[Thermodynamics] Heat Capacity for Polytropic Process

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  • #1
mef51
23
0
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?
 
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  • #2
mef51 said:
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).
 
  • #3
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##
$$
\delta Q = dU + \delta W \\
nC_\alpha dT = nC_vdT + PdV \\
nC_\alpha \triangle T = nC_v\triangle T + \int \frac {PV^\alpha}{V^\alpha}dV \\
nC_\alpha \triangle T = nC_v\triangle T + PV^\alpha\int \frac {1}{V^\alpha}dV \\
nC_\alpha \triangle T = nC_v\triangle T + PV^\alpha\int \frac {1}{V^\alpha}dV \\
nC_\alpha \triangle T = nC_v\triangle T + PV^\alpha \frac{1}{-\alpha + 1}V^{-\alpha + 1} \\
nC_\alpha \triangle T = nC_v\triangle T + PV \frac{1}{-\alpha + 1} \\
C_\alpha = C_v + \frac{PV}{n\triangle T} \frac{1}{-\alpha + 1} \\
C_\alpha = C_v + \frac{R}{1 - \alpha} \\
$$
 
  • #4
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:

[tex]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][/tex]

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

[tex]\left(\frac{V_i}{V_f}\right)^{\alpha -1}=\frac{T_f}{T_i}[/tex]

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

Substituting this into the first law gives:

[tex]nC_{\alpha}\Delta T=nC_v\Delta T-\frac{nR\Delta T}{\alpha -1}[/tex]

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

Chet
 
  • #5

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
 

FAQ: [Thermodynamics] Heat Capacity for Polytropic Process

What is heat capacity?

Heat capacity is a measure of the amount of heat energy needed to raise the temperature of a substance by one degree Celsius or Kelvin.

What is a polytropic process?

A polytropic process is a thermodynamic process in which the pressure and volume of a system are related by the equation P*V^n = constant, where n is a constant value.

How does the heat capacity change in a polytropic process?

In a polytropic process, the heat capacity can change depending on the value of n. For an ideal gas, the heat capacity is constant for a polytropic process with n = 0 or n = 1, but it varies for other values of n.

What is the equation for heat capacity in a polytropic process?

The equation for heat capacity in a polytropic process is C = (nR)/(n-1), where C is the heat capacity, R is the gas constant, and n is the polytropic index.

How does the heat capacity for a polytropic process compare to other types of thermodynamic processes?

The heat capacity for a polytropic process is generally higher than for an isothermal process and lower than for an adiabatic process. It is also dependent on the value of n, with lower values of n resulting in a higher heat capacity.

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