Deriving heat capacity using thermodynamics identity

In summary, the heat capacity formula C_V can be derived using the thermodynamic identity, which states that dS=(\frac{\partial{S}}{\partial{U}})_{PV}dU. By substituting this into the equation for dU and manipulating the terms, the formula C_V=T\frac{\partial{S}}{\partial{T}}_V=\frac{\partial{U}}{\partial{T}} can be obtained. This solution may be considered as satisfying the thermodynamic identity, although it may not be the most direct approach.
  • #1
ultimateguy
125
1

Homework Statement


Use the thermodynamic identity to derive the heat capacity formula [tex]C_V=T\frac{\partial{S}}{\partial{T}}_V[/tex]

Homework Equations


[tex]C_V=\frac{\partial{U}}{\partial{T}}[/tex]
[tex] T=\frac{\partial{U}}{\partial{S}}[/tex]
dU=TdS-PdV+[tex]\mu[/tex]dN

The Attempt at a Solution



I used [tex]C_V=\frac{\partial{U}}{\partial{S}}\frac{\partial{S}}{\partial{T}}=T\frac{\partial{S}}{\partial{T}}_V=\frac{\partial{U}}{\partial{T}}=C_V[/tex].
This is the only solution I can think of, but I don't think I used the thermodynamics identity, did I?
 
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  • #2
There are a distinct lack of thermal physicists on this website and with good reason as it is one of the most boring subjects ever. I think if you know that:

[tex] dS=(\frac{\partial{S}}{\partial{U}})_{PV}dU[/tex]

Then you can substitute the following into the equation for dU,

[tex] dU=(\frac{\partial{U}}{\partial{T}})_VdT[/tex]

and have a fiddle about, that should set you off on the right lines.
 
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  • #3
This may be a dumb question, but what's the difference between E and U?
 
  • #4
ultimateguy said:
This may be a dumb question, but what's the difference between E and U?

There is none. I apologise. I normally write E and I was trying to conform to your notation but obviously forgot half way through I will change it.
 
  • #5
The solution provided by the OP is correct.

Daniel.
 
  • #6
I'd have thought if they wanted a derivation from the thermodynamic identity then they'd want you to start from that. However the OP proof might be accepted.
 

What is the thermodynamics identity?

The thermodynamics identity is a mathematical equation that relates the heat capacity of a substance to its internal energy, enthalpy, and entropy. It is written as dU = TdS - PdV, where dU is the change in internal energy, T is the temperature, dS is the change in entropy, P is the pressure, and dV is the change in volume.

How is the heat capacity derived using the thermodynamics identity?

The heat capacity can be derived from the thermodynamics identity by rearranging the equation to solve for dU, which is equal to the heat added to the system. This can then be divided by the change in temperature to calculate the heat capacity.

Why is the thermodynamics identity useful in calculating heat capacity?

The thermodynamics identity is useful because it allows us to relate the change in internal energy to the change in temperature, which is directly related to the heat capacity. This allows us to calculate the heat capacity of a substance without having to measure the heat directly.

What factors can affect the accuracy of the heat capacity calculated using the thermodynamics identity?

Some factors that can affect the accuracy of the heat capacity calculated using the thermodynamics identity include the assumptions made in the derivation, the limitations of the ideal gas law, and any external factors that may affect the system, such as pressure or heat loss.

In what situations is the thermodynamics identity not applicable for calculating heat capacity?

The thermodynamics identity is not applicable for calculating heat capacity in situations where there is a phase change or a chemical reaction taking place, as these processes involve a change in the internal energy that is not solely due to a change in temperature.

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