Constant pressure or isobaric expansion of gas

[Soumalya]

It is said that in an isobaric expansion of a gas pressure remains constant throughout the expansion process.

Suppose we have a quantity of gas at initial pressure P1 and volume V2 in a piston cylinder arrangement.We heat it slowly such that it expands to obtain a state with pressure P2 and volume V2. Now if the process was isochoric P1=P2 and V2>V1.So the work done is said to be the displacement work done by the gas on the piston as ∫PdV=P(V2-V1) [as pressure remains constant].

But on the other hand for a system to expand there must be an imbalance in the external(surroundings) and internal(system)pressures.But in an isochoric process expansion takes place even if the pressure is said to be same at all instances of the process!

That is, if P1=P2=P(system pressure) how does the gas push the piston back?

Since the system was under equilibrium with the applied external pressure at state point 1 when external force was balanced by the internal pressure(force) by the gas on the piston multiplied by piston area.At state point 2 there is no change in pressure and hence internal pressure of the system is the same while the external pressure is still the same!Then how does the piston move back?

 

[DrClaude]

Hi Soumalya, welcome to PF!

You are right that for the piston to move, you need a $\delta P$. But conceptually, you can make it infinitesimally small, such that you can consider the pressure inside the piston as constant and equal to the outside pressure. It is similar to considering the reversible processes, where you take that the changes take place on a timescale that is slow with respect to the internal re-equilibration.

By the way, be careful not to confuse isobaric = constant pressure with isochoric = constant volume.

 

[256bits]

That is, if P1=P2=P(system pressure) how does the gas push the piston back?

Suppose the pressure of the gas was a result of the weight of a mass resting on the (frictionless ) piston.
And all forces are in balance - the weight of the mass pushing down on the piston counterbalancing the pressure P1 of the gas pushing up on the piston area.

If we heat the gas, for the piston to move upwards, the mass has to accelerate to a certain velocity in relation to how quickly the heat is being added. Once a steady state velocity of the mass ( and piston ) is achieved, you will have to agree that the forces again balance - the weight of the mass pushes down with as much force as the pressure P2 of the gas pushes up on the piston ( since the velocity is constant ). P1 must equal P2 throughout the constant velocity segment of the isobaric process.

 

[nayanm]

Also, some of the new-fangled experimental physical chemistry textbooks define an "infinitesimal" change in pressure to be such that the imprecision of your instrumentation is large enough to where it can't pick up such small changes/oscillations associated with the actually finite (though small) change in pressure.

This loosens the definition of such things as reversible and quasi-static processes to be applicable experimentally.

 

[yuiop]

It might help to imagine that the piston is initially locked in place before the heating takes place. After heating, the pressure in the cylinder is noticeably greater than the external pressure. The piston is released and the volume expands rapidly accelerating the piston. If the piston has mass, it will gain momentum and may even overshoot to the extent that the pressure inside the cylinder drops below the external pressure, but then the piston reverse direction and oscillates until it comes to its final equilibrium position at the new volume with pressure equalised. Now this is not how we normally do thermodynamics, as it usually requires very slow "quasi-static" changes, but visualised like this you can see that the average pressure inside the cylinder is the same as the pressure outside the cylinder and with a frictionless piston the pressure differential is infinitesimal. In practice it is impossible to have a perfectly frictionless piston and there will be a small dp.

 

[Soumalya]

If we heat the gas, for the piston to move upwards, the mass has to accelerate to a certain velocity in relation to how quickly the heat is being added. Once a steady state velocity of the mass ( and piston ) is achieved, you will have to agree that the forces again balance - the weight of the mass pushes down with as much force as the pressure P2 of the gas pushes up on the piston ( since the velocity is constant ). P1 must equal P2 throughout the constant velocity segment of the isobaric process.

What you say is the system would re-balance itself to a new equilibrium position after departing from the initial state properties where the weight of the mass is again balanced by the pressure force of the gas on the piston.

Since the process involves a transient phase in between two equilibrium states my precise question is how do we say the process is under constant pressure at all instances of observation?

Note: During the transient phase p_int>p_ext until p_int=p_ext when we say a new equilibrium state has been achieved.

Hence,a reversible process involves intermediate episodes of irreversibility when p_int>p_ext.

 

[Soumalya]

It might help to imagine that the piston is initially locked in place before the heating takes place. After heating, the pressure in the cylinder is noticeably greater than the external pressure. The piston is released and the volume expands rapidly accelerating the piston. If the piston has mass, it will gain momentum and may even overshoot to the extent that the pressure inside the cylinder drops below the external pressure, but then the piston reverse direction and oscillates until it comes to its final equilibrium position at the new volume with pressure equalised. Now this is not how we normally do thermodynamics, as it usually requires very slow "quasi-static" changes, but visualised like this you can see that the average pressure inside the cylinder is the same as the pressure outside the cylinder and with a frictionless piston the pressure differential is infinitesimal. In practice it is impossible to have a perfectly frictionless piston and there will be a small dp.

A small dP however infinitesimal it maybe is an imbalance in force which is mandatory for the piston the move.While we say and assume that for a reversible expansion process p_int=p_ext at all instances of observation it simply doesn't make sense as for p_int=p_ext the piston would never move according to laws of simple mechanics LOL.

What is being done is we are considering only the equilibrium states of the process and eradicating the transient part of the process in between equilibrium states when p_int>p_ext.

 

[Nugatory]

Since the process involves a transient phase in between two equilibrium states my precise question is how do we say the process is under constant pressure at all instances of observation?

How do we say it? Easy, we open our mouths and form the words "the process is under constant pressure at all instances of observation"... At least that's what I do .

But I expect that you're asking a different question, namely whether we believe it when we say it, and the answer to that question is "No, but for all practical purposes we can calculate as if we do"; and the other posters to this thread have pretty decent answers as to why this works for all practical purposes.

You'll find similar issues with just about any transition between states of macroscopic systems in equilibrium: Dig deep enough, remove enough of the idealizing assumptions, assume measurements of sufficient unrealistic precision, and you'll often find that the equilibrium is an approximation and the system is in fact moving between disequilibrium on one side and disequilibrium on the other side. Even the simple case of using a 9.8 Newton force to lift a 1kg object at a constant non-zero velocity is subject to these concerns: How did we accelerate the object to that speed in the first place, if the net force is zero?

 

[DrClaude]

To add to what Nugatory wrote, take into account also that there is no such thing as a precisely constant pressure. The pressure is always fluctuating around an average value.

 

[Soumalya]

Hi again,
I have another related doubt that's eating up my mind!Suppose we consider the piston cylinder arrangement again having some mass of gas in it.We observe the expansion of gas for two cases:

1. A reversible isobaric process

2. A reversible polytropic process

Reversible Isobaric Process:

Suppose the gas was at initially at pressure P₁ and occupying volume V₁.At this stage P_ext=P_int=P₁ so that the piston is stationary.Upon heating let's say the piston settles to a new position with an increase in volume to V₂ and pressure P₂ where it balances the external pressure again such that P₂=P_ext.

Since it's an isobaric process we must have P₁=P₂=P

Since both are equilibrium states internal gas pressure should also balance the external pressure and it's satisfied as P₁=P₂=P_ext

Note: P_ext is constantReversible Polytropic Process:

Again if we consider the expansion to follow polytropic path where the initial gas pressure was P₁ and volume V₁ the piston was stationary as P₁=P_ext.Now upon heating let's say the gas pressure becomes P₂ and volume V₂ when it reaches another equilibrium state.But for equilibrium we must have P₂=P_ext.Since external pressure remains constant we arrive to P₁=P_ext=P₂

But for a polytropic process we must have P₁≠P₂

So how can equilibrium states be achieved despite what I explained in a polytropic process?

 

[DrClaude]

What is the polytropic process you are considering? It seems to me that you are just again considering an isobaric process, since the pressure is the same.

 

[Soumalya]

I want to observe how an adiabetic process proceeds with equilibrium states against an external pressure.

 

[Soumalya]

Here's what I am confused about:

For a reversible process it should proceed in stages of equilibrium so that properties could be fixed corresponding to each equilibrium state such that a path could be obtained on a a graphical plot.

In an isobaric process all equilibrium states are such that P_int=P_ext.That is internal pressure was assumed to balance the exact external pressure at all equilibrium conditions such that the piston was stationary under a net vertical force as zero.

But if we consider a reversible adiabetic expansion of the gas then the change in pressure should follow the relation PV^γ=constant such that P₁V₁^λ=P₂V₂^λ.So the volume of the gas increases with an even more aggregative relative increase of pressure.

But if the piston has to proceed in steps of equilibrium for an adiabteic process we must have P_int=p_ext again for all equilibrium conditions.But in an adiabetic process let's say when the pressure was P₁ and the system was under equilibrium i.e, piston was stationary P₁=P_ext.

Again when another equilibrium state is reached let's say at P₂ the condition for the piston to remain stationary should give us P₂=P_ext.

But P₁≠P₂ hence for two successive equilibrium states for an adiabetic process P_ext should vary as internal gas pressure varies according to PV^λ=constant.

So my question is does the external pressure vary during a reversible adiabetic expansion or compression of a gas?

 

[DrClaude]

I think I understand your confusion. When you write $P_1 = P_\mathrm{ext}$ and $P_2 = P_\mathrm{ext}$, you are assuming that $P_\mathrm{ext}$ is constant, but it can't be! If you have an adiabatic process for an ideal gas in a cylinder with a piston, the only external driving you have is the pressure. If the pressure doesn't change, then nothing happens.

In other words, at the beginning you have $P_1 = P_\mathrm{ext}$. At the end of the adiabatic process, $P_1 \neq P_2$, according to $PV^\gamma = \textrm{const.}$. Since the system is then again at equilibrium, $P_2 = P_\mathrm{ext}'$. Therefore, $P_\mathrm{ext} \neq P_\mathrm{ext}'$: the external pressure has changed.

 

[Soumalya]

I think I understand your confusion. When you write $P_1 = P_\mathrm{ext}$ and $P_2 = P_\mathrm{ext}$, you are assuming that $P_\mathrm{ext}$ is constant, but it can't be! If you have an adiabatic process for an ideal gas in a cylinder with a piston, the only external driving you have is the pressure. If the pressure doesn't change, then nothing happens.

In other words, at the beginning you have $P_1 = P_\mathrm{ext}$. At the end of the adiabatic process, $P_1 \neq P_2$, according to $PV^\gamma = \textrm{const.}$. Since the system is then again at equilibrium, $P_2 = P_\mathrm{ext}'$. Therefore, $P_\mathrm{ext} \neq P_\mathrm{ext}'$: the external pressure has changed.

That's exactly what I meant!

Thanks DrClaude