Adiabatic or Isobaric process?

[Chestermiller]

I'd like to propose a series of thought experiments:
First let's take the opposite of the first post. A cylinder with a massless, frictionless piston contains gas at a pressure higher than surroundings. The piston is held by a pin. Then the pin is removed and the gas in the cylinder expands until the pressure in and out is the same or the piston hits a mechanical stop. The work done by the gas inside the cylinder on the surroundings is W = P_ext ΔV, let's say. We can now imagine that we keep reducing the external pressure, which reduces W, until W is zero when P_ext is zero. Here we see a nice and smooth transition to free expansion, where experiment has shown that W is indeed zero.
Now we'll turn the situation round (back to the original question of this thread): Now the pressure is higher outside than inside and the cylinder is suddenly compressed when we remove the pin holding the piston in place. If we now keep reducing the pressure inside the cylinder, is W also always P_ext ΔV? What happens when the pressure inside the cylinder is reduced to zero, so we have a free expansion of the surrounding gas into the volume initially occupied by the cylinder. Does W suddenly become zero when the internal pressure is zero?

How do you propose to keep reducing the pressure inside the cylinder in this series of experiments?

 

[Chestermiller]

I'd like to propose a series of thought experiments:
First let's take the opposite of the first post. A cylinder with a massless, frictionless piston contains gas at a pressure higher than surroundings. The piston is held by a pin. Then the pin is removed and the gas in the cylinder expands until the pressure in and out is the same or the piston hits a mechanical stop. The work done by the gas inside the cylinder on the surroundings is W = P_ext ΔV, let's say. We can now imagine that we keep reducing the external pressure, which reduces W, until W is zero when P_ext is zero. Here we see a nice and smooth transition to free expansion, where experiment has shown that W is indeed zero.
Now we'll turn the situation round (back to the original question of this thread): Now the pressure is higher outside than inside and the cylinder is suddenly compressed when we remove the pin holding the piston in place. If we now keep reducing the pressure inside the cylinder, is W also always P_ext ΔV? What happens when the pressure inside the cylinder is reduced to zero, so we have a free expansion of the surrounding gas into the volume initially occupied by the cylinder. Does W suddenly become zero when the internal pressure is zero?

The kind of calculations you are talking about for irreversible expansions can be done solely by applying the principles of thermodynamics only in a very limited number of cases where $P_{ext}$ is controlled from the outside to be constant, and only in cases where the total heat flow to the system is specified (and only for the change from the initial thermodynamic equilibrium state to the final thermodynamic equilibrium state). Otherwise, the calculations must involve detailed analysis of what is happening as a function of time and spatial position within the system. The latter us usually accomplished by employing computational fluid dynamics and heat transfer. The computational fluid dynamics and heat transfer software solves the detailed partial differential equations of continuity (conservation of mass), motion (conservation of momentum/Navier Stokes Equation), and heat transfer (conservation of energy).

 

[Philip Koeck]

I'm thinking of the same set-up as in the original post. Constant P_ext, Q = 0, both initial and final states are equilibrium states. The more interesting experiment is the compression. If we make a series of compression experiments, just as described in the original post, is the work always given by W = P_ext ΔV, no matter how small the pressure inside the cylinder is? What happens when the internal pressure goes to zero?

 

[Philip Koeck]

Consider the following problem:
Gaseous helium (assumed ideal) filled in a horizontal cylindrical vessel is separated from its surroundings by a massless piston. Both piston and cylinder are thermally insulating. The ambient pressure is suddenly tripled without changing the ambient temperature. How many times of its initial value will the volume of the helium become, when the piston finally stops?

Initial pressure and volume be P1 and V1. Now the ambient pressure is suddenly made P2 = 3 P1. The gas compresses under constant pressure P2, so the process must be isobaric. However the walls are also thermally insulating, so the process must also be Adiabatic. I find this confusing.
If I use the expression for adiabatic process i.e PV^\gamma =constant, then I don't get the answer at the back.
However, I can use 1st law to writeW=-P_2\Delta V=3/2 nR\Delta T=3/2\Delta (PV)=3/2(P_2V_2-P_1V_1) to get the correct answer.
I think the second way is fine. But why doesn't the first method work?

Could you tell us which book you took the example from?

 

[Chestermiller]

I'm thinking of the same set-up as in the original post. Constant P_ext, Q = 0, both initial and final states are equilibrium states. The more interesting experiment is the compression. If we make a series of compression experiments, just as described in the original post, is the work always given by W = P_ext ΔV, no matter how small the pressure inside the cylinder is? What happens when the internal pressure goes to zero?

The analysis is always valid as long as the mean free path of the molecules is small compared to the physical dimensions of the cylinder. For all practical purposes, that covers virtually all cases. If the pressure becomes low enough that the mean free path becomes comparable to the physical dimensions (or more), the statistics break down.

 

[chanchal pandey]

Consider the following problem:
Gaseous helium (assumed ideal) filled in a horizontal cylindrical vessel is separated from its surroundings by a massless piston. Both piston and cylinder are thermally insulating. The ambient pressure is suddenly tripled without changing the ambient temperature. How many times of its initial value will the volume of the helium become, when the piston finally stops?

Initial pressure and volume be P1 and V1. Now the ambient pressure is suddenly made P2 = 3 P1. The gas compresses under constant pressure P2, so the process must be isobaric. However the walls are also thermally insulating, so the process must also be Adiabatic. I find this confusing.
If I use the expression for adiabatic process i.e PV^\gamma =constant, then I don't get the answer at the back.
However, I can use 1st law to writeW=-P_2\Delta V=3/2 nR\Delta T=3/2\Delta (PV)=3/2(P_2V_2-P_1V_1) to get the correct answer.
I think the second way is fine. But why doesn't the first method work?

Consider the following problem:
Gaseous helium (assumed ideal) filled in a horizontal cylindrical vessel is separated from its surroundings by a massless piston. Both piston and cylinder are thermally insulating. The ambient pressure is suddenly tripled without changing the ambient temperature. How many times of its initial value will the volume of the helium become, when the piston finally stops?

Initial pressure and volume be P1 and V1. Now the ambient pressure is suddenly made P2 = 3 P1. The gas compresses under constant pressure P2, so the process must be isobaric. However the walls are also thermally insulating, so the process must also be Adiabatic. I find this confusing.
If I use the expression for adiabatic process i.e PV^\gamma =constant, then I don't get the answer at the back.
However, I can use 1st law to writeW=-P_2\Delta V=3/2 nR\Delta T=3/2\Delta (PV)=3/2(P_2V_2-P_1V_1) to get the correct answer.
I think the second way is fine. But why doesn't the first method work?

If the piston is not MASSLESS which is the case in real world , is it solution still relevant ? Bcoz seems like we can't do anything practical with it .

 

[Chestermiller]

If the piston is not MASSLESS which is the case in real world , is it solution still relevant ? Bcoz seems like we can't do anything practical with it .

It's just a limiting case (approximation) for a piston with small mass. Are you also saying anything we do with geometric figures (such as circles, rectangles, spheres) is not relevant to the real world bcuz there is no such thing as a perfect geometric figure in the real world?

 

[chanchal pandey]

It's just a limiting case (approximation) for a piston with small mass. Are you also saying anything we do with geometric figures (such as circles, rectangles, spheres) is not relevant to the real world bcuz there is no such thing as a perfect geometric figure in the real world?

well we can still perform the experiment whether the geometry of the given figure (circles , rectangles , spheres ) is perfect or not . But the issue whether the piston is massless or not is more important I think .

 

[Chestermiller]

well we can still perform the experiment whether the geometry of the given figure (circles , rectangles , spheres ) is perfect or not . But the issue whether the piston is massless or not is more important I think .

I guess we have a difference of opinion. I don't think it's any more important. If you want to design equipment and carry out experiments without using the laws of geometry (because it's too approximate), that's OK with me. I'm willing to give up massless pistons in everything I do if you're willing to never use geometry again to design equipment and calculate volumes, area, and lengths. Do we have a deal?

 

[Chestermiller]

If the piston is not MASSLESS which is the case in real world , is it solution still relevant ? Bcoz seems like we can't do anything practical with it .

Here's a better example. In real world, there is no such thing as an ideal gas, so is it solution still relevant ? Bcoz seems like we can't do anything practical with it .

 

[chanchal pandey]

Here's a better example. In real world, there is no such thing as an ideal gas, so is it solution still relevant ? Bcoz seems like we can't do anything practical with it .

Well I was about to bring Ideal gases into our conversation. But u did first. I have problem with ideal gas too . Whole Thermodynamics chapter(still in High school) is based on the assumption that for thermodynamic equations to work , the process must be Reversible ( means at every stage equilibrium state is present in the process) and reversible processes happen very very slow . And in real world most of the processes are irreversible (non-equilibrium at every stage of the process).
Example change in entropy = heat / change in temperature . This equation is only valid for REVERSIBLE processes and if I want to find out change in entropy in my room , I cannot apply this equation..

 

[Let'sthink]

All real processes are non-equilibrium or irreversible processes. So we cannot calculate work done or heat exchanged by using state variables and accompanied integration or differentiation mathematical techniques as the system does not have any meaningful property. In order to calculate the changes in state variables during irreversible (or real) processes which take the system form one equilibrium state to another. We construct or imagine a reversible process which takes the system from the given equilibrium state through a series of equilibrium states to the given final state of the system. For such processes we can calculate, or write expressions for work done or heat exchanged and then use laws of thermodynamics to get the changes between state variables of the system's final and initial equilibrium states. We do similar things in mechanics and other cases of difference in potential energy but we are not conscious of it.

 

[Philip Koeck]

We've had both opinions in this thread now: You can only calculate work for reversible processes. You can also calculate work for a sudden, irreversible process using W = P_ext ΔV.
If we go back to the original post, how would you replace the problem by a reversible process and calculate work. The problem I see is that you need the work to find out what the P, V and T of the final state actually are. As long as you don't know the final state, how do you choose the correct reversible process?

 

[Terry Bing]

Well I was about to bring Ideal gases into our conversation. But u did first. I have problem with ideal gas too . Whole Thermodynamics chapter(still in High school) is based on the assumption that for thermodynamic equations to work , the process must be Reversible ( means at every stage equilibrium state is present in the process) and reversible processes happen very very slow . And in real world most of the processes are irreversible (non-equilibrium at every stage of the process).
Example change in entropy = heat / change in temperature . This equation is only valid for REVERSIBLE processes and if I want to find out change in entropy in my room , I cannot apply this equation..

Well, you seem to have a problem with the whole of thermodynamics as such. And your concern, it seems, is that it is impractical. Now, I am no expert , but I see around me engines, air conditioners and refrigerators, all working fine. Of course, ideal gas and quasistatic process etc are approximate descriptions of nature. But can you tell me a description of nature that isn't approximate? Can you know anything for sure? Consider the example of geometry, for example, as mentioned by @Chestermiller .
Someone correct me if I am wrong, but my understanding is that as far as science is concerned, if it works, it is true. If it works for some specific scales, it is true for those scales. Whatever we say we know, are models we have built based on observations of nature, and they are tested by time. So it doesn't make sense to ask if anything is 'definitely true' or not, but if it works, it is a scientific truth.
Now, of course there are cases where an ideal gas assumption is a bad approximation (e.g. low temperature, high pressure). But they aren't completely useless. And non- ideal gases can be thought of as deviations from the ideal gas, in terms of the compressibility factor.

 

[Chestermiller]

We've had both opinions in this thread now: You can only calculate work for reversible processes. You can also calculate work for a sudden, irreversible process using W = P_ext ΔV.
If we go back to the original post, how would you replace the problem by a reversible process and calculate work. The problem I see is that you need the work to find out what the P, V and T of the final state actually are. As long as you don't know the final state, how do you choose the correct reversible process?

I agree with virtually all of this, except that, irrespective of whether the process is reversible or irreversible, the PV work is $W=\int{P_{ext}dV}$. The problem is that, for irreversible processes, only in special cases can we know in advance (in the absence of detailed gas dynamics calculations on the system) the relationship between $P_{ext}$ and V; one of these cases is if we control $P_{ext}$ so that it is constant.

 

[Chestermiller]

Well I was about to bring Ideal gases into our conversation. But u did first. I have problem with ideal gas too . Whole Thermodynamics chapter(still in High school) is based on the assumption that for thermodynamic equations to work , the process must be Reversible ( means at every stage equilibrium state is present in the process) and reversible processes happen very very slow . And in real world most of the processes are irreversible (non-equilibrium at every stage of the process).
Example change in entropy = heat / change in temperature . This equation is only valid for REVERSIBLE processes and if I want to find out change in entropy in my room , I cannot apply this equation..

Who says you can't find the change in entropy of your room? As @Philip Koeck correctly points out, all you need to know is the initial and final states, irrespective of whether the actual process from the initial state to the final state was irreversible. Didn't they teach you how to get the entropy change for an irreversible process?

 

[Philip Koeck]

We construct or imagine a reversible process which takes the system from the given equilibrium state through a series of equilibrium states to the given final state of the system. For such processes we can calculate, or write expressions for work done or heat exchanged and then use laws of thermodynamics to get the changes between state variables of the system's final and initial equilibrium states.

If we go back to the original post, how would you replace the problem by a reversible process and calculate work. The problem I see is that you need the work to find out what the V and T of the final state actually are. As long as you don't know the final state, how do you choose the correct reversible process?

 

[Let'sthink]

If we go back to the original post, how would you replace the problem by a reversible process and calculate work. The problem I see is that you need the work to find out what the V and T of the final state actually are. As long as you don't know the final state, how do you choose the correct reversible process?

Chester Miller has given answer to that: We forget thermodynamics and calculate integration of Pext which is given as 3P and calculate work done by environment on the system but in this case lasowe assume this is done quasi statically or with zero mass piston. Using first law we know this must increase internal energy of the system.Knowing that this is ideal gas and final state is equilibrium state with volume V2 and using Cp we can find the change in temperature.

 

[Philip Koeck]

Chester Miller has given answer to that: We forget thermodynamics and calculate integration of Pext which is given as 3P and calculate work done by environment on the system but in this case lasowe assume this is done quasi statically or with zero mass piston. Using first law we know this must increase internal energy of the system.Knowing that this is ideal gas and final state is equilibrium state with volume V2 and using Cp we can find the change in temperature.

Do you calculate W for a reversible process then?
My impression is that, if you use W = P_ext ΔV in this example, you are doing the calculation for an irreversible process.
I don't think it helps to imagine a quasi-static process as long as you don't even know the final state.
To get T from U for an ideal gas you would use C_V, not C_P.

 

[Let'sthink]

My impression is that, if you use W = Pext ΔV in this example, you are doing the calculation for an irreversible process.

No I am not calculating integral pdx, I am just using mechanics consideration. And even this mechanical work is to be assumed to be done quasi statically. We use work energy theorem of Mechanics to conclude that the energy of the system on which work has been done must increase its energy when the final equilibrium state is reached. To conclude the change in internal energy we will have to consider another reversible process of heating quasi statically to achieve the final state with the help of a reservoir of infinite thermal capacity passing through series of equlibrium states of increasing temperature.

 

[Chestermiller]

No I am not calculating integral pdx, I am just using mechanics consideration. And even this mechanical work is to be assumed to be done quasi statically. We use work energy theorem of Mechanics to conclude that the energy of the system on which work has been done must increase its energy when the final equilibrium state is reached. To conclude the change in internal energy we will have to consider another reversible process of heating quasi statically to achieve the final state with the help of a reservoir of infinite thermal capacity passing through series of equlibrium states of increasing temperature.

I have no understanding of what you are trying to say here, but I can tell you as one of Physics Forums experts on thermodynamics that I agree totally with what Philip Kosck and and Terry Bing have said. If what you are saying disagrees with them, then I too disagree with you.

 

[Let'sthink]

I have no understanding of what you are trying to say here, but I can tell you as one of Physics Forums experts on thermodynamics that I agree totally with what Philip Kosck and and Terry Bing have said. If what you are saying disagrees with them, then I too disagree with you.

It is too much conditional Sir. After all I had used your.earlier post, to write what I wanted, where in you said that we can assume that external pressure can be assumed to be held constant at 3P, then we can calculate the external work done. My submission is that this kind of quasi statistical integration is also present in the definition of potential energy in mechanics. Rather the work energy theorem it self talks about reversible work. I think it is not true that these considerations just pop up in thermodynamics alone but they are present in mechanics too. Having known that force varies as 1/r^2 our assertion that potential energy will be proportional to 1/r, does involve the idea of reversible work done.

 

[Chestermiller]

It is too much conditional Sir. After all I had used your.earlier post, to write what I wanted, where in you said that we can assume that external pressure can be assumed to be held constant at 3P, then we can calculate the external work done.
My submission is that this kind of quasi statistical integration is also present in the definition of potential energy in mechanics.

The compression process that occurs as a result of increasing the external pressure on the gas from P to 3P and then holding the pressure constant is neither quasi static nor reversible. It is a highly irreversible process, resulting in significant dissipation of mechanical energy to internal energy as a result of viscous friction. What does the work energy theorem say about processes in which friction is involved to dissipate mechanical energy?

 

[Let'sthink]

The compression process that occurs as a result of increasing the external pressure on the gas from P to 3P and then holding the pressure constant is neither quasi static nor reversible. It is a highly irreversible process, resulting in significant dissipation of mechanical energy to internal energy as a result of viscous friction. What does the work energy theorem say about processes in which friction is involved to dissipate mechanical energy?

For that work energy theorem says nothing, as work energy theorem is originally conceptualized for a point particle change in its KE and PE. In the problem posed by OP it was not written that P was made 3P in a gradual manner. It was written P was suddenly made 3P it is mentioned in the problem that the process is isobaric and finally it reaches an equilibrium state. Whether it is stated or not we know the process is irreversible and adiabatic. It is only externally isobaric. P as a parameter of the gas is just not defined during the process. So we need to use the information that it is ideal gas and guess the final state and then connect these by reversible processes to evaluate the final equilibrium state values using the tenets of thermodynamics.

Do you not now stand for your statement on this thread that external work done will be Pext*(V2 - V1)

There are totally three things I would like to say
1. What ever I have written on this thread nowhere you have said I am wrong. Now you say you do not understand what I am saying but if I disagree with others then you also disagree with me.

2. Idea of quasi static process is present in the definition of PE of a point particle too in Mechanics

3. There are two may I say modern definitions of "work", one related with with 'Force' and other related with energy. Former is historical and well known. The latter is work is that transfer of energy from one system to another where temperature difference is not directly involved. I do not know whether it is proper to quote books on this thread but as an indulgent student let me state here these two definitions were given first in mechanics chapter and then second in thermodynamics chapter in second edition of Halliday and Resnick Physics. But in the fifth edition of the same book with more authors both definitions are given in the chapter on mechanics it self.

 

[Chestermiller]

For that work energy theorem says nothing, as work energy theorem is originally conceptualized for a point particle change in its KE and PE. In the problem posed by OP it was not written that P was made 3P in a gradual manner. It was written P was suddenly made 3P it is mentioned in the problem that the process is isobaric and finally it reaches an equilibrium state. Whether it is stated or not we know the process is irreversible and adiabatic. It is only externally isobaric. P as a parameter of the gas is just not defined during the process. So we need to use the information that it is ideal gas and guess the final state and then connect these by reversible processes to evaluate the final equilibrium state values using the tenets of thermodynamics.

We don't need to guess anything. In this problem, during the deformation, 3P is the pressure of the gas at the piston face, and, from this we have enough information to precisely establish the final equilibrium state. We can precisely calculate the work done by the gas on the piston for this irreversible process, and we know that there is no heat transfer, so, from the first law of thermodynamics, we know the relationship between the work and the change in internal energy. We don't need to connect the initial and final states by a reversible process to evaluate the final equilibrium state; we already know the final thermodynamic equilibrium state for the irreversible process by applying the first law in conjunction with the ideal gas law.

Do you not now stand for your statement on this thread that external work done will be Pext*(V2 - V1)

Yes, I stand by this. This is work done by the gas on the piston face (which constitutes its surroundings), and it is also minus the work done by the piston face on the gas. If you want to call it external work, that's your prerogative. But it is really the work done by the gas on its surroundings.

 

[Let'sthink]

We don't need to guess anything. In this problem, during the deformation, 3P is the pressure of the gas at the piston face, and, from this we have enough information to precisely establish the final equilibrium state. We can precisely calculate the work done by the gas on the piston for this irreversible process, and we know that there is no heat transfer, so, from the first law of thermodynamics, we know the relationship between the work and the change in internal energy. We don't need to connect the initial and final states by a reversible process to evaluate the final equilibrium state; we already know the final thermodynamic equilibrium state for the irreversible process by applying the first law in conjunction with the ideal gas law.

I perfectly agree with this Sir. But if some one wants to know the work done in the irreversible process then we can connect by a reversible process and tell the work done in general especially in this case as the process is adiabatic.[/QUOTE]

 

[Chestermiller]

I perfectly agree with this Sir. But if some one wants to know the work done in the irreversible process then we can connect by a reversible process and tell the work done in general especially in this case as the process is adiabatic.

No Way. For this pair of initial and final states, there is no adiabatic reversible process that gives the same amount of work as for the present adiabatic irreversible path. The work for the present irreversible adiabatic process is $3P(V_2-V_1)$. I challenge you to specify an adiabatic reversible path that even goes between these same two end states (temperatures, pressures, and volumes). Every reversible path will have to involve some transfer of heat.

 

[Let'sthink]

I agree again Sir, But Cp(T2-T1), is it not a reversible heat transfer? Or you say it is irreversible heat transfer that will take place in a real process? It is reversible heat transfer through reservoirs of infinite thermal capacity at temperatures increasing steadily from T1 to T2 in infinite infinitesimal steps.Then using the fact that internal energy is a state function this heat transfer must be equal to the work done in the actual irreversible process. What I wish to state Sir is whenever we use integration to calculate either work done by the system or heat supplied to the system we make use of the concept of a reversible process only.

 

[Let'sthink]

I agree again Sir, But Cp(T2-T1), is it not a reversible heat transfer? Or you say it is irreversible heat transfer that will take place in a real process? It is reversible heat transfer through reservoirs of infinite thermal capacity at temperatures increasing steadily from T1 to T2 in infinite infinitesimal steps.Then using the fact that internal energy is a state function this heat transfer must be equal to the work done in the actual irreversible process. What I wish to state Sir is whenever we use integration to calculate either work done by the system or heat supplied to the system we make use of the concept of a reversible process only.

In other words whether in mechanics or thermodynamics the concept of quasi static and reversibility is required as a contrivance to let the 'mathematics' work!

 

[Philip Koeck]

Who says you can't find the change in entropy of your room? As @Philip Koeck correctly points out, all you need to know is the initial and final states, irrespective of whether the actual process from the initial state to the final state was irreversible. Didn't they teach you how to get the entropy change for an irreversible process?

I might have a way of solving the original problem via the entropy, but I'm not sure it's right. Anyway, the idea is this: The entropy of the surroundings is constant. If we imagine a reversible process between the same initial and final state, the entropy of the system is also constant (This might be be the problem.), since the total entropy has to be constant. Now we can use the Sackur-Tetrode equation to write down the entropy of the system for initial and final state and equate them. Then we also have the ideal gas law to eliminate the temperature in initial and final state. The result I get is V₂ p₂³ = V₁ p₁³. Does that make sense?