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Tian En
For an adiabatic process, Q = 0.
From the first law of thermodynamic,
ΔU = Q + W on the system
when Q = 0,
W = -PΔV,
then why is it that ΔU = Cv ΔT when Cv is meant for the constant volume? We know that when there is work done, the volume is changing, and making use of Cv sounds like an contradiction. Please enlighten me. Thank you.

mjc123
Homework Helper
Let's split it into two processes: Change temperature from T1 to T2 at constant volume ΔU1 = Cv ΔT
Change volume from V1 to V2 at constant temperature ΔU2 = 0
ΔU = ΔU1 + ΔU2 = Cv ΔT
As U is a state function, ΔU is independent of pathway.

Chestermiller
Mentor
Let's split it into two processes: Change temperature from T1 to T2 at constant volume ΔU1 = Cv ΔT
Change volume from V1 to V2 at constant temperature ΔU2 = 0
ΔU = ΔU1 + ΔU2 = Cv ΔT
As U is a state function, ΔU is independent of pathway.
This is true only for an ideal gas (since, for an ideal gas, the internal energy is a function only of temperature).

Tian En
Since U is a state function, ΔU independent of pathway, it means I can also use CpΔT = -PΔV, right?

Chestermiller
Mentor
Since U is a state function, ΔU independent of pathway, it means I can also use CpΔT = -PΔV, right?
No. $$\Delta U=nC_v\Delta T$$. On the other hand, $$\Delta H=nC_p\Delta T$$

Tian En
I see. I have yet to learn enthalpy.
let's say for an adiabatic process, Q = 0, ΔU = W on the system = -PΔV, and ΔU = nCvΔT = -PΔV.
If we keep the pressure P, number of moles n and molar specific heat capacity Cv constant and we compress the system by ΔV (-ve) and hence we have -PΔV (+ve) and therefore the ΔT (+ve). However, the ideal gas law PV = nRT => PΔV = nRΔT implies ΔV (+ve) and ΔT (+ve) for constant n, R and P.

Previously, we have seen that ΔV (-ve) => ΔT (+ve), but now ΔV (+ve) => ΔT (+ve) from ideal gas law. Is there a contradiction?

Chestermiller
Mentor
I see. I have yet to learn enthalpy.
let's say for an adiabatic process, Q = 0, ΔU = W on the system = -PΔV, and ΔU = nCvΔT = -PΔV.
If we keep the pressure P, number of moles n and molar specific heat capacity Cv constant and we compress the system by ΔV (-ve) and hence we have -PΔV (+ve) and therefore the ΔT (+ve). However, the ideal gas law PV = nRT => PΔV = nRΔT implies ΔV (+ve) and ΔT (+ve) for constant n, R and P.

Previously, we have seen that ΔV (-ve) => ΔT (+ve), but now ΔV (+ve) => ΔT (+ve) from ideal gas law. Is there a contradiction?
Please tell me how you think you are going to compress the gas is you keep the pressure constant and don't remove heat.

Tian En
So, this adiabatic equation no longer hold because by keeping the pressure constant, it violates the ideal gas law and by adding the term Q (-ve) to the equation, it becomes: ΔU = nCvΔT = -PΔV + Q (Isobaric equation)

Chestermiller
Mentor
So, this adiabatic equation no longer hold because by keeping the pressure constant, it violates the ideal gas law and by adding the term Q (-ve) to the equation, it becomes: ΔU = nCvΔT = -PΔV + Q (Isobaric equation)
I have no idea what you are talking about. All I can say is that, if you don't increase the pressure, you can't compress the gas (if the system is adiabatic).

HimanshuM2376
For an adiabatic process, Q = 0.
From the first law of thermodynamic,
ΔU = Q + W on the system
when Q = 0,
W = -PΔV,
then why is it that ΔU = Cv ΔT when Cv is meant for the constant volume? We know that when there is work done, the volume is changing, and making use of Cv sounds like an contradiction. Please enlighten me. Thank you.
∆U=Cv∆T comes from the fact that whatever heat is added to an ideal gas keeping it's volume constant will only increase its internal energy and for an ideal gas internal energy is a function of temperature only.
Now we imagine that the system is undergoing an adiabatic process in which the heat added to the system is zero, and as per our knowledge of first law we know that the change in internal energy is a point function. Therefore if the states 1 and 2 are defined then ∆U will remain the same irrespective of the path undergone by a system . So in an adiabatic process ∆U =Cv∆T = W.
Hope this helps you.

Tian En
Tian En
I have no idea what you are talking about. All I can say is that, if you don't increase the pressure, you can't compress the gas (if the system is adiabatic).
I was just trying to relate the formula to physical impossibility. When there are inconsistencies in the formula, it is impossible to happen physically. i.e. By keeping the pressure constant, it can never be adiabatic, because Q is no longer 0.
Hope this helps you.
It helps. Thank you. Since the change internal energy just as a function of temperature and previously I learnt that ΔU ≠ nCpΔT but ΔU = nCvΔT, is it possible to relate Cp and ΔT to the ΔU?

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Chestermiller
Mentor
I was just trying to relate the formula to physical impossibility. When there are inconsistencies in the formula, it is impossible to happen physically. i.e. By keeping the pressure constant, it can never be adiabatic, because Q is no longer 0.

It helps. Thank you. Since the change internal energy just as a function of temperature, I am looking at how to relate Cp and ΔT to the ΔU since previously I learnt that ΔU ≠ nCpΔT but ΔU = nCvΔT, is it possible?
##\Delta U=nC_p\Delta T## is just plain incorrect. If you learnt it that way, you learned it wrong.

Tian En
I like Serena
Homework Helper
It helps. Thank you. Since the change internal energy just as a function of temperature and previously I learnt that ΔU ≠ nCpΔT but ΔU = nCvΔT, is it possible to relate Cp and ΔT to the ΔU?
For an ideal gas we have Cp=Cv+R (see e.g. heat capacity on wiki), so ΔU = n(Cp-R)ΔT.

Tian En
Great, thank you.