Derive the expression for isothermal change in Constant Volume Heat Capacity

[eagerbeaver92]

Homework Statement

Derive the following expression for calculating the isothermal change in Constant Volume Heat Capacity:

(dC_v/dV)_T = T(d²P/dT²)_V

Homework Equations

The Attempt at a Solution

I have no idea, please help

 

[eagerbeaver92]

How about this?

assuming ideal gas:

P=nrt/V

then

(dP/dT) = R/V , ignoring moles

and

(d₂P/d²T) = 0

so (dC^v/dV)=T*0=0

 

[tomcue]

.

Sure, I can help you with this problem. The isothermal change in constant volume heat capacity, also known as the isochoric heat capacity, is defined as the change in heat capacity at constant volume with respect to temperature. Mathematically, it can be expressed as:

Cv = (dQ/dT)V

where Cv is the constant volume heat capacity, dQ is the change in heat, and dT is the change in temperature. Now, let's take the derivative of this equation with respect to volume at constant temperature:

(dCv/dV)T = (d/dV)(dQ/dT)V

Using the chain rule, we can rewrite this as:

(dCv/dV)T = (d/dT)(dQ/dV)T * (dT/dV)T

Since we are considering an isothermal process, the change in temperature (dT) is equal to zero. Therefore, the last term becomes zero and we are left with:

(dCv/dV)T = (d/dT)(dQ/dV)T

Now, we can use the Maxwell relation for the change in pressure with respect to temperature at constant volume, which is given by:

(dP/dT)V = (d2Q/dT2)V

Substituting this into our equation, we get:

(dCv/dV)T = (d/dT)(dQ/dV)T = (d/dT)(dP/dT)V = (d2P/dT2)V

And there you have it, we have derived the expression for calculating the isothermal change in constant volume heat capacity. I hope this helps! Let me know if you have any further questions.