How does Carnot Cycle Expand and Compress Isentropically?

[I<3NickTesla]

Homework Statement

I think I understand the first 3 steps of the Carnot cycle but not the 4th.

Homework Equations

The cycle here:
http://en.wikipedia.org/wiki/Carnot_cycle#Stages

The Attempt at a Solution

I understand that in stage 1, the gas expands by taking in heat from the hot reservoir and expanding to maintain constant temperature of the high temp reservoir.
Then in stage 2, the gas is allowed to do work on the surroundings by expanding more, this time cooling as it expands because it's not taking in any heat from the hot reservoir as it's adiabatic.
For stage 3, the cold reservoir takes away heat from the gas allowing it to compress it at constant temperature - the gas would cool if it was in contact with the cold reservoir at constant volume, but allowing it to compress keeps it at constant temperature.

Then in stage 4, the gas apparently compresses because the surroundings do work on it. But where is this work coming from? If the gas is at the cool temperature, it will be near enough at the same temperature as the surroundings so why would there be a pressure difference to allow it to compress?

 

[Chestermiller]

Homework Statement

I think I understand the first 3 steps of the Carnot cycle but not the 4th.

Homework Equations

The cycle here:
http://en.wikipedia.org/wiki/Carnot_cycle#Stages

The Attempt at a Solution

I understand that in stage 1, the gas expands by taking in heat from the hot reservoir and expanding to maintain constant temperature of the high temp reservoir.
Then in stage 2, the gas is allowed to do work on the surroundings by expanding more, this time cooling as it expands because it's not taking in any heat from the hot reservoir as it's adiabatic.
For stage 3, the cold reservoir takes away heat from the gas allowing it to compress it at constant temperature - the gas would cool if it was in contact with the cold reservoir at constant volume, but allowing it to compress keeps it at constant temperature.

Then in stage 4, the gas apparently compresses because the surroundings do work on it. But where is this work coming from?

We are part of the surroundings, and we can force the gas to compress (adiabatically) in stage 4. This can be done in a number of ways. One way is to gradually load a sequence of small weights onto the top of the piston. Another way is to gradually increase the force on the piston by hand, so that we personally are doing the work.

If the gas is at the cool temperature, it will be near enough at the same temperature as the surroundings so why would there be a pressure difference to allow it to compress?

Stage 4 is carried out adiabatically, and the temperature and pressure of the gas being compressed rises back up to the original temperature and pressure we started with at the beginning of stage 1. Then we can again carry out the isothermal expansion in stage 1 at the same temperature and pressure we did previously.

Chet

 

[I<3NickTesla]

We are part of the surroundings, and we can force the gas to compress (adiabatically) in stage 4. This can be done in a number of ways. One way is to gradually load a sequence of small weights onto the top of the piston. Another way is to gradually increase the force on the piston by hand, so that we personally are doing the work.

Stage 4 is carried out adiabatically, and the temperature and pressure of the gas being compressed rises back up to the original temperature and pressure we started with at the beginning of stage 1. Then we can again carry out the isothermal expansion in stage 1 at the same temperature and pressure we did previously.

Chet

Thanks for getting back so quickly.

I thought the point of a heat engine was to convert heat into work, so it seems weird if we have to do work on the system to make it function.

Is there some sort of feedback where the work done by the expansion is stored, then used to compress the gas again later?

 

[Chestermiller]

Thanks for getting back so quickly.

I thought the point of a heat engine was to convert heat into work, so it seems weird if we have to do work on the system to make it work.

Is there some sort of feedback where the work done by the expansion is stored, then used to compress the gas again later?

You are doing more work during the expansion steps (1 & 2) than you are expending during the compression steps (3 & 4). So the net effect is to absorb heat from the hot reservoir, transfer a smaller amount of heat to the cold reservoir, and use the difference to do a net amount of work. During the compression steps, you can be adding small weights to the top of the piston at various low elevations, and, during the expansion steps, you can be removing the weights from the piston at various higher elevations. The net effect of one cycle is that you have raised the entire series of weights to various higher elevations. Then you can repeat the process with a new set of weights.

Chet

 

[I<3NickTesla]

You are doing more work during the expansion steps (1 & 2) than you are expending during the compression steps (3 & 4). So the net effect is to absorb heat from the hot reservoir, transfer a smaller amount of heat to the cold reservoir, and use the difference to do a net amount of work. During the compression steps, you can be adding small weights to the top of the piston at various low elevations, and, during the expansion steps, you can be removing the weights from the piston at various higher elevations. The net effect of one cycle is that you have raised the entire series of weights to various higher elevations. Then you can repeat the process with a new set of weights.

Chet

So is the number given as the work provided by a Carnot engine the difference between the work done by the gas expanding and the work you have to do on the gas to compress it back to its original state?

 

[Chestermiller]

So is the number given as the work provided by a Carnot engine the difference between the work done by the gas expanding and the work you have to do on the gas to compress it back to its original state?

Yes.