release of uranium from tailing pond have been studied by many researchers using modeling approach [1418]. Groundwater contamination due to leaching of uranium and its progenies from tailing ponds is a major concern. This may lead to unfit the groundwater for human consumption. The naturally occurring U, Th, Ra and Rn nuclides transport in groundwater are mostly controlled by physicochemical processes of weathering, decay and sorption at the water-rock interface [19, 20]. Rotter et al. [21] developed a biogeochemical reactive transport model for uranium immobilization in both single and dual-porosity porous media to study the bioremediation efficiency on uranium. Recently, a decay-chain transport of uranium in a heterogeneous anisotropic saturated porous medium is performed [22]. This study identified the necessity of monitoring the short-lived progeny radionuclides apart from their long-lived parents in the groundwater in the vicinity of uranium tailing ponds.
It is observed that many modeling studies are performed on transport of uranium in the saturated porous medium. The modeling studies to investigate the transport of radionuclide in unsaturated condition are very limited. Qian et al. [23] simulated the vertical transport of Sr90 in the unsaturated Chinese loess under artificial rain conditions. Merk [24] studied the radionuclide transport from the concrete rubble when the radionuclide dissolves in the infiltrated water using HYDRUS-1D. Sanchez and Thorne [25] presented a mathematical model to study the behavior of U238 series radionuclide entering soils in solution and their uptake by plants along with the decay chain of uranium in soil-plant system under different hydrological regimes. Nair et al. [26] developed a multi-compartmental source term model to assess the radionuclide leaching in a saturated or unsaturated porous medium such as a uranium tailing ponds.
Vadose zone thickness can extend several meters and the soil hydraulic parameters are nonlinear [2731]. Few studies indicated the vadose zone thickness can extend up to 50 m [32]. The flow and transport of radionuclide is caused by unsaturated matrix permeability, porosity and sorption coefficient [33]. Further, experimental and modeling study confirms that the radionuclide migration in unsaturated zone is highly dependent on unsaturated van Genuchten parameter [34]. The modeling as well as experimental studies in field scale also discussed earlier on the migration of radionuclide in large vadose zone thickness [3537]. Moreover, the transport behavior of uranium from the tailing pond has extremely affected under various soil types such as sand, silt, clay, sediments and sandstone [19, 20, 23, 24].
The mill tailings are expected to include all the radionuclide in the 238U decay series after the uranium extraction from ore [22]. The 238U decay chain contains four long-lived radionuclide (238U, 234U, 230Th, 226Ra); five short-lived nuclides (234Th, 222Rn, 210Pb, 210Bi, 210Po); and five very short-lived nuclides (234mPa, 218Po, 214Pb, 214Bi, 214Po). On the other hand, 222Rn can escape from the ore and other short-lived progenies may decay to innocuous levels during the ore extraction processes. Thus, primarily uranium tailing ponds consists of long-lived radionuclide of 238U, 234U, 230Th, 226Ra. Subsequently, ingrowths of progenies will happen in the tailing ponds as well as in the percolated water.
Many modeling studies have been investigated on the migration of various radionuclides in the unsaturated porous medium. However, all long-lived, short-lived and very short-lived 238U nuclide transport in unsaturated porous medium has not been analyzed extensively. Since, the uranium tailing ponds are located in the soil surface a suitable numerical model has to be developed for vertical transport of multi-species radionuclide incorporating the nonlinear soil hydraulic parameters under unsaturated condition. Hence, the motivation of the present work is to develop a one-dimensional decay chain transport model to predict the concentration of radionuclide from uranium tailing pond due to vertical leaching under unsaturated condition. Further, the developed model is extended to evaluate the transport of radionuclide in different soils which are generally available beneath the uranium tailing pond.

### 2. Physical System and Governing Equations

Uranium tailings ponds are generally located in unsaturated zones [26]. The vertical transport by soil water, of radionuclide from uranium tailings ponds, is of major concern in assessing the vulnerability of groundwater contamination. Fig. 1 illustrates the conceptual understanding of uranium transport in an unsaturated zone.
The equation of mass conservation can be coupled with the Darcy’s law to produce the Richard’s equation which governs the water flow in an unsaturated porous medium [3840] that is used for this study is given in Eq. (1):
##### (1)
$C(h)∂h∂t=∂∂zK(∂h∂z)-∂K∂z$
Where: $C(h)=∂θw∂h$ is specific moisture capacity (1/cm); θw is the volumetric water content (cm3/cm3); h is the pressure head (cm); K is the unsaturated hydraulic conductivity (cm/s); t is the time (s); z is the vertical coordinate (cm) positive downward.
In unsaturated porous medium, the part of pore space volume is filled by air. Hence, the moisture content θw is generally smaller than the porosity. The water pathway modeling in unsaturated porous system is considerably complicated as compared with saturated porous system. The water content (θw), pressure head (h) and hydraulic conductivity (K) are highly correlated in the unsaturated case. The hydraulic conductivity in unsaturated porous medium is not only depends on the matrix material but also on the local pressure head [24]. Widely acknowledged empirical relationships were arrived by van Genuchten [41] and described in Eq. (2) and (3).
##### (2)
$K(h)=Ks[θw-θrθs-θr]1/2 [1-(1-[θw-θrθs-θr]1/η)η]2$
##### (3)
$θw=θr+(θs-θr) [1+(α|h|)β]-η$
where η is an empirical constant affecting the shape of the retention curve, θs is the saturated water content, θr is the and residual water content, α and β are both empirical constants.
The decay chain transport model for the parent radionuclide for a one-dimensional and homogeneous unsaturated porous medium can be represented as [22] expressed in Eq. (4):
##### (4)
$R1∂N1∂t=∂∂z(D∂N1∂z)-v∂N1∂z-R1λ1N1$
where N1 is the concentration of the parent radionuclide in the aqueous phase (atoms L−3 or mol); D is the hydrodynamic dispersion coefficient (L2T−1) (D = αL×v); αL is the longitudinal dispersivity (L) can be produced by different pore sizes along the flow paths and/or by tortuosity; v is the pore water velocity (L/T) can be obtained by dividing the Darcy velocity by the effective porosity (which is expressed as v = q/θw); q is obtained from Darcy’s law $q=-K[∂(h-z)∂z]$; λ1 is the radioactive decay constant (T−1).
Similarly, the decay chain transport equation for the ith member of the decay chain can be written [22] as given in Eq. (5):
##### (5)
$Ri∂Ni∂t=∂∂z(D∂Ni∂z)-v∂Ni∂z-RiλiNi+Ri-1λi-1Ni-1$
where i > 1 to M; M is the number of total nuclides involved in the decay chain. The ingrowth of the progenies from the previous parent radionuclide is included in the last term of Eq. (5). The sorption of radionuclides is described by linear and reversible. Hence, the retardation factor (Ri) of the radionuclide ‘i’, which can be expressed in Eq. (6):
##### (6)
$Ri=1+kdi ρbθw$
where kdi is the distribution coefficient of radionuclide ‘i’ (L3M−1); ρb is the bulk density of the porous material (M L−3). Fig. 2 illustrates the schematic representation of solving water flow and transport of radionuclides in an unsaturated porous medium.

### 2.1. Initial and Boundary Conditions

Initially the unsaturated soil is assumed as dry condition. An average infiltration rate of 10, 3, 0.75 m/y was taken as top boundary for sand and silt soil with zero flux is used as the bottom boundary condition for water flow model. At the beginning of the numerical experiment, the unsaturated soil is assumed to be free from contamination, and subsequently the initial values of all radionuclides concentration are assumed to be zero. The concentration of 238U and 234U in the tailing pond is 0.129 Bq/L [2] considered as top boundary and all other radionuclides concentration in the tailing pond is assumed as zero. The concentration gradient for all radionuclide is assumed to be at zero for the bottom boundary condition. The graphical representation of boundary conditions for one-dimensional domain is shown in Fig. 3.

### 2.2. Numerical Scheme

In this present study, the transport of radionuclides from uranium tailing pond along with water flow have been described by a set of partial differential equations. The water flow equation has been solved by implicit finite difference scheme with explicit linearization of K(h) and C(h) to solve Eq. (1) described by Van Dam and Feddes [42].
##### (7)
$hnj+1-hnjΔt=kn-(1/2)j(Δhn-(1/2)j+1Δz+1)-kn+(1/2)j(Δhn+(1/2)j+1Δz+1)cnjΔz$
where subscript ‘n’ is the node number and superscript ‘j’ is the time level.
Further, the transport Eq. (4) and (5) were solved by implicit finite difference method. The value of cell width in the vertical direction (Δz) is assumed to be 0.1 m and the time interval (Δt) is assumed to be 500 s in order to meet the numerical stability.

### 2.3. Verification of the Model

A numerical model for radionuclide transport from uranium tailings pond in the unsaturated soil system has been developed. The developed model is verified against well-known analytical solution before applying to the uranium transport from uranium tailing pond. The developed decay chain transport model is validated by comparing the analytical solution proposed by Bauer et al. [43] by considering four species and its corresponding retardation factors. The results obtained for this verification exercise is presented in Fig. 4. The model verification results show that the present model results are found to be good matching with that of Bauer et al. [43].

### 3. Results and Discussion

The developed model is applied to a hypothetical uranium tailing pond. The soil properties used for this study are shown in Table 1. The radionuclides used in the decay chain transport model and their properties are given in Table 2. Further, the developed model is extended to various soil types and the corresponding soil hydraulic properties are presented in Table 3.
Fig. 5 provides the spatial variation of water content and the corresponding concentration profile of 238U after 1, 10 and 50 d in the unsaturated subsurface system in the presence and absence of sorption process. It is observed that the volumetric water content profile abruptly reduced from 0.207 (cm3/cm3) to nearly 0.1 (cm3/cm3) within the shallow depth (up to 50 cm) in one day. However, the water content profile remains constant up to 350 cm and 1,450 cm for 10 d and 50 d, respectively, i.e., the unsaturated system up to these depths continuously receives water and further reduces to the low water content (0.1 cm3/cm3). Further, it is observed from Fig. 5 that the movement of 238U seems similar to the profile of water content after one day in the absence of sorption process. The concentration of 238U starts at 0.129 Bq/L from the soil surface (uranium tailing pond) and is reduced to zero at 40 cm depth after 1 d. Conversely, significant delay is observed in 238U concentration profile (250 cm depth after 10 d and 900 cm depth after 50 d) as compared with the corresponding water content profile (350 cm depth after 10 d and 1,450 cm depth after 50 d). Additionally, the results of simulation indicate that there is strong delay in movement of 238U in the presence of sorption. This is because of the distribution capacity of 238U on soil particles (5.00 × 10−4 L/mg) which hold huge amount of 238U concentration as solid phase, which retard the movement 238U in the aqueous phase. Due to the sorption process the concentration of 238U becomes zero form 0.129 Bq/L from its source at 1.5, 2.5 and 5 cm depth after 1, 10 and 50 d, respectively. A closer observation between the presence and absence of sorption process implies that there is approximately 99% reduction in depth of penetration on 238U concentration with the sorption process. It can be concluded from Fig. 5 that the water content variation in the unsaturated soil is directly affecting the movement of 238U concentration. The presence of sorption leads to retard the movement of 238U concentration, which actually aids to minimize the 238U concentration in the deeper zone.
Fig. 6 shows the evolution of the water content for different infiltration velocity and its associated 238U transport in unsaturated soil in the presence and absence of sorption process for 10 d. The water content starts at 0.17 m3/m3 from the soil surface and gradually reduced to 0.1 m3/m3 at the depth of approximately 100 cm for the infiltration rate of 1 m/y with the saturated hydraulic conductivity of 2,907 m/y. Moreover, it can be observed from Fig. 6 that the water contents at the soil surface are 0.17, 0.207 and 0.22 m3/m3 when the infiltration rates are 1, 10 and 20 m/y, respectively. The higher infiltration rate eventually attempts the water content to move deeper in the unsaturated porous media. For example, as the infiltration velocity increases (10 m/y and 20 m/y), a relatively large amount of water content is retained in the soil surface (0.207 m3/m3), and subsequently the depth of movement of water content is also increased (maximum water content is observed up to the depth of 250 cm and 500 cm for the infiltration rate of 10 m/y and 20 m/y, respectively). Further, the results show that the concentration of 238U is reduced from 0.129 Bq/L at the soil surface to zero at the depth of 75 cm, 250 cm and 400 cm for the infiltration rate of 1, 10 and 20 m/y in the absence of sorption process. This shows the typical water content variation in the unsaturated soil is influencing the transport of 238U and the variation in infiltration is also impacting the depth of penetration. The similar trent is also observed for 238U transport in the presence of sorption process with considerable delay. This delay can be explained by the retradation induced by the solid phase of 238U during the sorption phenomina. It can be concluded from Fig. 6 that, the behavior of variation in the water content distributions during various infiltration rates is significantly affecting the movement of 238U in the unsaturated sub-surface system.
Fig. 7 provides the spatial distribution of 238U and its progenies in the absence of sorption process in the unsaturated sandy soil with the assumed infiltration rate of 10 m/y during 1, 5 and 10 y. It is observed from Fig. 7 that there is a maximum concentration (0.129 Bq/L) observed for 238U and 234U at the soil surface (uranium tailing pond) and it is further reduced to zero concentration as the depth increases. Conversely, all other progenies initially start from very low concentration (nearly 0 Bq/L) at the soil surface and gradually increase to their corresponding peak concentration with depth and finally become zero concentration followed by its asymptoticity. The order of peak concentrations is observed such as 238U = 234U > 230Th > 226Ra > 234Th > 210Pb > 210Bi > 222Rn > 210Po. As soon as the concentration of 238U is available, the decay starts, gets converted into 234Th, and subsequently into 234U concentration. As 234U is abundantly available from the source at the soil surface; its concentration profile is similar to 238U. Further, the decay of 234U produces the remaining progenies in the decay chain reaction. The lowest 210Po concentration is due to the last product in the decay chain and relatively its own high decay rate (1.83 y−1). It is keenly observed from Fig. 7 that, the second lowest peak concentration of 222Rn due to the highest decay rate (6.6 × 101 y−1) among all the uranium progenies in the uranium decay chain reaction. All the results present in Fig. 5 are independent of sorption process and the variations in the concentrations are observed only based on the respective decay rate. It is further observed that all the 238U and its progenies are available up to the depth of 800 m after 10 y and dangerous for the deeper zone in the unsaturated sandy soil. Thus it can be concluded from Fig. 7 that the concentration of 238U and 234U are continuously arrived the deeper depth from the soil surface (uranium tailing pond) for longer time interval and an enhanced peak concentrations are observed at a various depths (50, 250 and 550 m) at various time interval (1, 5 and 10 y) for all other 238U progenies except 234Th concentration.
The spatial distribution of 238U and its progenies during various intervals in the presence of sorption process in unsaturated sandy soil has been illustrated in Fig. 8. Since the sorption process of all the 238U and its progenies is considered in this simulation, a significant delay in movement and the reduction in peak are observed in all 238U and its progenies concentration. It is observed that there are approximately one to two order of magnitude reductions in peaks for all species except 238U and 234U from Fig. 7 and 8 at 10 y. This peak reduction is due to the consideration of sorption process. Moreover, the maximum concentration of 238U and 234U in the soil surface is also abruptly reduced when depth increases, but which is moving long depth in the case of absence of sorption process. It is further observed that the peak concentration of 222Rn is also reduced to one order even though the distribution coefficient is zero as compared to all other species. This reduction happens due to the distribution coefficients of the previous species which reduce its corresponding concentration in the aqueous phase of the decay chain and in turn, lead to the considerable reduction in peak 222Rn concentration. Essentially, the sorption process hinders the movement of all the species in the aqueous phase due to its corresponding distribution coefficient which helps to attract the species from the aqueous phase to the solid phase. The order of peak concentrations in the sorption process is similar as the absence of sorption process such as 238U = 234U > 230Th > 226Ra > 234Th > 210Pb > 210Bi > 222Rn > 210Po except the order of reduction in peak concentration for 238U and its progenies. Further, it is observed from Fig. 8 that the concentration is virtually negligible after 1 y and 5 y as compared with 10 y. Moreover, the depth of penetration of 238U and its progenies is reduced to less than 10 m and the peak is observed at nearly 2 m depth due to the presence of sorption process, whereas the 238U and its progenies move up to 800 m and the peak is observed at nearly 550 m depth when the sorption process is not considered after 10 y. Since, the half-lives of the parent radionuclides are very long [22], it is great possibility that the concentration of 238U and its progenies can move in deeper depth even though the sorption process is considered.
Fig. 9 provides the spatial distribution of 238U and its progenies in the absence of sorption process in the unsaturated silty soil with the assumed infiltration rate of 3 m/y during 1, 5 and 10 y. The water content is varied from 0.438 m3/m3 at the soil surface and gradually reduced to 0.099 m3/m3 at the depth of approximately 12 m, 50 m and 100 m (not shown in the Fig. 9) for the infiltration rate of 3 m/y with the saturated hydraulic conductivity of 22 m/y after 1, 5 and 10 y, respectively. It can be noted that the concentration of 238U and 234U is following similar trend with maximum inflow concentration of 0.129 Bq/L at the soil surface and reducing to zero at 10, 45 and 90 m depth after 1, 5 and 10 y, respectively. As compared with the sandy soil (Fig. 7) the concentration movement with respect to the depth is significantly reduced in the silty soil, i.e., the concentration of 238U and its progenies are become zero at the depth of 100, 450 and 800 m in the sandy soil whereas it is reduced to 10, 45 and 90 m in the silty soil (approximately 90% reduction in movement). Similarly, the peak concentrations are also observed in the shallow depths in the silty soil as compared with sandy soil (For example, the peak concentration of 230Th is 50 m in the silty soil and 550 m in the sandy soil for 10 y). This reduction in movement in silty soil is primarily due to the decrease of two orders in hydraulic conductivity and other changes in soil hydraulic parameters such as water content, pressure head and van Genuchten parameters (Table 3) as compared with the sandy soil. However, the order of peak concentrations is relatively similar which is observed in the sandy soil such as 238U = 234U > 230Th > 226Ra > 234Th > 210Pb > 210Bi > 222Rn > 210Po. Further, it is keenly observed that the maximum concentration of 238U and 234U are reducing with respect to depth in the silty soil, while the same (maximum) concentrations are observed for larger depths in sandy the soil (Fig. 7). For instance, the concentration of 238U and 234U are constant up to the depth of 250 and 550 m and slowly becoming zero in the sandy soil (Fig. 7), whereas the maximum concentration starts declining immediately just below the soil surface (uranium tailing pond as source) in the silty soil (Fig. 9) due to its corresponding soil hydraulic properties. Thus, it can be concluded from Fig. 9 that the 238U and its progenies in the absence of sorption process in the unsaturated silty soil is significantly reduced its depth of movement predominantly due to the reduced unsaturated hydraulic conductivity and other soil hydraulic parameters such as water content, pressure head and van Genuchten parameters, which in turn, the enhance availability of 238U and its progenies are in the shallow depths. However, the concentration of 238U and its progenies are speared up to 90 m depth after 10 y, which shows that the groundwater aquifers may get affected if the water table is shallow.
Fig. 10 describes the spatial distribution of 238U and its progenies in the presence of sorption process in the unsaturated silty soil. The 238U and 234U concentrations are released from the uranium tailing pond at the soil surface which have higher concentration (0.129 Bq/L) and become zero at 1 m after 1 y but very marginal downward movement is observed after 5 and 10 years. Similarly, other progenies of 238U are also shown a low depth of transport in the unsaturated silty soil. This significant retardation in movement on 238U and its progenies is predominantly happened by the sorption process of all the uranium species and the low hydraulic conductivity of soil. Approximately 90% reduction in the depth of penetration is observed in the sorption process when compared Fig. 10 with 9; (zero concentration is observed nearly at 1, 3 and 5 m in the presence of sorption process and 10, 45 and 90 m in the absence of sorption process after 1, 5 and 10 y); on the other hand almost 99% reduction in depth of penetration is observed in silty soil (Fig. 10) as compared with the sandy soil (Fig. 8) in the presence of sorption process (i.e., zero concentration is observed nearly at 1, 3 and 5 m in the silty soil and 100, 450 and 800 m in the in sandy soil with the sorption process after 1, 5 and 10 y). Moreover, it is observed from Fig. 9 and 10 that the peak concentration of all the uranium species except 238U and 234U are reduced nearly one order in the presence of sorption process as compared with the absence of sorption phenomena. It can be concluded from Fig. 10 that the concentration of 238U and its progenies vanish at very shallow depths (approximately before 10 m) and offer a positive impact to stop the groundwater contamination.

### 4. Conclusions

In the present study, a numerical model is developed to describe the transport of 238U and its progenies in the vadose zone from uranium tiling pond situated on the soil surface. The one-dimensional coupled water flow and 238U and its progenies decay chain reaction is developed using implicit finite difference scheme. In addition, this study is also extended to perform the movement of 238U and its progenies concentration in different soils such as sand and silt. From the developed model, the following conclusions are arrived:
1. The vertical downward transport of 238U and its progenies from uranium tailing pond at soil surface is highly dependent on the water content variation in the unsaturated soil and its other soil hydraulic properties such as unsaturated hydraulic conductivity, pressure head and van Genuchten parameters.

2. The downward movement of 238U and its progenies from the uranium tailing pond results in higher concentrations at the shallow depth irrespective of sorption process in the silty soil, whereas considerable increase in concentrations is observed at the deeper depth in the sandy soil. This is due to the high hydraulic conductivity accompanied by the increase in other soil hydraulic properties which yield a larger depth of movement in the sandy soil as compared with the silty soil.

3. While, silt transmit a relatively lesser uranium concentration down the ground surface, the sand or alluvium transport a relatively larger fraction of uranium vertically downwards and hence, it is more likely that this particular geological unit has more probability of potentially contaminating the saturated groundwater system. On the contrary, with reference to root zone uptake, it is the silty formations that retain the uranium in a relatively larger amount and subsequently contaminate the root zone in the unsaturated zone. Further, sorption plays a vital role in deciding the resultant retardation, and in turn, the total migration pathway. Thus, a coupled effect of sorption and a given geological unit decide the resultant uranium transport in a given subsurface environment.

In over all, the presence of sorption process in the silty soil significantly helps to avert the groundwater contamination, especially at deeper depths and for larger span. Future studies can be intended to integrate other mechanisms such as evaporation, root uptake, surface runoff, degradation in uranium pond and other climatic conditions to achieve the combined effect of all the coupled processes on the equivalent system to develop a comprehensive model.

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##### Fig. 1
Pictorial representation of uranium transport in unsaturated porous media.
##### Fig. 2
Schematic representation of solving water flow and transport of radionuclides in an unsaturated porous medium.
##### Fig. 3
Graphical representation of boundary conditions for one-dimensional domain.
##### Fig. 4
Validation of one-dimensional multi-species transport with Bauer et al., (2001) at T = 3,000 d; λ = (7.0, 5.0, 4.5, 3.8) × 10−4 d−1; R = (5.3, 1.9, 1.2, 1.3); v = 1 m/d; C0 = (100.0, 0.0, 0.0, 0.0) mmol.
##### Fig. 5
Effect of water content on 238U concentration in unsaturated porous media in the presence and absence of sorption process.
##### Fig. 6
Effect of infiltration rate on the transport of 238U in unsaturated porous media for the period of 10 d.
##### Fig. 7
Concentration profile for 238U and its progenies during various intervals without considering sorption process in unsaturated sandy soil.
##### Fig. 8
Concentration profile for 238U and its progenies during various intervals in the presence of sorption process in unsaturated sandy soil.
##### Fig. 9
Concentration profile for 238U and its progenies during various intervals without considering sorption process in unsaturated silty soil.
##### Fig. 10
Concentration profile for 238U and its progenies during various intervals in the presence of sorption process in unsaturated silty soil.
##### Table 1
Hydrodynamic Properties and Transport Parameters for an Unsaturated Zone
Parameter Notation Value Reference
Soil type Sand [44]
Van Genuchten parameter (m−1) α 3.35 [44]
Van Genuchten parameter β 2.0 [44]
Van Genuchten parameter η 0.5 [44]
Saturated water content (m3/m3) θs 0.381 [44]
Residual water content (m3/m3) θr 0.102 [44]
Saturated hydraulic conductivity (m/y) Ks 2,907 [44]
Soil bulk density (g/m3) ρb 1.56 × 106 [45]
Longitudinal dispersivity (m) αL 0.5 [45]
##### Table 2
Uranium Decay Chain Parameters [22]
Nuclide Half-life, T1/2 (y) Decay coefficient λi (y−1) Distribution coefficient, kdi (L/mg)
238U 4.51 × 109 1.54 × 10−10 5.00 × 10−4
234Th 6.60 × 10−2 1.05 × 101 2.00 × 10−3
234U 2.48 × 105 2.79 × 10−6 5.00 × 10−4
230Th 8.00 × 104 8.66 × 10−6 2.00 × 10−3
226Ra 1.60 × 103 4.33 × 10−4 5.00 × 10−4
222Rn 1.05 × 10−2 6.6 × 101 0.00
210Pb 2.20 × 101 3.15 × 10−2 3.00 × 10−4
210Bi 1.37 × 106 5.06 × 10−7 3.00 × 10−4
210Po 3.79 × 10−1 1.83 × 100 1.50 × 10−4
##### Table 3
Hydraulic Properties and Van Genuchten Fitting Parameters for Various Types of Soil [46]
Soil type % of sand % of clay α β θr θs Ks
Sand 92.7 2.9 14.5 2.68 0.045 0.43 2,602
Silt 5.8 9.5 1.6 1.37 0.034 0.46 22
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text:
# Short-circuit of compound Boolean expressions

Short-circuit evaluation (aka. minimal evaluation or McCarthy evaluation) is when the remaining arguments in a Boolean operation are executed or evaluated only if the previous operations are true. Fortran does not provide short-circuit evaluations. This allows for optimizations such as parallel evaluation. Some compilers do provide an extension to provide short-circuit evaluation but to keep your code standard-conforming you should assume all sections of an expression may or may not be evaluated and executed, including function calls in the expression.

The solution is to use nested conditionals. For example:

val=0.0
if(val.ne.0 .and. (top/val.gt.10.0) )then ! DO NOT DO THIS
write(*,*)'met conditions'
endif

Should be written as

val=0.0
if(val.ne.0 )then
if (top/val.gt.10.0) )then
write(*,*)'met conditions'
endif
endif

So any time you have a compound Boolean expression (anywhere, not just in an IF(3f) expression) make sure all subexpressions are safe to evaluate or change the expression to conditionally executed sections.

# Common errors …

IF ( INDX .GT.0 .AND. ARRAY(INDX) .EQ. 'xxx' ) THEN ...

If INDX can be out of range for array ARRAY this could cause such problems as getting a run-time out-of-bounds error. This is true not just in IF statements but anywhere a compound Boolean expression is used …

I=MERGE (A(ii), 0, ii .GE. 1)

# By the way …

Some new forms of short-circuiting conditional expressions are planned for Fortran 202x.