Chapter 2, Problem 52P

The ideal gas equation of state is very simple, but its range of applicability is limited. A more accurate but complicated equation is the Van der Waals equation of state given by

   $P=\frac{RT}{V-b}-\frac{a}{V^2}$

where a and b are constants depending on critical pressure and temperatures of the gas. Predict the coefficient of compressibility of nitrogen gas at $T=175\text{K}$ and $V=0.00375{\text{m}}^3/\text{kg}$ , assuming the nitrogen to obey the Van der Waals equation of state. Compare your result with the ideal gas value. Take $a=0.175{\text{m}}^6\cdot {\text{kPa/kg}}^2$ and $b=0.00138{\text{m}}^3/\text{kg}$ for the given conditions. The experimentally measured pressure of nitrogen is 10,000 kPa.

To determine

To compare: the coefficient of compressibility of nitrogen gas by using ideal gas equation and Van der Waals equation.

Answer to Problem 52P

The coefficient of compressibility of nitrogen gas by using ideal gas equation is calculated as $13850.667\text{kPa}$.

The coefficient of compressibility of nitrogen gas by using Van der Waals equation is calculated as $9787.712\text{kPa}$.

Error occurred by comparing is $41.5\%$.

Explanation of Solution

Given information:

The temperature of the nitrogen gas is $175\text{K}$, specific volume is given as $0.00375{\text{m}}^{\text{3}}/\text{kg}$, the value for $a$ is $0.175{\text{m}}^{\text{6}}\cdot \text{kPa}/{\text{kg}}^2$, the value for $\mathrm{b}$ is given as $0.00138{\text{m}}^{\text{3}}/\text{kg},$ and the pressure is given as $10000\text{kPa}$.

Write the expression for Van der Waals equation.

   $P=\frac{RT}{v-\mathrm{b}}-\frac{a}{v^2}$    ...... (I)

Here, constants depending on critical pressure and temperature are $a$ and $\mathrm{b}$, pressure is $P$, gas constant is $R$, specific volume is $v,$ and temperature is $T$.

Write the expression for coefficient of compressibility for real gas equation.

   $k_1=-v{\left(\frac{\partial P}{\partial v}\right)}_T$    ...... (II)

Here, rate of change of pressure with respect to specific volume at temperature $T$ is ${\left(\frac{\partial P}{\partial v}\right)}_T$.

Substitute $\frac{RT}{v-\mathrm{b}}-\frac{a}{v^2}$ for $P$ in the Equation (II).

   $\begin{array}{c}{k}_1=-v\left(\frac{\partial \left(\frac{RT}{v-\mathrm{b}}-\frac{a}{v^2}\right)}{\partial v}\right)\\ =-\frac{vRT}{{\left(v-\mathrm{b}\right)}^2}+\frac{2a}{v^2}\end{array}$    ...... (III)

Substitute $\frac{vRT}{{\left(v-\mathrm{b}\right)}^2}-\frac{2a}{v^2}$ for $k$ in the Equation.(II).

   $\begin{array}{l}\frac{vRT}{{\left(v-\mathrm{b}\right)}^2}-\frac{2a}{v^2}=-v{\left(\frac{\partial P}{\partial v}\right)}_T\\ {\left(\frac{\partial P}{\partial v}\right)}_T=\frac{2a}{v^3}-\frac{RT}{{\left(v-\mathrm{b}\right)}^2}\end{array}$

Write the expression ideal gas Equation.

   $\begin{array}{l}Pv=RT\\ P=\frac{RT}{v}\end{array}$    ...... (IV)

Write the expression for coefficient of compressibility for ideal gas at constant pressure.

   $k_2=P$    ...... (V)

Substitute

Write the expression for percentage error.

   $\%\text{error}=\frac{k_2-k_1}{k_1}\times 100$    ...... (VII)

Refer to Table A-1 “Molar mass, gas constant and ideal gas specific heats of some substances” to obtain the value for gas constant as $0.2968\text{kJ}/\text{kg}\cdot \text{K}$

Substitute $0.175{\text{m}}^{\text{6}}\cdot \text{kPa}/{\text{kg}}^2$ for $a$, $0.2968\text{kJ}/\text{kg}\cdot \text{K}$ for $R$, $175\text{K}$ for $T$, $0.00375{\text{m}}^{\text{3}}/\text{kg}$ for $v$ and $0.00138{\text{m}}^{\text{3}}/\text{kg}$ for $\mathrm{b}$ in the Equation (III).

   $\begin{array}{c}{k}_1=\frac{2\left(0.175{\text{m}}^{\text{6}}\cdot \text{kPa}/{\text{kg}}^2\right)}{{\left(0.00375{\text{m}}^{\text{3}}/\text{kg}\right)}^2}-\frac{\left(0.00138{\text{m}}^{\text{3}}/\text{kg}\right)\left(0.2968\text{kJ}/\text{kg}\cdot \text{K}\right)\left(175\text{K}\right)}{{\left(0.00375{\text{m}}^{\text{3}}/\text{kg}-0.00138{\text{m}}^{\text{3}}/\text{kg}\right)}^2}\\ =\left(24888.88\text{kPa}\right)-34676.6\left(\text{kJ}/{\text{m}}^{\text{3}}\right)\left(\frac{1\text{kNm}}{1\text{kJ}}\right)\\ =24888.88\text{kPa}-34676.6\left(\text{kN}/{\text{m}}^{\text{2}}\right)\left(\frac{1\text{kPa}}{1\text{kN}/{\text{m}}^{\text{2}}}\right)\\ =9787.72\text{kPa}\end{array}$

Substitute $0.2968\text{kJ}/\text{kg}\cdot \text{K}$ for $R$, $175\text{K}$ for $T$ and, $0.00375{\text{m}}^{\text{3}}/\text{kg}$ for $v$ in the Equation (VI).

   $\begin{array}{c}{k}_2=\frac{\left(0.2968\text{kJ}/\text{kg}\cdot \text{K}\right)\left(175\text{K}\right)}{0.00375{\text{m}}^{\text{3}}/\text{kg}}\\ =\left(13850.667\text{kJ}/{\text{m}}^{\text{3}}\right)\left(\frac{1\text{kN}\cdot \text{m}}{1\text{kJ}}\right)\\ =13850.667\text{kN}/{\text{m}}^{\text{2}}\left(\frac{1\text{kPa}}{1\text{kN}/{\text{m}}^{\text{2}}}\right)\\ =13850.667\text{kPa}\end{array}$

Substitute $13850.667\text{kPa}$ for $k_2$ and $9787.72\text{kPa}$ for $k_1$ in the Equation (VII).

   $\begin{array}{c}\%\text{error}=\frac{\left(13850.667\text{kPa}\right)-\left(9787.72\text{kPa}\right)}{\left(9787.72\text{kPa}\right)}\times 100\\ =\frac{4062.94\text{kPa}}{9787.72\text{kPa}}\times 100\\ =0.4151\times 100\\ =41.5\%\end{array}$

Conclusion:

The coefficient of compressibility of nitrogen gas by using ideal gas equation is calculated as $13850.667\text{kPa}$.

The coefficient of compressibility of nitrogen gas by using Van der Waals equation is calculated as $9787.712\text{kPa}$.

Error occurred by comparing is $41.5\%$.