# Thermodynamics problem - circular process

• solidbastard
In summary, the problem involves a cycle of an ideal gas with a total energy given by U = αT^4. The first part of the problem asks for the change of total energy from point 1 to 2, which can be solved using the first law of thermodynamics and the ideal gas law. The second part involves determining the heat acquired on the path from point 3 to 1, which can be solved using the relationship between ΔT, Cv and ΔU. The problem is similar to another problem with a different expression for U, U = cvT, and involves determining the change of total energy from point 2 to 3 and the heat acquired on the path from point 3 to 1.f

#### solidbastard

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I am having trouble attacking this problem properly and getting the right solution, which I do not have. (only know I did it wrong on exam).

Anyway, there is picture attached this is a cycle of some working material that has a total energy of U= alpha*T^4

I need to determine:

1. Change of total energy from 1 to 2.

2. Heat that came from 3 to 1.
For the first, I view it as adiabatic expansion, so there is no change of heat, only the work done. And for the second question, it is a constant volume, so there is no mechanical work that is done.

To solve it is going to use this or?
Code:
$(\frac{\partial U}{\partial V})\cdot dV + (\frac{\partial U}{\partial T})\cdot dT + (\frac{\partial U}{\partial p})\cdot dp$

So not sure if in total calculation should take in mind changes in p, T and V or what? Becuase T is changing always?

Anyway, thanks for help if anyone has solution :

#### Attachments

• problem_thermodynamics_cycleprocess.jpg
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I need to determine:

1. Change of total energy from 1 to 2.

2. Heat that came from 3 to 1.
For the first, I view it as adiabatic expansion, so there is no change of heat, only the work done. And for the second question, it is a constant volume, so there is no mechanical work that is done.
Hi. Welcome to PF!
It would help to give us all the details. In order to quantify the total energy we would to know more than youhave provided. I assume you were told it is an ideal gas and can determine Cv and Cp.

This is a first law problem. Just use ## Q = \Delta U + \int PdV##

For the isothermal expansion from 1 to 2 is there any change in U? So what is Q?

For the constant volume heating from 3 to 1, what is the change in T? (hint: how is ΔU related to ΔT?)

AM

Hi. Welcome to PF!
It would help to give us all the details. In order to quantify the total energy we would to know more than youhave provided. I assume you were told it is an ideal gas and can determine Cv and Cp.

This is a first law problem. Just use ## Q = \Delta U + \int PdV##

For the isothermal expansion from 1 to 2 is there any change in U? So what is Q?

For the constant volume heating from 3 to 1, what is the change in T? (hint: how is ΔU related to ΔT?)

AM

Yeah, problems is about ideal gas of total energy ## U = \alpha \cdot T^4 ##

I know for isothermal expansion that there is no change in U. So, with the first law ## Q = \int PdV##

And for constant volume part from 3 to 1, should I maybe try change in T with this relation. Where of course part of dV falls of, because there is no change?
##(\frac{\partial U}{\partial V})\cdot dV + (\frac{\partial U}{\partial T})\cdot dT + (\frac{\partial U}{\partial p})\cdot dp##

Thanks for the help very much!

Yeah, problems is about ideal gas of total energy ## U = \alpha \cdot T^4 ##
This is not the expression for U. That may be the source of your difficulty. T is a measure of the average translational kinetic energy of the molecules so U is proportional to T.

I know for isothermal expansion that there is no change in U. So, with the first law ## Q = \int PdV##
hint: to calculate, substitute for P using the ideal gas law.

And for constant volume part from 3 to 1, should I maybe try change in T with this relation. Where of course part of dV falls of, because there is no change?
Yes. Just use the relationship between ΔT, Cv and ΔU.

AM

What is the exact statement of the problem (not your interpretation)?

What is the exact statement of the problem (not your interpretation)?

In cycle shown on the picture working material of total energy ## U = \alpha \cdot T^4 ## is give.

Determine:
Change of total energy on the way 1 to 2.
Heat acquired on the way 3 to 1.

And I assume it is also ideal gas.

And I got one more problem that is almost all the same, and goes like this (picture is the same):
In cycle shown on the picture working material is ideal gas of total energy ## U = c_v \cdot T##

Determine:
Change of total energy on the way 2 to 3.
Heat acquired on the way 3 to 1.

In cycle shown on the picture working material of total energy ## U = \alpha \cdot T^4 ## is give.

Determine:
Change of total energy on the way 1 to 2.
Heat acquired on the way 3 to 1.

And I assume it is also ideal gas.

And I got one more problem that is almost all the same, and goes like this (picture is the same):
In cycle shown on the picture working material is ideal gas of total energy ## U = c_v \cdot T##

Determine:
Change of total energy on the way 2 to 3.
Heat acquired on the way 3 to 1.
This is the exact statement, word-for-word?

This is the exact statement, word-for-word?
Yes. :)

Yes. :)
In my judgment, there is not sufficient information provided to consider the path from 1 to 2.

Hi. Welcome to PF!
It would help to give us all the details. In order to quantify the total energy we would to know more than youhave provided. I assume you were told it is an ideal gas and can determine Cv and Cp.

This is a first law problem. Just use ## Q = \Delta U + \int PdV##

For the isothermal expansion from 1 to 2 is there any change in U? So what is Q?

For the constant volume heating from 3 to 1, what is the change in T? (hint: how is ΔU related to ΔT?)

AM
Um..is the path 1 to 2 isothermal ?
The OP has traced out a straight line instead of the hyperbolic curve for isotherms.Can we still consider it to be isothermal ?
I think some information has been left out.

Um..is the path 1 to 2 isothermal ?
The OP has traced out a straight line instead of the hyperbolic curve for isotherms.Can we still consider it to be isothermal ?
I think some information has been left out.

Yeah. Consider it isothermal. It is little bad drawing out there. :)

Yeah. Consider it isothermal. It is little bad drawing out there. :)
Ah, then its alright.

In cycle shown on the picture working material of total energy ## U = \alpha \cdot T^4 ## is give.

Determine:
Change of total energy on the way 1 to 2.
Heat acquired on the way 3 to 1.

And I assume it is also ideal gas.

And I got one more problem that is almost all the same, and goes like this (picture is the same):
In cycle shown on the picture working material is ideal gas of total energy ## U = c_v \cdot T##

Determine:
Change of total energy on the way 2 to 3.
Heat acquired on the way 3 to 1.
In the first part, this is obviously not an ideal gas. We need to know how P is related to V at constant T. The diagram appears to show ##P \propto 1/V##. Is that accurate? We really need to see the exact and complete wording of the problem.

AM

Chestermiller
Yes. :)
Pants on fire

shihab-kol