Thermodynamics: Finding the molar heat capacity

[Saitama]

Homework Statement

An ideal gas has a molar heat capacity $C_V$ at constant volume. Find the molar heat capacity of this gas as a function of its volume $V$, if the gas undergoes the following process: $T=T_0e^{\alpha V}$ here $T_0$ and $\alpha$ are constants.

Homework Equations

The Attempt at a Solution

Work done by gas: $\displaystyle W=\int PdV$
From ideal gas equation: $PV=nRT=nRT_0e^{\alpha V}$. Hence
$$W=\int_{V_1}^{V_2} \frac{nRT_0 e^{\alpha V}}{V}dV$$
But I cannot integrate this.

 

[Basic_Physics]

no work is done if the volume is kept constant - sorry I jumped too quickly on this.

 

[Saitama]

no work is done if the volume is kept constant

Yes but the question does not state that the volume is constant during the process.

 

[Basic_Physics]

It might help to integrate by parts.

 

[Saitama]

It might help to integrate by parts.

I plugged this in wolfram alpha: http://www.wolframalpha.com/input/?i=integrate+e^(ax)dx/x

I don't think this question requires me to use Exponential Integral as shown by wolfram alpha.

 

[ehild]

Ask yourself: What is heat capacity?

ehild

 

[Saitama]

Ask yourself: What is heat capacity?

ehild

Heat required to change the temperature of a body by a given amount.

I approached the problem the same way as I have done for the polytropic processes. I calculate the work done in those processes through integration in terms of $\Delta T$, change in temperature. I then write the change in internal energy in terms of $\Delta T$ and plug them in $Q=W+\Delta U$. Rewriting Q in terms of heat capacity, I obtain the answer. Is it incorrect to use the same approach here?

 

[ehild]

Heat required to change the temperature of a body by a given amount.

Change by what amount? Can you formulate the definition of heat capacity mathematically?

I approached the problem the same way as I have done for the polytropic processes. I calculate the work done in those processes through integration in terms of $\Delta T$, change in temperature. I then write the change in internal energy in terms of $\Delta T$ and plug them in $Q=W+\Delta U$. Rewriting Q in terms of heat capacity, I obtain the answer. Is it incorrect to use the same approach here?

How do you write Q (the total amount of heat during a process) in terms of heat capacity, when it changes during the process? Q depends on V and you integrate with respect to V.

Have you solved the problem? What is your solution?

ehild

 

[Saitama]

Change by what amount? Can you formulate the definition of heat capacity mathematically?

Sorry ehild, but I am less familiar with the technical terms.

$$C=\frac{\Delta Q}{n\Delta T}$$

How do you write Q (the total amount of heat during a process) in terms of heat capacity, when it changes during the process? Q depends on V and you integrate with respect to V.

Is it wrong to write $Q=nC\Delta T$ where C is the molar heat capacity?

 

[ehild]

C can depend on other state variables. Here it depends on V, and also on T as T is function of V. The total heat Q taken in a process is not necessarily proportional to ΔT. The heat taken by the system depends also on the process. How do your lecture notes or books define the molar heat capacity? Certainly not as you said that "Heat required to change the temperature of a body by a given amount." If the given amount is 218 K, and you need 1000 J heat to change the temperature by that amount than the heat capacity is 1000 J?


ehild

 

[Saitama]

C can depend on other state variables. Here it depends on V, and also on T as T is function of V. The total heat Q taken in a process is not necessarily proportional to ΔT. The heat taken by the system depends also on the process. How do your lecture notes or books define the molar heat capacity? Certainly not as you said that "Heat required to change the temperature of a body by a given amount." If the given amount is 218 K, and you need 1000 J heat to change the temperature by that amount than the heat capacity is 1000 J?


ehild

Sorry, I just looked up heat capacity in my book. It defines heat capacity as C=Q/ΔT.

I am still not sure how to begin with the problem.

 

[ehild]

The same way as you define speed as ds/dt, you can define heat capacity at a given state as small amount of heat divided by small change of temperature C=δQ/dt . The First Law of Thermodynamics says that dU=δQ+δW, that is, δQ=dU-δW. The small amount of heat taken is equal to the sum of the infinitesimal change of internal energy + infinitesimal amount of work.

dU=CvdT, δW=-pdV - write this in terms of V and dT. ehild

 

[Saitama]

δW=-pdV - write this in terms of V and dT.

I can replace P with $nRT/V=nRT_0e^{\alpha V}/V$ and $dV$ with $dT/(T_0 \alpha e^{\alpha V}$

$$dW=\frac{nRdT}{\alpha V}$$

 

[ehild]

Very good, plug into the equation δQ=dU-δW and find C=δQ/dt. (Recall that it is an ideal gas, so dU=Cvdt.)

ehild

 

[Saitama]

Very good, plug into the equation δQ=dU-δW and find C=δQ/dt. (Recall that it is an ideal gas, so dU=Cvdt.)

ehild

$$CdT=C_VdT+\frac{nRdT}{\alpha V}$$
I can cancel dT but what about n?

 

[ehild]

The molar heat capacity is the heat capacity of one mole gas. n=1.

 

[Saitama]

The molar heat capacity is the heat capacity of one mole gas. n=1.

Completely missed that, thanks!