Thermodynamics problem: Heat capacity of a Gas

[Krushnaraj Pandya]

Homework Statement

In a certain process, a gas absorbs Q amount of heat and performs kQ amount of work, the molar heat capacity of the gas in terms of R, k and γ(Cp/Cv) is?

Homework Equations

U=Q+W
U=nCvdT
Q=CdT

The Attempt at a Solution

replacing U and Q with the above formulas and W from the given information gives C = (CvdT + kQ)/dT How do I proceed further?

 

[Chestermiller]

Homework Statement

In a certain process, a gas absorbs Q amount of heat and performs kQ amount of work, the molar heat capacity of the gas in terms of R, k and γ(Cp/Cv) is?

Homework Equations

U=Q+W
U=nCvdT
Q=CdT

The Attempt at a Solution

replacing U and Q with the above formulas and W from the given information gives C = (CvdT + kQ)/dT How do I proceed further?

This question makes no sense to me. The heat capacity is supposed to be a physical property of the material, independent of any process that the material is subjected to.

Was this the exact statement (word for word) of the problem? What book did this problem come from?

 

[Krushnaraj Pandya]

This question makes no sense to me. The heat capacity is supposed to be a physical property of the material, independent of any process that the material is subjected to.

Was this the exact statement (word for word) of the problem? What book did this problem come from?

well, I saw a formula that for a polytropic process with index n, the heat capacity is C=Cv + R/1-x.
I have written the exact wording, the answer given is R/(γ-1)(1-k) the book is a preparatory module for the JEE Advanced examination in India

 

[Chestermiller]

Well, I can help you solve this problem but, with the following caveat: I think it is a very bad idea to define heat capacity in terms of the temperature change and heat added for a specified process path. The heat capacity is supposed to be a physical property of the material being processed, independent of process path. Defining it in terms of heat (process path) causes nothing but confusion for thermodynamics students later on. For example, when the student studies adiabatic expansions and compressions of gases, the heat Q is zero, but there is a temperature change. So the student calculates that the heat capacity of the gas is zero. Instant confusion! With this warning, I will now help you solve this.

You indicated that the change in internal energy is given by $$\Delta U=C_v\Delta T=Q-W$$where W is the work done by the gas on its surroundings. If you substitute W from the problem statement into this equation, what do you get for the temperature change $\Delta T$?

 

[Navin]

Well, I can help you solve this problem but, with the following caveat: I think it is a very bad idea to define heat capacity in terms of the temperature change and heat added for a specified process path. The heat capacity is supposed to be a physical property of the material being processed, independent of process path. Defining it in terms of heat (process path) causes nothing but confusion for thermodynamics students later on. For example, when the student studies adiabatic expansions and compressions of gases, the heat Q is zero, but there is a temperature change. So the student calculates that the heat capacity of the gas is zero. Instant confusion! With this warning, I will now help you solve this.

Wait they taught/are teaching me the same thing in school too !

And we defined Heat capacity in terms of the process the gas undergoes and we also defined that C_ad(heat capacity of gas for adiabetic process) is Zero !

Dont tell me all that is wrong !

And in the JEE Advance/NEET/AIIMS Prepetory module my school provided me with we have tonnes of questions,all talking about heat capacity in terms of the process it undergoes !

Is my life a lie ?,

 

[Chestermiller]

Wait they taught/are teaching me the same thing in school too !

And we defined Heat capacity in terms of the process the gas undergoes and we also defined that C_ad(heat capacity of gas for adiabetic process) is Zero !

Dont tell me all that is wrong !

And in the JEE Advance/NEET/AIIMS Prepetory module my school provided me with we have tonnes of questions,all talking about heat capacity in terms of the process it undergoes !

Is my life a lie ?,

I hate to bust your balloon, but that's not the way we do it in thermodynamics. In thermodynamics, we define heat capacity as a physical property of the material, in terms of the equilibrium thermodynamic functions U and H:
$$C_v=\left(\frac{\partial U}{\partial T}\right)_V\tag{1}$$
$$C_p=\left(\frac{\partial H}{\partial T}\right)_P\tag{2}$$
These definitions reduce to the old definition in terms of heat only for a process at constant volume or a process at constant pressure. If we don't adopt these new definitions, it leads to great confusion and ambiguity.

That being said, you still need to pass your course. So you need to "play the game" the way they expect you to play play the game. Just bear in mind that it is not the correct way to do it in thermodynamics. They have basically done you a disservice by teaching it to you in this way. One very good thermodynamics book is Fundamentals of Engineering Thermodynamics by Moran et al; here, when they introduce heat capacity, they go immediately to the definition given by Eqns. 1 and 2.

 

[Navin]

I hate to bust your balloon, but that's not the way we do it in thermodynamics. In thermodynamics, we define heat capacity as a physical property of the material, in terms of the equilibrium thermodynamic functions U and H:
$$C_v=\left(\frac{\partial U}{\partial T}\right)_V\tag{1}$$
$$C_p=\left(\frac{\partial H}{\partial T}\right)_P\tag{2}$$
These definitions reduce to the old definition in terms of heat only for a process at constant volume or a process at constant pressure. If we don't adopt these new definitions, it leads to great confusion and ambiguity.

That being said, you still need to pass your course. So you need to "play the game" the way they expect you to play play the game. Just bear in mind that it is not the correct way to do it in thermodynamics. They have basically done you a disservice by teaching it to you in this way. One very good thermodynamics book is Fundamentals of Engineering Thermodynamics by Moran et al; here, when they introduce heat capacity, they go immediately to the definition given by Eqns. 1 and 2.

...*sigh*

Well unfortunately i have to play the game to get into a good college and stuff !

On the bright side we are going to learn chemical thermodynamics later in the year in chemistry.
Crossing my fingers that they teach us the truth there !

Thank you so much though Chestermiller ! Atleast my ignorance has been shed !

 

[Krushnaraj Pandya]

And in the JEE Advance/NEET/AIIMS Prepetory module my school provided me with we have tonnes of questions,all talking about heat capacity in terms of the process it undergoes !

same here. which coaching?

 

[Krushnaraj Pandya]

Well, I can help you solve this problem but, with the following caveat: I think it is a very bad idea to define heat capacity in terms of the temperature change and heat added for a specified process path. The heat capacity is supposed to be a physical property of the material being processed, independent of process path. Defining it in terms of heat (process path) causes nothing but confusion for thermodynamics students later on. For example, when the student studies adiabatic expansions and compressions of gases, the heat Q is zero, but there is a temperature change. So the student calculates that the heat capacity of the gas is zero. Instant confusion! With this warning, I will now help you solve this.

You indicated that the change in internal energy is given by $$\Delta U=C_v\Delta T=Q-W$$where W is the work done by the gas on its surroundings. If you substitute W from the problem statement into this equation, what do you get for the temperature change $\Delta T$?

I solved it, thank you very much :D
The truth is everyone here has to "play the game" because that's the only way to score well in highly competitive objective exams which in the end decide our grad schools.
Once I get there, I'll definitely refer the book you mentioned and relearn things the right way

 

[Navin]

same here. which coaching?

I wouldn't like to tell the name because a loooot of my classmates use this site and in the state i live in there are only 3 good coaching institutes so i will be easily identified by them , and i wouldn't like that to happen ! I wouldn't mind PM-ing you the name though if you want

 

[Navin]

Oh and what's the answer you have got ?

 

[Krushnaraj Pandya]

Oh and what's the answer you have got ?

R/(γ-1)(1-k)
It was a very simple substitution