Andrew Mason said:
It depends on how fast the compression is. A compression wave moves pretty quickly though a gas.
That is true, of course. But how do you measure that pressure exerted by the piston? ie. Pext? It depends on the pressure of the gas and the mass and speeds of the gas molecules. If there are no gas molecules at all, there is no force exerted by the piston so Pext=0
It makes perfect sense. You are making a valiant effort to analyse the physics of a dynamic process. But I am not sure it is useful. While the internal and external pressures are necessarily the same at the interface between the piston and the gas in the cylinder, this does not tell us what either pressure is. It is really complicated. Really complicated.
AM
"It depends on how fast the compression is. A compression wave moves pretty quickly though a gas. "
For a compression or expansion within a cylinder, this is what it takes for the pressure, temperature, and density to vary with position, and, in particular, for the pressure at the inside face of the piston to differ from the pressures at other locations within the gas. More precisely, the characteristic time for the compression or expansion multiplied by the speed of a sound in the gas must be on the same order or smaller than the length of the cylinder. Probably, this is why, in practice, if the piston is frictionless and the time scale for expansion or compression is relatively large (by this criterion), the compression or expansion can be regarded as reversible, the gas pressure within the cylinder will be nearly uniform; also, the gas pressure on the inside face of the piston p
ext will match the gas pressure everywhere else in the cylinder.
"That is true, of course. But how do you measure that pressure exerted by the piston? ie. P
ext? It depends on the pressure of the gas and the mass and speeds of the gas molecules."
Lets suppose that, by brute force, you can control the motion of the outside face of the piston as a function of time (i.e., you control its velocity) and you can also measure the force required to impose this motion, (This way you don't have to make measurements of pressure within the cylinder). If the mass of the piston is negligible, then this force divided by the area of the piston must be equal to pressure of the gas on the inside face of the piston (by Newton's second law). But, what if the mass of the piston is not negligible. Then, if you do a force balance on the piston, you find that:
p_{ext} A= p_{gipf}A=F+m\frac{dv}{dt}=p_{opf}A+m\frac{dv}{dt}
where p_{ext} is the force per unit area exerted by the inside piston face on the gas, p_{gipf} is the pressure of the gas at the inside face of the piston, m is the mass of the piston, F is the imposed force on the outside piston face, p_{opf} is the imposed force per unit area on the outside piston face (F/A) and v(t) is the imposed velocity variation of the piston.
If we take the above equation, multiply by the velocity v, and integrate between the initial and final equilibrium states, we obtain:
W = \int p_{gipf}dV=\int p_{opf}dV
According to this equation, it doesn't matter whether we calculate the work between the two equilibrium states by using the pressure history at the inside face of the piston or the pressure history at the outside face of the piston. You get the same result either way. This is because, the change in kinetic energy of the piston is zero between the two equilibrium states.
"It makes perfect sense. You are making a valiant effort to analyse the physics of a dynamic process. But I am not sure it is useful. While the internal and external pressures are necessarily the same at the interface between the piston and the gas in the cylinder, this does not tell us what either pressure is. It is really complicated.
Really complicated."
The only reason I discussed the dynamics was to provide a rationale for how the pressure of the gas would vary within the cylinder during an irreversible compression or expansion. As far as the pressure at the interface between the piston and the gas in the cylinder are concerned, I showed in the previous paragraph how the pressure histories on both faces of the piston could be measured, and how they could be used to calculate the irreversible work. For obvious reasons, in calculating the irreversible work, my preference would be to employ the time history of the force on the outside face of the piston.
