[Thermodynamics] Heat Capacity for Polytropic Process

[mef51]

Consider an ideal gas. For a polytropic process we have $PV^n = const$. Different values of $n$ will represent different processes; for example isobaric ($n=0$), isothermal ($n=1$), and isochoric ($n=\infty$).

The Wikipedia article on polytropic processes states that the specific heat for a polytropic process is given by
$C_n = C_v \frac{\gamma - n}{1 - n}$ where $\gamma = \frac{C_P}{C_v}$

We can see that this expression for $C_n$ reduces to what we expect for the different values of n..
For example if $n = 0$ (isobaric) we have $C_0 = C_v \frac{\gamma}{1} = C_v \frac{C_P}{C_v} = C_P$.

How is the equation $C_n = C_v \frac{\gamma - n}{1 - n}$ derived?

 

[Chestermiller]

Consider an ideal gas. For a polytropic process we have $PV^n = const$. Different values of $n$ will represent different processes; for example isobaric ($n=0$), isothermal ($n=1$), and isochoric ($n=\infty$).

The Wikipedia article on polytropic processes states that the specific heat for a polytropic process is given by
$C_n = C_v \frac{\gamma - n}{1 - n}$ where $\gamma = \frac{C_P}{C_v}$

We can see that this expression for $C_n$ reduces to what we expect for the different values of n..
For example if $n = 0$ (isobaric) we have $C_0 = C_v \frac{\gamma}{1} = C_v \frac{C_P}{C_v} = C_P$.

How is the equation $C_n = C_v \frac{\gamma - n}{1 - n}$ derived?

I can help you with the derivation of this equation. But, before I do, please confirm that this is not a homework problem (or re-introduce it in one of the homework forums).

 

[mef51]

It's not a homework problem.
I had an attempt at it and this is what I got: (I used $\alpha$ as the polytropic index instead of $n$)

Using: $\delta Q = nC_\alpha dT$ and $dU = nC_vdT$
$$
\delta Q = dU + \delta W \\
nC_\alpha dT = nC_vdT + PdV \\
nC_\alpha \triangle T = nC_v\triangle T + \int \frac {PV^\alpha}{V^\alpha}dV \\
nC_\alpha \triangle T = nC_v\triangle T + PV^\alpha\int \frac {1}{V^\alpha}dV \\
nC_\alpha \triangle T = nC_v\triangle T + PV^\alpha\int \frac {1}{V^\alpha}dV \\
nC_\alpha \triangle T = nC_v\triangle T + PV^\alpha \frac{1}{-\alpha + 1}V^{-\alpha + 1} \\
nC_\alpha \triangle T = nC_v\triangle T + PV \frac{1}{-\alpha + 1} \\
C_\alpha = C_v + \frac{PV}{n\triangle T} \frac{1}{-\alpha + 1} \\
C_\alpha = C_v + \frac{R}{1 - \alpha} \\
$$

 

[Chestermiller]

It looks like you had the right idea, and your last equation is correct, but I'm not able to understand how you got to that point because of notational issues. For the work W, I got:

$$W=-\frac{P_iV_i}{(\alpha -1)}\left[\left(\frac{V_i}{V_f}\right)^{\alpha -1}-1\right]=-\frac{nRT_i}{(\alpha -1)}\left[\left(\frac{V_i}{V_f}\right)^{\alpha -1}-1\right]$$

Also, by combining the ideal gas law with the polytropic equation, I get:

$$\left(\frac{V_i}{V_f}\right)^{\alpha -1}=\frac{T_f}{T_i}$$

Combining these two equations, I get:
$$W=-\frac{nR}{(\alpha -1)}(T_f-T_i)$$

Substituting this into the first law gives:

$$nC_{\alpha}\Delta T=nC_v\Delta T-\frac{nR\Delta T}{\alpha -1}$$

Cancel the n and the delta T from both sides, reduce the right hand side to a common denominator, and factor the C_v from the right side, and you obtain the desired relationship.

Chet

 

[sardar]

The equation $C_n = C_v \frac{\gamma - n}{1 - n}$ is derived from the first law of thermodynamics, which states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system. For an ideal gas, the internal energy is a function of temperature only, and the work done is given by $PdV$. Therefore, the first law can be written as:

$$dU = C_VdT = \delta Q - PdV$$

where $C_V$ is the specific heat at constant volume, $dT$ is the change in temperature, $\delta Q$ is the heat added to the system, and $PdV$ is the work done by the system.

For a polytropic process, we have $PV^n = const$, which can be rewritten as $P = \frac{const}{V^n}$. Substituting this into the first law equation, we get:

$$C_VdT = \delta Q - \frac{const}{V^n}dV$$

Rearranging and integrating both sides, we get:

$$\int_{C_V}^{C_n}\frac{dQ}{T} = -\int_{V_1}^{V_2}\frac{const}{V^n}dV$$

where $C_n$ is the specific heat for the polytropic process and $V_1$ and $V_2$ are the initial and final volumes of the process. Using the definition of specific heat, $C = \frac{dQ}{dT}$, and integrating, we get:

$$C_n\ln\frac{T_2}{T_1} = -const\int_{V_1}^{V_2}\frac{dV}{V^n}$$

Solving for $C_n$, we get:

$$C_n = C_V\frac{T_2}{T_1}\frac{\ln\frac{T_2}{T_1}}{\ln\frac{V_2}{V_1}} = C_V\frac{T_2}{T_1}\frac{\ln\frac{T_2}{T_1}}{\ln\frac{P_2}{P_1}} = C_V\frac{T_2}{T_1}\frac{\ln\frac{T