Ideal Gas law Partial derivative

[Punkyc7]

This is a question from my calculus book that i thought was interesting, its not homework but I am 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?

 

[SammyS]

\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)\ ?

 

[Punkyc7]

it should be NR. But the way it is written shouldn't it be T N^2R^2/(PV)

 

[sjb-2812]

it should be NR. But the way it is written shouldn't it be T N^2R^2/(PV)

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

 

[bpt3235]

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