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}


closed as off-topic by Cyclohexanol., Zain Patel, Claude Leibovici, user91500, TastyRomeo Mar 3 '17 at 8:18

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Cyclohexanol., Claude Leibovici, user91500
If this question can be reworded to fit the rules in the help center, please edit the question.


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

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


Not the answer you're looking for? Browse other questions tagged or ask your own question.