- #1
Coffee_
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Let's say that we have some canonical ensemble where I know that the heat capacity is given by
##C_{V}=\alpha(N,V) T^{n}##
Since ##C_{V}=T\frac{\partial S(T,V)}{\partial T}## I know that
##S(V,T)=\frac{1}{n} \alpha(N,V) T^{n} + f(N,V) ##
Where the function ##f(N,V)## has to do with the fact that I'm only taking the derivative wrt. to T and can lose such additional terms in general.
I also know that ##S(V,0)=0## which means that ##f(N,V,)=0## which means I can always use this trick to find the entropy if the heat capacity is known. Obviously if ##C_{V}## is an uglier function of T we'd have to integrate and so on.
##QUESTION##: When is it okay to do such a reasoning and when isn't it?
##C_{V}=\alpha(N,V) T^{n}##
Since ##C_{V}=T\frac{\partial S(T,V)}{\partial T}## I know that
##S(V,T)=\frac{1}{n} \alpha(N,V) T^{n} + f(N,V) ##
Where the function ##f(N,V)## has to do with the fact that I'm only taking the derivative wrt. to T and can lose such additional terms in general.
I also know that ##S(V,0)=0## which means that ##f(N,V,)=0## which means I can always use this trick to find the entropy if the heat capacity is known. Obviously if ##C_{V}## is an uglier function of T we'd have to integrate and so on.
##QUESTION##: When is it okay to do such a reasoning and when isn't it?