# Heat capacity under constant pressure or volume question

ho
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I have encountered a problem at the university in which there is a thermally isolated container of constant volume in which the number of particles and temperature change with time(the temperature increases). The change in particle number ensures constant pressure. The question was to find the heat that is needed to change that temperature.
So, naturally I would conclude from the beginning that δQ=0 since it's thermally isolated but then again, particles flow out of the system. So, I used δQ=C*dT.

Our professor said that C(the heat capacity) should be that of a process of constant pressure. But, I don't quite understand why. His reasoning was that the system in reality is not in constant volume since particles flow out of it. But, my reasoning is that while particles do flow out, the remaining particles spread out in the original volume of the container and it's just the density of the gas that changes.

Could someone explain to me what is the correct reasoning here?

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Chestermiller
Mentor
Please provide the exact wording of the problem.

hmmm27
Gold Member
.

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Please provide the exact wording of the problem.
A thermally isolated container contains ideal has of increasing temperature. Its volume is constant and the process is adiabatic. Also, there is a tiny hole so that the inside of the container is in contact with the outside of it which has pressure of 1 atm. We need to find the heat that causes that temperature change.
So, to solve the problem I essentially used that the pressure inside the container is always 1atm and that particles are exchanged in order for the temperature to change. I think there is no other way since the process is adiabatic and isochoric.
But, as my professor pointed out, by solving it through the relation δQ=C*dT with C the heat capacity of the gas. I have solved the problem but my professor pointed out that C should be of constant pressure rather than constant volume. I don't understand that. Since both pressure and volume are constant, why should I use the heat capacity under constant pressure? His argument was that since particles are flowing out of the system, its volume changes. But, the volume is constant as stated in the problem.

Assumedly it's a tank with a baggie on the end for overflow gas.

In which case you'll probably be heating all the gas, unless there's a one-way valve. So - constant pressure: yes; constant volume, no
Constant volume is stated from the beginning. It's a given of the problem. Also given is that it's adiabatic and the container has a hole that allows contact with air of P=1atm.

Chestermiller
Mentor
A thermally isolated container contains ideal has of increasing temperature. Its volume is constant and the process is adiabatic. Also, there is a tiny hole so that the inside of the container is in contact with the outside of it which has pressure of 1 atm. We need to find the heat that causes that temperature change.
If the process is adiabatic, how can the temperature be increasing? Also, if gas is escaping through a hole, the number of moles of gas in the container is changing.

If the process is adiabatic, how can the temperature be increasing? Also, if gas is escaping through a hole, the number of moles of gas in the container is changing.
Well, that's the whole point of the problem. The number of particles is changing and so the temperature changes.

hmmm27
Gold Member
It makes at least a bit of sense if it isn't described as adiabatic. Heat the tank, the air warms, some warm air escapes, apply calculus.

Chestermiller
Mentor
It makes at least a bit of sense if it isn't described as adiabatic. Heat the tank, the air warms, some warm air escapes, apply calculus.
Adiabatic means that there is no heat transfer. So, it makes no sense.

Adiabatic means that there is no heat transfer. So, it makes no sense.
Well, that's the problem I was given..

Adiabatic means that there is no heat transfer. So, it makes no sense.

It makes at least a bit of sense if it isn't described as adiabatic. Heat the tank, the air warms, some warm air escapes, apply calculus.
PV=k(NT). Because PV is constant, while T is changing, so does N. So, also (NT) is constant and so is the energy of the container because its described as E=3/2k(NT).

hmmm27
Gold Member
Oh, I see... start off with a cold chamber... still not adiabatic unless there's an outer chamber.

Chestermiller
Mentor
PV=k(NT). Because PV is constant, while T is changing, so does N. So, also (NT) is constant and so is the energy of the container because its described as E=3/2k(NT).
I agree with you and not your professor. You should be using Cv and not Cp, but that is because the focus should be on the internal energy of the remaining gas in the container, and not the enthalpy. Also, the gas which leaves the tank takes energy with it, and the gas that has left was at a lower temperature than the gas that has remained. I will be back later to provide a more complete analysis of this of this problem.

Chet

Chestermiller
Mentor
The open system (control volume) version of the first law of thermodynamics applied to this particular problem gives:
$$\frac{dU}{dt}=\frac{dQ}{dt}-\dot{m}h\tag{1}$$where U is the internal energy of the gas remaining in the container, Q is the cumulative amount of heat added, ##\dot{m}## is the molar flow rate out of the container, and h is the enthalpy per mole of the stream exiting the container. The internal energy U is given by:$$U=nC_v(T-T_0)\tag{2}$$ where n(t) is the number of moles of gas remaining in the container at time t, T(t) is the gas temperature in the tank at time t, and ##T_0## is the arbitrary datum temperature for zero internal energy. The number of moles in the container at time t is given by:$$n=\frac{PV}{RT}\tag{3}$$, where P is the the pressure (1 atm.) and V is the volume of the container. The enthalpy per mole of the stream leaving the container at time t is given by: $$h=C_v(T-T_0)+Pv=C_v(T-T_0)+RT\tag{4}$$ where v is the molar volume of the exit stream, which is equal to RT/P. The molar flow rate ##\dot{m}## out of the container is equal to minus the rate of change of the number of moles of gas inside the container:$$\dot{m}=-\frac{dn}{dt}=\frac{PV}{RT^2}\frac{dT}{dt}\tag{5}$$If we substitute Eqns. 2-5 into Eqn. 1, we obtain:
$$\frac{dQ}{dt}=C_p\frac{PV}{RT}\frac{dT}{dt}\tag{6}$$where ##C_p## is the molar heat capacity at constant pressure ##(=C_v+R)##. We can immediately integrate Eqn. 6 subject to the initial condition ##T=T_i## at t = 0 to obtain:
$$Q=C_p\frac{PV}{R}\ln{(T/T_i)}\tag{7}$$

The open system (control volume) version of the first law of thermodynamics applied to this particular problem gives:
$$\frac{dU}{dt}=\frac{dQ}{dt}-\dot{m}h\tag{1}$$where U is the internal energy of the gas remaining in the container, Q is the cumulative amount of heat added, ##\dot{m}## is the molar flow rate out of the container, and h is the enthalpy per mole of the stream exiting the container. The internal energy U is given by:$$U=nC_v(T-T_0)\tag{2}$$ where n(t) is the number of moles of gas remaining in the container at time t, T(t) is the gas temperature in the tank at time t, and ##T_0## is the arbitrary datum temperature for zero internal energy. The number of moles in the container at time t is given by:$$n=\frac{PV}{RT}\tag{3}$$, where P is the the pressure (1 atm.) and V is the volume of the container. The enthalpy per mole of the stream leaving the container at time t is given by: $$h=C_v(T-T_0)+Pv=C_v(T-T_0)+RT\tag{4}$$ where v is the molar volume of the exit stream, which is equal to RT/P. The molar flow rate ##\dot{m}## out of the container is equal to minus the rate of change of the number of moles of gas inside the container:$$\dot{m}=-\frac{dn}{dt}=\frac{PV}{RT^2}\frac{dT}{dt}\tag{5}$$If we substitute Eqns. 2-5 into Eqn. 1, we obtain:
$$\frac{dQ}{dt}=C_p\frac{PV}{RT}\frac{dT}{dt}\tag{6}$$where ##C_p## is the molar heat capacity at constant pressure ##(=C_v+R)##. We can immediately integrate Eqn. 6 subject to the initial condition ##T=T_i## at t = 0 to obtain:
$$Q=C_p\frac{PV}{R}\ln{(T/T_i)}\tag{7}$$
Yes, his was also the solution. From your previous comment, the temperature is rising so we are supposed to conclude that particles are added in the system.
Also, as I do not know about enthalpy yet, I did not understand your argument about using Cv over Cp. I don't suggest using Cv instead of Cp, I am just saying that I do not understand why distinguish between the two in a problem where both pressure and volume remain constant.
Also, thanks for the detailed analysis.

Chestermiller
Mentor
Actually, as the contents of the container is heated, molecules are leaving the container, not being added.

Eqn. 6 of my previous post really tells an important story. Written is a slightly different way, it reads:
$$dQ=\frac{PV}{RT}C_pdT=nC_pdT$$But this is the equation that you would get if the gas was being heated at constant pressure (as your professor contended), not at constant volume. But, how can this be if the container is constant volume? There is a simple physically intuitive way of resolving this issue. Imagine that, at time t, the temperature of the gas in the container is T and the number of moles is n. Picture an imaginary membrane surrounding the gas within the container at this time. Now imagine that, during the time interval between t and t + dt, you add a small amount of heat dQ to the gas. The temperature of the gas in the container will rise by dT, and, because the pressure is constant (i.e., the gas is in contact with the atmosphere through the hole in the container), the membrane that contains the n moles of gas will have to bulge out through the hole in the container to allow the volume to increase slightly. So the net effect is that, over the time interval between t and t + dt, the gas that was in the container at time t expands at constant pressure as a result of adding dQ. Once the time interval dt is over, we put a new membrane around the gas in the container at time t + dt, and start over.

Hope this helps.

Actually, as the contents of the container is heated, molecules are leaving the container, not being added.

Eqn. 6 of my previous post really tells an important story. Written is a slightly different way, it reads:
$$dQ=\frac{PV}{RT}C_pdT=nC_pdT$$But this is the equation that you would get if the gas was being heated at constant pressure (as your professor contended), not at constant volume. But, how can this be if the container is constant volume? There is a simple physically intuitive way of resolving this issue. Imagine that, at time t, the temperature of the gas in the container is T and the number of moles is n. Picture an imaginary membrane surrounding the gas within the container at this time. Now imagine that, during the time interval between t and t + dt, you add a small amount of heat dQ to the gas. The temperature of the gas in the container will rise by dT, and, because the pressure is constant (i.e., the gas is in contact with the atmosphere through the hole in the container), the membrane that contains the n moles of gas will have to bulge out through the hole in the container to allow the volume to increase slightly. So the net effect is that, over the time interval between t and t + dt, the gas that was in the container at time t expands at constant pressure as a result of adding dQ. Once the time interval dt is over, we put a new membrane around the gas in the container at time t + dt, and start over.

Hope this helps.
But,shouldn't we analyze what is happening in the system using something like a control volume that is inside the container since we are concerned only for what is inside of it? Also, from PV=NkT, because PV is constant, isn't also NT? So, as the temperature rises, the number of molecules decreases.

Chestermiller
Mentor
But,shouldn't we analyze what is happening in the system using something like a control volume that is inside the container since we are concerned only for what is inside of it?
That's what my previous analysis did. It treated the container as a control volume, and allowed for gas escaping. But, it the end, it gave exactly the same answer.
Also, from PV=NkT, because PV is constant, isn't also NT? So, as the temperature rises, the number of molecules decreases. w
The number of molecules inside the container decreases, but the number of molecules that exited the container is contained within the bulge in the imaginary membrane outside the container. The imaginary membrane allows us to focus on a closed system in which the total number of molecules is constant during the time interval dt.

Chestermiller
Mentor
In my judgment, this problem involves way too much of a conceptual leap for a novice student such as yourself to be subjected to. You should not feel bad that you have been struggling with it. I think your professor should have held off until you had learned about the control volume version of the first law.

Chet

That's what my previous analysis did. It treated the container as a control volume, and allowed for gas escaping. But, it the end, it gave exactly the same answer.

The number of molecules inside the container decreases, but the number of molecules that exited the container is contained within the bulge in the imaginary membrane outside the container. The imaginary membrane allows us to focus on a closed system in which the total number of molecules is constant during the time interval dt.
So, we can solve the problem with a moving control volume where the particle number does not change and we can also solve it by considering a static control volume(that of the inside of the container) with changing particle number, right? But, because in reality the volume does change(like in the moving control volume case), we must use Cp rather than Cv. Did I get it right?

In my judgment, this problem involves way too much of a conceptual leap for a novice student such as yourself to be subjected to. You should not feel bad that you have been struggling with it. I think your professor should have held off until you had learned about the control volume version of the first law.

Chet
Well, the course is considered as being an advanced undergraduate thermodynamics and statistical physics course, but the lectures give too simple examples for us to be able to solve problems like this. In any way, in the end I learned a lot..

Chestermiller
Mentor
So, we can solve the problem with a moving control volume where the particle number does not change and we can also solve it by considering a static control volume(that of the inside of the container) with changing particle number, right? But, because in reality the volume does change(like in the moving control volume case), we must use Cp rather than Cv. Did I get it right?
Yes. Very nice.

• Joker93
Yes. Very nice.
Thanks a lot for the help!

Yes. Very nice.
Can I also ask something else? When particles are extracted from the system, why is work equal to zero? Or, put differently, when the control volume deforms, isn't the work being done on the system not equal to zero?
In your solution, I think that you implied that the work being done on the system is zero. Also, my professor's solution also uses that because the full solution uses the fact that δQ=Cp*dT.

@Chestermiller I think I got it: Because we are talking about an ideal gas where its particles are not interacting(the forces between them are nonexistent), the work that the particles do to each other as they get out of the container is zero. Thus, the change in temperature is due to the flow of particles that flow outside the container which causes a heat flow(or just heat) outside the container.
Did I get it right?
If I did, then if the gas was not ideal, then we should have also considered changes in work of the system. And because (I think) the process is irreversible, we couldn't find exactly how much heat was exchanged, right?

Chestermiller
Mentor
Can I also ask something else? When particles are extracted from the system, why is work equal to zero? Or, put differently, when the control volume deforms, isn't the work being done on the system not equal to zero?
In your solution, I think that you implied that the work being done on the system is zero. Also, my professor's solution also uses that because the full solution uses the fact that δQ=Cp*dT.
This equation does not imply that the work being done is zero. For that to be the case, there would have to be a Cv in the equation, not a Cp. The gas in the container is doing work to force the gas ahead of it out of the container. So most of the gas in the container is doing work at constant pressure. This is all easier to visualize if you imagine that the gas is a continuum rather then being comprised of individual molecules (particles). Do you think you can do that?