Partial Derivative Question Relating to Chemistry

Question

The Ideal Gas Law PV=nrT shows the relationship between pressure P, volume V, and temperature T of a gas, where n is the number of moles of the gas and r is the universal gas constant. Prove \begin{equation} \frac{\partial V}{\partial T} \times \frac{\partial T}{\partial P} \times \frac{\partial P}{\partial V} = -1 \end{equation}

Answer 1

The functions you are considering are the following:

$$V(T,P) = \frac{nrT}{P}, \quad T(V,P) = \frac{PV}{nr}, \quad P(V,T) = \frac{nrT}{V}. $$

This implies

$$\frac{\partial V}{\partial T} =\frac{nr}{P}, \quad \frac{\partial T}{\partial P} =\frac{V}{nr}, \quad \frac{\partial P}{\partial V} = -\frac{nrT}{V^2}. $$

Multiplying these three quantities you get

$$\frac{\partial V}{\partial T} \frac{\partial T}{\partial P} \frac{\partial P}{\partial V} = -\frac{nrT}{PV} = -1.$$