Question

Prove cp-cv=R derivation PDF for written exam

Answer 1

HEY Buddy.....!! here is ur answer

There are two main specific heat capacities for a gas. CPCP is the specific heat capacity at constant pressure, CVCV is the specific heat capacity at constant volume.

The specific heat capacity at constant pressure is greater than the specific heat capacity at constant volume by the amount of work done in expansion.

The relation given in the question holds for an ideal gas when the internal energy does not depend on the volume. RR is the Gas constant .

Here is a brief derivation of the relation :

For an ideal gas, evaluating the partial derivatives above according to the equation of state where R is the gas constant for an ideal gas

PV=nRTPV=nRT

CP−CV=T(∂P∂T)V,n(∂V∂T)P,nCP−CV=T(∂P∂T)V,n(∂V∂T)P,n

P=nRTV⇒(∂P∂T)V,n=nRVP=nRTV⇒(∂P∂T)V,n=nRV

V=nRTP⇒(∂V∂T)P,n=nRPV=nRTP⇒(∂V∂T)P,n=nRP

substituting

T(∂P∂T)V,n(∂V∂T)P,n=T(nRV)(nRP)=(nRTV)(nRP)=P(nRP)=nRT(∂P∂T)V,n(∂V∂T)P,n=T(nRV)(nRP)=(nRTV)(nRP)=P(nRP)=nR

this equation reduces simply to Mayer's relation,

CP,m−CV,m=R

I hope it will be helpful for you...!!

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