Relationship between Heat Capacity Cv for Different Models

[A Story of a Student]

Homework Statement

Derived a relationship between heat capacity C_V described as ideal and van der waals gas.

Relevant Equations

Ideal gas/ Van der waals gas equations; Equation of heat capacity, and any thermodynamic equation that may be helpful

I think the C_V for van der waals gas will be larger than ideal gas since it‘s a more precise description. However, for the relationship I cannot come up with a specific equation.

 

[mfig]

Homework Statement:: Derived a relationship between heat capacity C_V described as ideal and van der waals I think the C_V for van der waals gas will be larger than ideal gas since it‘s a more precise description. However, for the relationship I cannot come up with a specific equation.

Instead of guessing, why not work it out and compare? Start with what you know about each model and go from there. If you are stuck, post what you have so far and people will help.

 

[Chestermiller]

What is the general equation for dU as a function of dT and dV?

 

[A Story of a Student]

What is the general equation for dU as a function of dT and dV?

$$dU=C_V dT+\left(\frac{\partial U}{\partial V}\right)_{T}dV$$

 

[Chestermiller]

$$dU=C_V dT+\left(\frac{\partial U}{\partial V}\right)_{T}dV$$

Are you also familiar with the following relationship?: $$\left(\frac{\partial U}{\partial V}\right)_T=-\left[P-T\left(\frac{\partial P}{\partial T}\right)_V\right]$$

 

[A Story of a Student]

Are you also familiar with the following relationship?: $$\left(\frac{\partial U}{\partial V}\right)_T=-\left[P-T\left(\frac{\partial P}{\partial T}\right)_V\right]$$

Yes! We have derived that as a homework assignment. However, how do I go from this to heat capacity?
Btw thanks for the help!

 

[mfig]

Yes! We have derived that as a homework assignment. However, how do I go from this to heat capacity?
Btw thanks for the help!

Substitute that expression, evaluated with VdW equation, into the one in post #4 and integrate.

 

[Chestermiller]

Yes! We have derived that as a homework assignment. However, how do I go from this to heat capacity?
Btw thanks for the help!

Substitute the van der waals equation into the right hand side of that relationship, and then take the partial derivative of both sides of the resulting equation with respect to T at constant V. What do you get?

 

[A Story of a Student]

I have tried to do that, I got $$dU=C_VdT+\frac{n^2a}{V^2}dV$$ and the integration (which I am not sure if this is correct) gives me $$\Delta U=C_V\Delta T-\frac{n^2a}{V}$$ Is so far the calculation correct?

 

[Chestermiller]

The equation is supposed to be in terms of specific volumes, not volumes, so lose the n's. I get:
$$\left(\frac{\partial U}{\partial V}\right)_T=-\frac{a}{V^2}$$So taking the partial derivative of this with respect to T at constant V yields: $$\frac{\partial ^2 U}{\partial V\partial T}=\frac{\partial ^2 U}{\partial T\partial V}=\frac{\partial C_v}{\partial V}=0$$What does this tell you about how Cv at a finite specific volume for a van der Waals gas compares with Cv for the same gas in the ideal gas limit of infinite specific volume.

 

[A Story of a Student]

specific volumes, not volumes, so lose the n's. I get:

​

Sorry I did not follow. For $$-[P-T\left(\frac{\partial P}{\partial T}\right)_V]=-[\frac{nRT}{V-nb}-\frac{n^2a}{V^2}-T\frac{nR}{V-nb}]=\frac{n^2a}{V^2}$$ right? Even dropping the n will give positive results.

So taking the partial derivatives with respect to T means Cv for vdw gas does not change with volume? This equation is derived based on the model of vdw gas. For ideal gas, dU/dV=0. That's all I've gotten so far...

 

[A Story of a Student]

Substitute that expression, evaluated with VdW equation, into the one in post #4 and integrate.

If the integration of post 9 were right, does it imply that for $$\Delta T C_{V,ideal}=\Delta U$$ since dU/dV=0; for vdw gas, we have an extra positive term added on the dU side that makes it bigger?
$$\Delta T C_{V,vdw}=\Delta U+\frac{n^2a}{V^2}$$

 

[mfig]

...the integration (which I am not sure if this is correct) gives me $$\Delta U=C_V\Delta T-\frac{n^2a}{V}$$ Is so far the calculation correct?

With indefinite integration, you would get:

$U = C_v T - a/V$

What this means is that the plot of U vs. T for some fixed volume VdW gass (think of a plot of U as a function of T for a fixed volume) is offset from the plot of U vs. T at that fixed volume for an IG, but the lines are parallel, i.e., they have the same slope. The slope of such a plot (U vs. T for fixed V) is some thermodynamic property (hint, hint)!

 

[Chestermiller]

Then you’re missing a factor of n in front of the Cv.

 

[Chestermiller]

A vdw gas approaches ideal gas at large specific volumes. So Cv for a vdw gas of any volume is equal to Cv for the same gas in the ideal gas limit.