Why Do Heat Capacities Use Derivatives of Entropy in Their Formulas?

[dapias09]

Hi all,

I'm working with the heat capacities definition and I have got a confusion. I don't understand why we can express them like

Cp = T(∂S/∂T)p
Cv = T(∂S/∂T)v

I know that Cp=(dQ/dT)p = (∂H/∂T)p with H equal to TdS + VdP and Cv=(dQ/dT)v = (∂U/∂T)v with U equal to TdS + PdV,

My guess: If I begin for instance with the enthalpy, H, and I constrain it to constant pressure I get just the TdS term, that is the thermodynamic definition of "Heat (Q)" . So I get the definition of the heat capacity if I derive Q respect to T, or TdS respect to T (is the same thing). Doing it, I get:

dQ/dT = (TdS)/dT = Td^2S/d^2T + dS/dT.

An expression very different of the definition.

Can anyone help me?

Thanks in advance.

 

[Andrew Mason]

Hi all,

I'm working with the heat capacities definition and I have got a confusion. I don't understand why we can express them like

Cp = T(∂S/∂T)p
Cv = T(∂S/∂T)v

I know that Cp=(dQ/dT)p = (∂H/∂T)p with H equal to TdS + VdP

You cannot equate TdS to dQ unless dQ = dU + PdV, and that is true only if the process is reversible. Generally, you have to use the first law: dQ = dU + ∂W.

and Cv=(dQ/dT)v = (∂U/∂T)v with U equal to TdS + PdV,

I think you mean dU = TdS - PdV. Again, that is only true if it is a reversible process.

My guess: If I begin for instance with the enthalpy, H, and I constrain it to constant pressure I get just the TdS term, that is the thermodynamic definition of "Heat (Q)" . So I get the definition of the heat capacity if I derive Q respect to T, or TdS respect to T (is the same thing). Doing it, I get:

dQ/dT = (TdS)/dT = Td^2S/d^2T + dS/dT.

I don't follow what you are doing. Why would it not just be T(dS/dT)? Again, this is true only in a reversible process.

AM

 

[juanrga]

Hi all,

I'm working with the heat capacities definition and I have got a confusion. I don't understand why we can express them like

Cp = T(∂S/∂T)p
Cv = T(∂S/∂T)v

I know that Cp=(dQ/dT)p = (∂H/∂T)p with H equal to TdS + VdP and Cv=(dQ/dT)v = (∂U/∂T)v with U equal to TdS + PdV,

My guess: If I begin for instance with the enthalpy, H, and I constrain it to constant pressure I get just the TdS term, that is the thermodynamic definition of "Heat (Q)" . So I get the definition of the heat capacity if I derive Q respect to T, or TdS respect to T (is the same thing). Doing it, I get:

dQ/dT = (TdS)/dT = Td^2S/d^2T + dS/dT.

An expression very different of the definition.

Can anyone help me?

Thanks in advance.

( ∂Q/∂T )p = ( T∂S/∂T )p = T (∂S/∂T)p

Cp = T (∂S/∂T)p = (∂Q/∂T)p
Cv = T (∂S/∂T)v = (∂Q/∂T)v

 

[pabloenigma]

My guess: If I begin for instance with the enthalpy, H, and I constrain it to constant pressure I get just the TdS term, that is the thermodynamic definition of "Heat (Q)" . So I get the definition of the heat capacity if I derive Q respect to T, or TdS respect to T (is the same thing). Doing it, I get:

dQ/dT = (TdS)/dT = Td^2S/d^2T + dS/dT.

An expression very different of the definition.

The problem you have encountered in the italicized step is mathematical.
(TdS)/dT is T(dS/dT).But in the mentioned step,you have computed the derivative of (T(dS/dT)).
Read up on differentials.

 

[dapias09]

Thanks everybody,

pabloenigma you are right, your advice was very useful!

 

[dapias09]

Thank you