Question

The Vander Waals equation for n moles of a gas is \(\left( {P + \frac{{{n^2}a}}{{{V^2}}}} \right)\left( {V - nb} \right) = nRT\) where P is the pressure, V is volume and T is the temperature of the gas. The constant R is the universal gas constant and \(a\) and \(b\)are positive constants that are characteristics of a particular gas. Calculate \(\frac{{\partial T}}{{\partial P}}\)and \(\frac{{\partial P}}{{\partial V}}\).

Short answer

A partial differential equation (PDE) is a mathematical equation that establishes relationship between the partial derivatives of a multivariable function.

Step-by-step solution

Given data

Given: \(\left( {P + \frac{{{n^2}a}}{{{V^2}}}} \right)\left( {V - nb} \right) = nRT\)

Where P\( \to \)Pressure

V\( \to \)Volume

T\( \to \)Temperature

R\( \to \)Gas constant

a, b\( \to \)Positive constant

Differentiate with respect to T

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

\(T = \frac{{\left( {P + \frac{{{n^2}a}}{{{V^2}}}} \right)\left( {V - nb} \right)}}{{nR}}\)

\(\frac{{\partial T}}{{\partial P}} = \frac{{V - nb}}{{nR}}\frac{\partial }{{\partial P}}\left( {P + \frac{{{n^2}a}}{{{V^2}}}} \right)\)

\(\frac{{\partial T}}{{\partial P}} = \frac{{V - nb}}{{nR}}\)

Differentiate with respect to V.

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

\(P + \frac{{a{n^2}}}{{{V^2}}} = \frac{{nRT}}{{V - nb}}\)

\(\frac{{\partial P}}{{\partial V}} = - \frac{{nRT}}{{{{(V - nb)}^2}}} + \frac{{2{n^2}a}}{{{V^2}}}\)

Therefore,\(\frac{{\partial T}}{{\partial P}} = \frac{{V - nb}}{{nR}}\)

\(\frac{{\partial P}}{{\partial V}} = - \frac{{nRT}}{{{{(V - nb)}^2}}} + \frac{{2{n^2}a}}{{{V^2}}}\).