Thermodynamic derivation involving heat capacities

In summary, the question asks to derive a relationship between Cp-Cv and the isothermal compressibility (∂p/∂V)T and the coefficient of thermal expansion (∂V/∂T)p using the intensive entropy S as a function of T and V. The recommended steps include taking the partial derivative of S with respect to T while holding p constant, using a Maxwell relation to remove the partial derivative containing S, and applying the triple product rule. The resulting equation is Cp-Cv = -T(∂p/∂V)T(∂V/∂T)2p. The poster is seeking a thorough explanation of how to do this and acknowledges that they may have overlooked something.
  • #1
mcdonkdik
4
0
I have the answer to this question but I'm finding it hard making sense of it...

Q5) Dervive a relationship relating Cp-Cv to the isothermal compressibility (∂p/∂V)T and the coefficient of thermal expansion (∂V/∂T)p. Hint: consider the intensive entropy S as a function of T and V.

So I've started with S(T, V):

dS = (∂S/∂T)dT + (∂S/∂V)dV

Apparently we take the partial derivative wrt T while holding p(pressure) constant.. then we use a Maxwell relation to remove the partial derivative containing S. Then we use the triple product rule for something.

We end up with:

Cp - Cv = -T(∂p/∂V)T(∂V/∂T)2p



I'd really appreciate it if someone could give me a thorough explanation of how to do this.

Many thanks!
 
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  • #2
Hi mcdonkdik, welcome to PF. Nobody's going to solve your problem for you, but if you work through the recommended steps (which constitute the entire solution already!) and show where you get hung up, you'll likely get helpful comments.
 
  • #3
Mapes said:
Hi mcdonkdik, welcome to PF. Nobody's going to solve your problem for you, but if you work through the recommended steps (which constitute the entire solution already!) and show where you get hung up, you'll likely get helpful comments.

It's ok, I'm being dumb. This post can be deleted!

Thnx
 

1. What is thermodynamic derivation?

Thermodynamic derivation is the process of mathematically deriving thermodynamic equations and relationships using fundamental principles and laws, such as the laws of thermodynamics.

2. What does heat capacity mean?

Heat capacity is a measure of the amount of heat required to raise the temperature of a substance by a certain amount. It is often denoted as C and has units of energy per unit temperature (e.g. J/K).

3. How are heat capacities related to thermodynamics?

Heat capacities are important in thermodynamics because they describe how a substance responds to changes in temperature. They are used to calculate the amount of heat needed to raise the temperature of a substance, as well as to determine the change in internal energy and enthalpy of a system.

4. What is the difference between specific heat and molar heat capacity?

Specific heat is the amount of heat required to raise the temperature of 1 gram of a substance by 1 degree Celsius, while molar heat capacity is the amount of heat required to raise the temperature of 1 mole of a substance by 1 degree Celsius. Molar heat capacity takes into account the number of moles of a substance, making it a more useful measure for thermodynamic calculations.

5. How are heat capacities experimentally determined?

Heat capacities can be experimentally determined by measuring the change in temperature of a substance when a known amount of heat is added. This can be done using a calorimeter, which is a device that measures the transfer of heat between a system and its surroundings. The heat capacity can then be calculated using the equation Q = mCΔT, where Q is the heat added, m is the mass of the substance, C is the heat capacity, and ΔT is the change in temperature.

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