Heat Transfer in Ideal Gases: An Example of Path Dependence

[hiba fatima]

< Mentor Note -- Posts split off from an old thread to preserve very good helpful responses...>[/color]

how is heat path dependent?prove with an example?

 

[rude man]

I've read a solution of a problem, in which there are two different gases in a container, initally at equilibrium and separated by an adiabatic fix wall. At some time, this wall is changed by a diathermic mobile wall, so the equilibrium point changes. You have to find the final state of the gases, given the initial volumes, temperatures and pressions.

Then the solution says: we will divide the process in two parts: first an isochoric process and after that and isothermal one. But it is clear that pressure and temperature evolve at the same time.
I'm making a bit of a mess here: is it path dependent or indpendent? In a generic process, when is it path independent and when it is not? And why?

Consider an ideal gas at initial state (p1,V1,T1). We want to change the state to (p2,V2,T2).

We can do this in many ways, but let's pick two in particular :
1. we can change the temperature at constant V1 (isochoric) until p = p2, then, holding p constant, change the temperature some more until V = V2. We have reached (p2,V2,T2).
2. we can change the temperature at constant p1 (isobaric) until V = V2, then, holding V constant, change temperature some more until p = p2. We have again reached (p2,V2,T2).
In both cases we went from state (p1,V1,T1) to (p2,V2,T2).
Can you compute the heat and work required in both processes?

 

[hiba fatima]

thanx ... but you did not clear the point that the heat is path dependent...means we go to the same state in both cases...but what is the difference in both cases that makes the heat path dependent...

 

[rude man]

thanx ... bt u did not clear the point that the heat is path dependent...means we go to the same state in both cases...but what is the difference in both cases that makes the heat path dependent...

The difference is in one case we change p first and then V, in the other we change V first and then p. The sequence makes it different. Compute W and Q for both cases and you will see.

 

[hiba fatima]

i could not do it :(

 

[rude man]

i could not do it :(

What is the area under a p-V curve going from a state 1 to a state 2?

 

[Chestermiller]

I assume you are dealing with ideal gases. If the gases are in a rigid container, how much external work do the combination of gases do on the surroundings? Assuming that the container is adiabatic, how much heat passes through the walls of the container. From the first law of thermodynamics, what is the change in internal energy for the combination of gases when the system finally equilibrates? If the wall is mobile and diathermal, how do the temperatures and pressures in the two compartments compare at the final equilibrium state?

Chet