# Maximum temperature reached by gas in expansion

1. Feb 22, 2017

### vishnu 73

1. The problem statement, all variables and given/known data
1 mole of ideal gas with internal energy U= 3/2 RT , expands from initial volume Vi = 1/10 Vo following the equation p=(− po / Vo ) V +po
.
Find
(a) the highest temperature reached by the gas during the expansion and
(b) the maximum amount of heat taken in by the gas.

2. Relevant equations
pv = nrt

3. The attempt at a solution
p=(− po / Vo ) V +po
pV=(− po / Vo ) V2 +poV
(pV/nr)=( (− po / Vo ) V2 +poV)/nr = T
taking derivative of T with respect to v gets
dT/dx= ((− po / Vo ) 2V +po)/nr
setting derivative to 0 yields
V = Vo/2
plugging back into equation for temperature gets
maximum temperature = poV0/4nr

this is what i got is the right method or is there a way to get a numerical anser

2. Feb 22, 2017

### kuruman

The best you can do is express the maximum temperature as a number times the initial temperature.

3. Feb 22, 2017

### vishnu 73

ok then how do i do part b

4. Feb 22, 2017

### kuruman

Use the first law.

5. Feb 22, 2017

### vishnu 73

you mean llike Δu + w =q

6. Feb 22, 2017

### vishnu 73

as Δu is easy but w = ∫(− po / Vo ) V +po dv
so solving the integral do i plug in v as the v obtained for maximum temperature or what?
is most heat added in to reach the highest temperature in this case

Last edited: Feb 22, 2017
7. Feb 22, 2017

### Staff: Mentor

You calculated that the highest temperature is 1/4 the initial temperature. How can that be if the initial temperature is one of the states that it passes through? Regarding application of the first law, why do't you just run the calculation and see what you get?

8. Feb 23, 2017

### vishnu 73

no in this question po and vo are not the initial states they are just some constants
initial volume = 1/10 vo
initial pressure can be calculated using the above equation
thus maximum temperature reached is 100/36 Ti

now my my question is
is maximum heat added to bring the system to state with highest temperature?
if that is so then the problem is very easy hope you understand my question?

9. Feb 23, 2017

### Staff: Mentor

Oops. Sorry. My mistake.
The question is kind of ambiguous. I would just solve the problem as a function of V and see how it plays out. Let the math do the work for you.

10. Feb 25, 2017

### vishnu 73

(− po / Vo ) V +po dv = w.
[ (− p0 / V0 ) V2/2 + p0V ]vivf

while vi is known what do i plug in for vf is it the volume that gives maximum temperature or what or should i differentiate the above integral to find when the derivative of work is zero and plug in that v. thanks!!

11. Feb 25, 2017

### Staff: Mentor

I would plug in the volume that gives the maximum temperature.

12. Feb 25, 2017

### vishnu 73

but how can we be sure that yields maximum works and is the other method just as equally correct
and btw thanks for the fast replies

13. Feb 25, 2017

### Staff: Mentor

Who said anything about maximum work?

14. Feb 25, 2017

### vishnu 73

no because q = Δu + w
so isnt maximum q added when both these quantities are maximum am i wrong please correct me if i am

15. Feb 25, 2017

### Staff: Mentor

Isn't there also a change in internal energy?

16. Feb 25, 2017

### vishnu 73

yeah change in internal energy is automatically maximum when the temperature is maximum but how are we supposed to know that is the case for work done by the gas too

17. Feb 25, 2017

### Staff: Mentor

The problem is worded poorly. They want you to calculate the heat when the temperature reaches its maximum.

18. Feb 25, 2017

### vishnu 73

ooh okay how did you arrive at that thanks anyways

19. Feb 25, 2017

### Staff: Mentor

It's the only thing that makes sense to me.

20. Feb 25, 2017

### kuruman

What if you used the First Law to find an expression for the heat Q in terms of independent variable V and constants p0 and V0? Then you can find the value of V at which the heat that has entered the gas up to that point reaches a maximum before it starts going down, i.e. before heat starts being removed from the gas. The answer is a simple fraction of V0.