How to get $\left(\frac{\delta S}{\delta p}\right)_T$ from $\left(\frac{\delta S}{\delta p}\right)_V$?
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S can either be a function of p,Tp,T or p,Vp,V(since p,T,Vp,T,V are related by an equation of state). So we can relate the derivatives by the chain rule
S(p,V)=S(p,V(p,T))S(p,V)=S(p,V(p,T))
(∂S∂p)T=(∂S∂p)V+(∂S∂V)p(∂V∂p)T(∂S∂p)T=(∂S∂p)V+(∂S∂V)p(∂V∂p)T
So if you know everything on the right hand side you have the answer. (∂V∂p)T(∂V∂p)Tisn't on your list of quantities you can use, but you can use
(∂p∂V)T=1(∂V∂p)T.
S(p,V)=S(p,V(p,T))S(p,V)=S(p,V(p,T))
(∂S∂p)T=(∂S∂p)V+(∂S∂V)p(∂V∂p)T(∂S∂p)T=(∂S∂p)V+(∂S∂V)p(∂V∂p)T
So if you know everything on the right hand side you have the answer. (∂V∂p)T(∂V∂p)Tisn't on your list of quantities you can use, but you can use
(∂p∂V)T=1(∂V∂p)T.
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It is well known that there are the black hole horizon and the cosmological horizon for the Reissner–Nordström–de Sitter (RN-dS) spacetime.
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