# Ideal Gas law Partial derivative

This is a question from my calculus book that i thought was interesting, its not homework but im curious to how you go about showing it.

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T (∂P/∂T)(∂V/∂T)=NR

We know PV=NRT

so if we take a partial how does the T end up on the other side?

## Answers and Replies

SammyS
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$$\frac{\partial P}{\partial T} = \frac{\partial }{\partial T} \left(\frac{NRT}{V}\right)= \frac{NR}{V}$$

$$\frac{\partial V}{\partial T} = \frac{\partial }{\partial T} \left(\frac{NRT}{P}\right)= \frac{NR}{P}$$

So, what is $$T\ \left(\frac{\partial P}{\partial T}\right)\left(\frac{\partial V}{\partial T}\right)\ ?$$

it should be NR. But the way it is written shouldnt it be T N^2R^2/(PV)

it should be NR. But the way it is written shouldnt it be T N^2R^2/(PV)

But if PV = NRT, what does NRT/PV equal?

T(∂P/∂T)(∂V/∂T)=NR

T(NR/V)(NR/P)=[T(NR)^2]/PV but PV equals NRT

[T(NR)^2]/NRT ==> NR

you just needed the substitution for PV