Chapter 8, Problem 1P

Perform the same computation as in Sec. 8.1, but for ethyl alcohol $\left(a=12.02\text{ and }b=0.08407\right)$ at a temperature of 375 K and p of 2.0 atm. Compare your results with the ideal gas law. Use any of the numerical methods discussed in Chaps. 5 and 6 to perform the computation. Justify your choice of technique.

To determine

The modal volume using Ideal gas law and Van der waals equation for ethyl alcohol $\left(a=12.02,b=0.08407\right)$ at a temperature of 375 K and $p$ of 2.0 atm where the both the equations are given by,

Ideal Gas equation states,

$pV=nRT$

Here, $p$ is the absolute pressure, $V$ is the volume, $n$ is the number of moles, $T$ is the absolute temperature and $R$ is the universal gas constants.

Van der waals equation states,

$\left(p+\frac{a}{v^2}\right)\left(v-b\right)=RT$

Here, $v=\frac{V}{n}$ is the modal volume and $a,b$ are empirical constants depending upon the gas.

Answer to Problem 1P

Solution:

Using van der waals equation the modal volume $\underset{\_}{0.3644374484}$ and using ideal gas equation $\underset{\_}{\text{15}\text{.385125}}$.

Van der waals equation is more appropriate to calculate the modal volume.

Explanation of Solution

Given Information: For ethyl alcohol $a=12.02,b=0.08407$ and the temperature is $T=375\text{K}$, the number of moles is $p=2\text{atm}$. The universal gas constant $R=0.082054\text{ L atm/(mol K)}$

Consider Van der waals equation,

$\left(p+\frac{a}{v^2}\right)\left(v-b\right)=RT$

Therefore, the modal volume $v$ will be the roots of the equation $f\left(v\right)=0$ where,

$f\left(v\right)=pv+\frac{a}{v}-\frac{ab}{v^2}-\left(RT+pb\right)$

Hence, differentiating,

$f^{\prime }\left(v\right)=p-\frac{a}{v^2}+\frac{2ab}{v^3}$

Therefore, for ethyl alcohol $a=12.02,b=0.08407$ with $T=375\text{K}$ and $P=2$, the modal volume can be computed using the Newton Raphson method by following iteration technique, $v_{i+1}=v_i-\frac{f\left(v_i\right)}{f^{\prime }\left(v_i\right)}$ for $a=12.02,b=0.08407,T=375,P=2$

Construct a MATLAB code “Code_97924_8_1P_a.m” to perform the mentioned iteration.

function Code_97924_8_1P_a()

% tolerance used in the iteration process.

Tol = 1e-10;

%the parameters of the equation is written in the following way

a = 12.02;

b = 0.08407;

R = 0.082054;

T = 375;

p = 2;

% the function and its derivative is written in the following way as % a function of v.

f = @(v) p*v + a/v - a*b/v - (R*T + p*b);

fd = @(v) p - a/(v^2) + 2*a*b/(v^3);

% Finding the initial guess of the root of f(v) = 0.

v = 0:0.001: 1; k = 0;

fun = zeros(1,1); Guess = zeros(1,1);

for i = 1: 1: length(v)

fun(i) = f(v(i));

if i>1 && fun(i-1)*fun(i) < 0

k = k + 1; Guess(k) = v(i);

end

end

clear v

% Modifying the root using Newton Rapson Method

v(1) = Guess(1); i = 1;

while (1)

v(i+1) = v(i) - f(v(i))/fd(v(i));

Er = abs(v(i+1)-v(i))/abs(v(i));

if Er<=Tol, break, end

i = i + 1;

end

% Print the desired root in the screen.

fprintf('The Modal Volume %10.10f\n',v(end));

end

The output of the function is,

Therefore, using van der waals equation the modal volume of the ethyl alcohol at $T=375\text{K}$ and $P=2\text{atm}$ is $0.3644374484$.

Using Ideal gas equation the modal volume,

$v=\frac{V}{n}=\frac{RT}{p}$

Therefore, for ethyl alcohol at $T=375\text{K}$ and $P=2\text{atm}$, $\begin{array}{c}v=\frac{0.082054\times 375}{2}\\ =\text{15}\text{.385125}\end{array}$

Therefore, using Ideal gas equation the modal volume of the ethyl alcohol at $T=375\text{K}$ and $P=2\text{atm}$ is $\text{15}\text{.385125}$.

Interpretation:

As Ethyl alcohol is not an ideal gas hence Ideal gas equation is not appropriate to calculate the modal volume so the van deer waals equation should be imposed to calculate the modal volume.