Chapter 1, Problem 1.39E

Interpretation Introduction

Interpretation:

The expression for (δp / δV)_{T, n} for a van der Waals gas and for the virial equation in terms of volume is to be stated.

Concept introduction:

The ideal gas law considered the molecules of a gas as point particles with perfectly elastic collisions among them in nature. This works importantly well for gases at dilution and at low pressure in many experimental calculations. But the gas molecules are not performing as point masses, and there are situations where the properties of the gas molecules have measurable effect by experiments. Thus, a modification of the ideal gas equation was coined by Johannes D. van der Waals in 1873 to consider size of molecules and the interaction forces among them. It is generally denoted as the van der Waals equation of state.

Answer to Problem 1.39E

The expression for ${\left(\frac{\delta p}{\delta V}\right)}_{T,n}$ a vander Waals gas and for the virial equation in terms of volume is as follows;

${\left(\frac{\delta p}{\delta V}\right)}_{T,n}=\frac{-nRT}{{(V-nb)}^2}+\frac{2a{n}^2}{V^3}$

Explanation of Solution

The ideal gas law considered the gas molecules a point particles which interact only with their containers but not with each other. They do not take up space in the container and are not attracted and repelled by any other gas particles in the container and do not lose the kinetic energy during the collisions. The ideal gas equation can be represented as;

$PV=nRT\cdot \cdot \cdot \text{(1)}$

Vander Waals equation is based on the reasons that non-ideal gases do not obey the ideal gas equation. Notably, the van Waals equation improves the ideal gas law by adding two significant terms in the ideal gas equation: one term is to account for the volume of the gas molecules and another term is introduced for the attractive forces between them. The non-ideal gas equation represented as;

$\left(p+\frac{a{n}^2}{V^2}\right)\left(V-nb\right)=nRT\cdot \cdot \cdot (2)$

In the above equation,

$\begin{array}{l}\left(p+\frac{a{n}^2}{V^2}\right)\left(V-nb\right)\\ \text{(Correction term introduced}\text{(Correction term introduced}\\ \text{for molecular attraction)}\text{for volume of molecules)}\end{array}$

‘a’ and ‘b’ are called as van der Waals constants and ‘a’ represents the pressure correction and it is related to the magnitude and strength of the interactions between gas particles. Similarly, ‘b’ describes the volume correction and it is having relationship to the size of the gas particles. When the value of V is large and ‘n’ is comparatively small, both corrections become negligible. Thus, the van der Waals equation contracts to the ideal gas equation PV = nRT. Such situations are for a gas having relatively lower number of gas molecules, occupying in a relatively larger volume i.e., the gas is at lower pressure. Generally, at low pressures, the correction term ‘a’ is more predominant than the correction term for volume ‘b’. At high pressures, the correction term for the volume of the gas molecules is predominant since, the molecules are incompressible and occupies a notable fraction of the total volume of gas.

$\begin{array}{l}\left(p+\frac{a{n}^2}{V^2}\right)\left(V-nb\right)=nRT\\ \left(p+\frac{a{n}^2}{V^2}\right)=\frac{nRT}{\left(V-nb\right)}\\ p=\frac{nRT}{\left(V-nb\right)}-\frac{a{n}^2}{V^2}\cdot \cdot \cdot (3)\end{array}$

On taking the derivative of this expression with respect to temperature, we can get the expression for ${\left(\frac{\delta p}{\delta V}\right)}_{T,n}$

Both the term in the above expression has a volume as a variable so the derivative of pressure with respect to volume will give two terms. Thus,

$\begin{array}{l}{\left(\frac{\delta p}{\delta V}\right)}_{T,n}={\left[\frac{\delta }{\delta V}\left(\frac{nRT}{V-nb}-\frac{a{n}^2}{V^3}\right)\right]}_{T,n}\\ =\frac{-nRT}{{(V-nb)}^2}+\frac{2a{n}^2}{V^3}\end{array}$

Thus, variation of pressure with respect to temperature having the volume ‘V’ with them. Varying the quantities of volume will result in the variation of $\frac{dp}{dV}$ term.

Conclusion

The expression for ${\left(\frac{\delta p}{\delta V}\right)}_{T,n}$ for a van der Waals gas and for the virial equation in terms of volume is stated.