Solving for λ in an Adiabatic Process: Applying the Ideal Gas Equation"

[S_Flaherty]

Thermodynamics - adiabatic process

Homework Statement

The question is: Consider a hypothetical ideal gas with internal energy U = NkT_o(T/T₀)^α+1, where T_o and α are positive constants. Show that in an adiabatic process, V*exp[(1+1/α)(T/T_o)^α] = constant.

Homework Equations

PV^γ = constant
γ = C_p/C_v
C_p = C_v + Nk

The Attempt at a Solution

I'm pretty sure that I'm supposed to show that [(1+1/α)(T/T_o)^α] is equal to γ and since PV^γ = constant, V*exp[(1+1/α)(T/T_o)^α] = constant. When I try to solve it though I can't get the solution to come out. I differentiate U to get C_v = Nk(1+α)(T/T_o)^1+α. When I plug that into γ I get γ = 1 + 1/[(1+α)(T/To)^α]. Either I'm just not simplifying it enough and the answer is correct, or I solved for λ incorrectly, or my equations are incorrect. I don't know which it is though.

 

[nasu]

Homework Statement

The question is: Consider a hypothetical ideal gas with internal energy U = NkT_o(T/T₀)^α+1, where T_o and α are positive constants. Show that in an adiabatic process, V*exp[(1+1/α)(T/T_o)^α] = constant.

Homework Equations

PV^γ = constant
γ = C_p/C_v
C_p = C_v + Nk

The Attempt at a Solution

I'm pretty sure that I'm supposed to show that [(1+1/α)(T/T_o)^α] is equal to γ and since PV^γ = constant, V*exp[(1+1/α)(T/T_o)^α] = constant. When I try to solve it though I can't get the solution to come out. I differentiate U to get C_v = Nk(1+α)(T/T_o)^1+α. When I plug that into γ I get γ = 1 + 1/[(1+α)(T/To)^α]. Either I'm just not simplifying it enough and the answer is correct, or I solved for λ incorrectly, or my equations are incorrect. I don't know which it is though.

The equation
PV^γ = constant
is not valid for this problem. It follows from the usual internal energy for ideal gas,
U(T)=nCvT

Here you have a different function U(T) and you have to find the relationship between volume and temperature. (for the "usual" ideal gas this will be TV^γ-1 = constant )
You can use the first law in conjunction with the equation of state to do this.

 

[ehild]

Homework Statement

The question is: Consider a hypothetical ideal gas with internal energy U = NkT_o(T/T₀)^α+1, where T_o and α are positive constants. Show that in an adiabatic process, V*exp[(1+1/α)(T/T_o)^α] = constant.

Homework Equations

PV^γ = constant
γ = C_p/C_v
C_p = C_v + Nk

The Attempt at a Solution

I'm pretty sure that I'm supposed to show that [(1+1/α)(T/T_o)^α] is equal to γ and since PV^γ = constant, V*exp[(1+1/α)(T/T_o)^α] = constant. When I try to solve it though I can't get the solution to come out. I differentiate U to get C_v = Nk(1+α)(T/T_o)^1+α. When I plug that into γ I get γ = 1 + 1/[(1+α)(T/To)^α]. Either I'm just not simplifying it enough and the answer is correct, or I solved for λ incorrectly, or my equations are incorrect. I don't know which it is though.

Is PV^γ = constant when Cv depends on T?

ehild

 

[S_Flaherty]

The equation
PV^γ = constant
is not valid for this problem. It follows from the usual internal energy for ideal gas,
U(T)=nCvT

Here you have a different function U(T) and you have to find the relationship between volume and temperature. (for the "usual" ideal gas this will be TV^γ-1 = constant )
You can use the first law in conjunction with the equation of state to do this.

I'm not really sure what you mean, can you explain it more?

 

[S_Flaherty]

Is PV^γ = constant when Cv depends on T?

ehild

I'm guessing it's not, but I don't know what it should be then.

 

[Chestermiller]

I'm not really sure what you mean, can you explain it more?

For an ideal gas,
dU=C_vdT
PV=RT
From the first law, for an adiabatic reversible process, how is dU related to PdV?

 

[S_Flaherty]

For an ideal gas,
dU=C_vdT
PV=RT
From the first law, for an adiabatic reversible process, how is dU related to PdV?

dU = -PdV, so C_v = -PdV/dT right?

 

[ehild]

U is given as function of T. PV=kNT is valid for the ideal gas, and also the First Law is valid. For an adiabatic process dU=-PdV. Use P=kNT/V, and integrate.

ehild

 

[S_Flaherty]

U is given as function of T. PV=kNT is valid for the ideal gas, and also the First Law is valid. For an adiabatic process dU=-PdV. Use P=kNT/V, and integrate.

ehild

Ok, so I get U = -kNT(ln(V))

 

[haruspex]

κ
I'm pretty sure that I'm supposed to show that [(1+1/α)(T/T_o)^α] is equal to γ and since PV^γ = constant, V*exp[(1+1/α)(T/T_o)^α] = constant.

No, that would not follow, so I don't think you want to show that [(1+1/α)(T/T_o)^α] is equal to γ. Instead, try raising V*exp[(1+1/α)(T/T_o)^α] to the power of γ, using the expression for γ that you derived.

 

[S_Flaherty]

No, that would not follow, so I don't think you want to show that [(1+1/α)(T/T_o)^α] is equal to γ. Instead, try raising V*exp[(1+1/α)(T/T_o)^α] to the power of γ, using the expression for γ that you derived.

So that makes (V*exp[(1+1/α)(T/T_o)^α])^{1+1/(1+α)(T/T_o)α}

I'm not sure what to do with that

 

[nasu]

I'm not really sure what you mean, can you explain it more?

Well, I don't understand what part you don't understand.

But the idea is "forget gamma". And "forget pv^gamma". Does not apply here.

1. From first law applied to adiabatic process you have:
dU=pdV
You have U(T) so find dU.

2. You have PV=nRT so you can eliminate p on the right hand side:
pdV= nRTdV/V

So you will have an equation relating V and T. Integrate (after separating variables) and you'll find that exponential relationship.

 

[Chestermiller]

You have the equation of U as a function of T, and you know know that
$$C_v=\frac{\partial U}{\partial T}$$
Just differentiate the equation for U with respect to T, and write
$$dU=C_vdT=\frac{\partial U}{\partial T}dT=-PdV$$
Then, just substitute the ideal gas law for P, and integrate.

 

[nasu]

You have the equation of U as a function of T, and you know know that
$$C_v=\frac{\partial U}{\partial T}$$
.

This is a "hypothetical" ideal gas.
Cv=∂U/∂T is valid for a "real" ideal gas.

The whole point here is that U(T) is not given by
dU=CvdT but by that other, more complicated formula.
If he does what you suggest he'l get just the usual
$$TV^{\gamma -1 }= constant$$ and not the formula required by the problem.

But the method will work. This is what I tried to explain as well.
Just use
$$dU=Nk(\alpha +1) (T/T_0)^{\alpha} dT$$.

There is no need to introduce Cv or gamma.

 

[DrClaude]

This is a "hypothetical" ideal gas.
Cv=∂U/∂T is valid for a "real" ideal gas.

I have to disagree. Cv=∂U/∂T follows simply from the definition of heat capacity,
$$
C = \frac{Q}{\Delta T}
$$
by considering a constant volume (hence $W=0$), without invoking an ideal gas.

 

[nasu]

I have to disagree. Cv=∂U/∂T follows simply from the definition of heat capacity,
$$
C = \frac{Q}{\Delta T}
$$
by considering a constant volume (hence $W=0$), without invoking an ideal gas.

Did I say anything that seem to contradict your statement? I just meant just that you don't need Cv to solve the problem. It does not appear in this problem.
Oh, I see. I used partial derivatives.

I meant that dU=CvdT may not apply to other systems.
It is valid only for some systems, like ideal gas in the "proper" definition.


So dU=Nk(α+1)(T/T0)^α dT
You don't need to define or use a specific heat to solve the problem.
Sorry for the confusion.

 

[Chestermiller]

This is a "hypothetical" ideal gas.
Cv=∂U/∂T is valid for a "real" ideal gas.

The whole point here is that U(T) is not given by
dU=CvdT but by that other, more complicated formula.
If he does what you suggest he'l get just the usual
$$TV^{\gamma -1 }= constant$$ and not the formula required by the problem.

But the method will work. This is what I tried to explain as well.
Just use
$$dU=Nk(\alpha +1) (T/T_0)^{\alpha} dT$$.

There is no need to introduce Cv or gamma.

This is exactly what I was suggesting. I brought the heat capacity into the picture because I felt the OP would feel more comfortable with it. For this particular ideal gas, C_v is not independent of temperature, but is given by:
$$C_v=Nk(\alpha +1) (T/T_0)^{\alpha}$$
Are you uncomfortable with an ideal gas heat capacity that varies with temperature. A temperature-dependent heat capacity is part of the definition of an ideal gas that we engineers use.

 

[nasu]

Are you uncomfortable with an ideal gas heat capacity that varies with temperature. A temperature-dependent heat capacity is part of the definition of an ideal gas that we engineers use.

Is this a question?
I don't feel any discomfort about temperature variation of Cv or about Cv in general. Even Cp it's bearable, despite all these pressure variations.

 

[Chestermiller]

Is this a question?
I don't feel any discomfort about temperature variation of Cv or about Cv in general. Even Cp it's bearable, despite all these pressure variations.

Oops. I left out the question mark. Thank you for serving as the grammar police enforcer.

Getting back to the thread, I think we are (and were) totally in agreement on how this problem should be solved. Of course, for an ideal gas, C_p is also a function only of temperature.
Chet

 

[nasu]

I agree that we are in agreement.
It was not intended as police work. Just curious.

 

[DrClaude]

Did I say anything that seem to contradict your statement?

Yes, you did, which is why I wanted to point it out. You said:

This is a "hypothetical" ideal gas.
Cv=∂U/∂T is valid for a "real" ideal gas.

I don't see how to read this other than Cv=∂U/∂T is valid only for a "real" ideal gas, not for this "hypothetical" ideal gas. This statement is not correct, as Cv=∂U/∂T is universally valid, except if a phase transition occurs. This may not have been what you were thinking when you wrote that, but I wanted to make things clear.

 

[ehild]

Ok, so I get U = -kNT(ln(V))

U is given in the OP, and it is explicitly independent on the volume, it is function of T only. But V depends on T. You have to find the relationship between T and V in an adiabatic process, when dU=-PdV. From here, you get a differential equation relating V and T, that you have to integrate. No need to mix gamma in.

ehild

 

[nasu]

Yes, you did, which is why I wanted to point it out. You said:

I don't see how to read this other than Cv=∂U/∂T is valid only for a "real" ideal gas, not for this "hypothetical" ideal gas. This statement is not correct, as Cv=∂U/∂T is universally valid, except if a phase transition occurs. This may not have been what you were thinking when you wrote that, but I wanted to make things clear.

Yes, I realized that.
As I already said in my post.