#### Chestermiller

Mentor

- 19,348

- 3,838

A friend of mine and I have been discussing how to apply the first law of thermodynamics to analyze the quasi static expansion of an ideal gas in a cylinder featuring a piston having both mass and friction (with the cylinder). We have identified two different systems that can be used in the analysis:

1. gas alone as system, with piston as part of the surroundings

2. gas and piston (and cylinder) as system

We wanted to see how the analysis using the first law of thermodynamics would play out in each of these approaches, knowing that, ultimately, the answer would have to come out the same in each case. So here is my specification of the problem:

There is a vertical insulated cylinder cross sectional area A containing an ideal gas. There is a piston of mass m situated above the gas. The piston is insulated at the top and has negligible heat capacity so that any frictional heat generated by piston contact with the cylinder is ultimately transferred to the gas (rather than the surroundings). The frictional force F is always constant and present both initially and finally. This can be achieved if we say that the coefficient of static friction is equal to the coefficient of kinetic friction, and that, in the initial and final states, the piston is at the verge of slipping (This is an idealization which does not detract from what we are trying to achieve in our analysis). We have a force f applied downward on the piston which decreases gradually (quasi statically) from ##f_i## in the initial state of the system to ##f_f## in the final state of the system; this is how we bring about the desired volume increase. There are n moles of ideal gas in the cylinder, initially at temperature ##T_i##, and initially at mechanical equilibrium with the mass of the piston, the frictional force F, and the outside downward force ##f_i##.

I am now going to stop and allow my friend to make comments and suggestions about the problem description. Is this basically what we had in mind? (Others are invited to participate).

Chet

1. gas alone as system, with piston as part of the surroundings

2. gas and piston (and cylinder) as system

We wanted to see how the analysis using the first law of thermodynamics would play out in each of these approaches, knowing that, ultimately, the answer would have to come out the same in each case. So here is my specification of the problem:

There is a vertical insulated cylinder cross sectional area A containing an ideal gas. There is a piston of mass m situated above the gas. The piston is insulated at the top and has negligible heat capacity so that any frictional heat generated by piston contact with the cylinder is ultimately transferred to the gas (rather than the surroundings). The frictional force F is always constant and present both initially and finally. This can be achieved if we say that the coefficient of static friction is equal to the coefficient of kinetic friction, and that, in the initial and final states, the piston is at the verge of slipping (This is an idealization which does not detract from what we are trying to achieve in our analysis). We have a force f applied downward on the piston which decreases gradually (quasi statically) from ##f_i## in the initial state of the system to ##f_f## in the final state of the system; this is how we bring about the desired volume increase. There are n moles of ideal gas in the cylinder, initially at temperature ##T_i##, and initially at mechanical equilibrium with the mass of the piston, the frictional force F, and the outside downward force ##f_i##.

I am now going to stop and allow my friend to make comments and suggestions about the problem description. Is this basically what we had in mind? (Others are invited to participate).

Chet

Last edited: