Is the energy of this system conserved?

[rtareen]

TL;DR Summary

A gas is contained in an insulated cylinder with a movable piston onto of which is a lead shot with mass. The thermal reservoir at the bottom allows for heat the be transferred to the gas.

In this system (consisting of just the gas) heat is transferred to the gas by means of a reservoir. So this means energy is added to the system. Does this necessarily mean that the work done on the lead shot due to an increased pressure will equal the heat that enters the gas? We are assuming everything is ideal. The book does not really say what would happen. How can we know? What dictates how much work the gas will do?

 

[kuruman]

I think it is safe to ask what would happen if, starting from equilibrium, you add (or remove) a piece of lead shot and wait until the system reaches equilibrium again. That will be an isothermal compression (or expansion). You can calculate the final pressure from the added weight, then use the ideal gas law to find the final volume and do the integral to find the work done by the gas.

 

[russ_watters]

In this system (consisting of just the gas) heat is transferred to the gas by means of a reservoir. So this means energy is added to the system. Does this necessarily mean that the work done on the lead shot due to an increased pressure will equal the heat that enters the gas? We are assuming everything is ideal. The book does not really say what would happen. How can we know? What dictates how much work the gas will do?

Be organized and specific about what is changing and what isn't and think it through:

• It is being heated, so is the temperature of the gas changing?
• It is a sealed piston, so is the mass of gas changing?
• The piston moves, so is the volume of gas changing?
• The weight of the bucket of shot isn't changing, so are you sure the pressure is increasing?

An answer of "yes" is a change in energy and a "place" where energy is going. You didn't actually develop a clear-cut problem here, so I'm going to assume the unknown you are trying to solve for is how far the cylinder moves, given a fully defined starting state and a given amount of heat addition. Along the way, you account for where the energy goes.

What kind of book did that come from and what is the context behind this? It feels like it should be part of a chapter in a thermodynamics book, on constant pressure heat addition:
https://en.wikipedia.org/wiki/Isobaric_process

 

[russ_watters]

I think it is safe to ask what would happen if, starting from equilibrium, you add (or remove) a piece of lead shot and wait until the system reaches equilibrium again. That will be an isothermal compression (or expansion). You can calculate the final pressure from the added weight, then use the ideal gas law to find the final volume and do the integral to find the work done by the gas.

I don't think that matches the problem the OP is trying to solve, nor do I think it is true that that process would be isothermal given that the piston is insulated.

 

[kuruman]

I don't think that matches the problem the OP is trying to solve, nor do I think it is true that that process would be isothermal given that the piston is insulated.

The piston is insulated but in thermal contact with the heat reservoir. As shown in the figure, heat may be exchanged with the reservoir. I would agree, though, that an isothermal process is not necesarily what OP is trying to solve. OP asserts that "heat is transferred to the gas by means of a reservoir". However, there is no mention whether the heat transfer is taking place isothermally, isobarically or in some other manner. I like the isothermal process because it has the simple feature that whatever heat is added to the gas is equal to the work done by the gas on the piston.

 

[rtareen]

The piston is insulated but in thermal contact with the heat reservoir. As shown in the figure, heat may be exchanged with the reservoir. I would agree, though, that an isothermal process is not necesarily what OP is trying to solve. OP asserts that "heat is transferred to the gas by means of a reservoir". However, there is no mention whether the heat transfer is taking place isothermally, isobarically or in some other manner. I like the isothermal process because it has the simple feature that whatever heat is added to the gas is equal to the work done by the gas on the piston.

Be organized and specific about what is changing and what isn't and think it through:

• It is being heated, so is the temperature of the gas changing?
• It is a sealed piston, so is the mass of gas changing?
• The piston moves, so is the volume of gas changing?
• The weight of the bucket of shot isn't changing, so are you sure the pressure is increasing?

An answer of "yes" is a change in energy and a "place" where energy is going. You didn't actually develop a clear-cut problem here, so I'm going to assume the unknown you are trying to solve for is how far the cylinder moves, given a fully defined starting state and a given amount of heat addition. Along the way, you account for where the energy goes.

What kind of book did that come from and what is the context behind this? It feels like it should be part of a chapter in a thermodynamics book, on constant pressure heat addition:
https://en.wikipedia.org/wiki/Isobaric_process

This is from Halliday and Resnick. I am self-studying Chapter `18. This section introduces the first law of thermodynamics. I haven't yet gotten to the point of isobaric or isothermal transfer. I don't know what those are.

Yes, the reservoir is adding heat to the gas.
The piston can move up and down, so the volume of the gas can change.
I am assuming the pressure increases because if work is done on the lead there needs to be a net force upward.

This example is supposed to give us the relationship between heat and work. But does not explicitly say what that relationship is. All it says is that heat can be added to the gas. And that the gas can do work on the lead if some of the mass is removed. But I want to know whether or not the gas will do an amount of work equal to the amount of heat added in the case there is no change to the mass of the lead, which is not explained by the book.

 

[jbriggs444]

However, there is no mention whether the heat transfer is taking place isothermally, isobarically or in some other manner. I like the isothermal process because it has the simple feature that whatever heat is added to the gas is equal to the work done by the gas on the piston.

I do not see how it can be isothermal in the OP's scenario.

For slow changes (negligible acceleration), the pressure in the cylinder is constant due to the piston with the lead shot. So, by inspection we have an isobaric expansion.

If we are pumping in heat and keeping pressure constant, volume will increase and temperature must rise.

Edit:

The OP asks:

Does this necessarily mean that the work done on the lead shot due to an increased pressure will equal the heat that enters the gas?

[emphasis mine]
How can there be an increased pressure if we are talking about an isobaric expansion?!

My answer is that we are talking about a slowly changing equilibrium. It is "quasi-static". At some point early in the process there is a tiny increase in pressure. Just enough to get the piston moving. Then pressure can return to the baseline. The piston moves slowly and steadily upward as the expanding gasses do work and the temperature knob is slowly turned up. Then when we stop turning the knob, there is a tiny decrease in pressure as the piston coasts to a stop.

[And we ignore any possible oscillations as things settle down to the final state]

 

[rtareen]

I do not see how it can be isothermal in the OP's scenario.

For slow changes (negligible acceleration), the pressure in the cylinder is constant due to the piston with the lead shot. So, by inspection we have an isobaric expansion.

If we are pumping in heat and keeping pressure constant, volume will increase and temperature must rise.

Great! So what does this mean in terms of energy? Does the gas keep the added energy or will it exit the gas by doing work on the lead mass?

 

[jbriggs444]

Great! So what does this mean in terms of energy? Does the gas keep the added energy or will it exit the gas by doing work on the lead mass?

I've edited some additional verbiage into the prior post which may be useful.

To directly answer the question: Why can't it be a little of both?

 

[rtareen]

I've edited some additional verbiage into the prior post which may be useful.

To directly answer the question: Why can't it be a little of both?

You say that the gas will do work as the knob turn up. The pressure increases, and this translates to energy leaving the gas through work. Then when we stop turning the knob the piston slowly comes to a stop. So I think you're implying that the gas does an amount of work proportional to the amount of heat added. So you answered part of the question, whether there will be work. But you didn't explicitly say whether this work will be equal to the amount of energy gained by the gas through heat transfer. In other words
$\Delta E_{int} = Q - W = 0$?

 

[etotheipi]

For slow changes (negligible acceleration), the pressure in the cylinder is constant due to the piston with the lead shot. So, by inspection we have an isobaric expansion.

I think @kuruman was referring to a process in which individual lead shots are removed, one by one, causing the external pressure (and thus internal pressure, if the process is quasi-static) to decrease very gradually. For it to be isothermal, $dQ = p dV$ but we also have $d(pV) = pdV + Vdp = nR dT = 0$ and consequently $dQ = -Vdp$. So for any increment (decrement) in pressure due to removing a lead shot, we can always find a corresponding amount of heat $dQ$ to add so that the process is isothermal. Would that work?

 

[kuruman]

You say that the gas will do work as the knob turn up. The pressure increases, and this translates to energy leaving the gas through work. Then when we stop turning the knob the piston slowly comes to a stop. So I think you're implying that the gas does an amount of work proportional to the amount of heat added. So you answered part of the question, whether there will be work. But you didn't explicitly say whether this work will be equal to the amount of energy gained by the gas through heat transfer. In other words
$\Delta E_{int} = Q - W = 0$?

As long as the gas expands, the work $W$ done by the gas is positive. The first law of thermodynamics always holds true. To figure out whether the heat added to the gas is greater than or less than the work done by the gas, we need to know how this heat is added. If it is added at constant temperature, the work done by the gas is equal to the heat added so no net change in the internal energy. That's the dividing line. If heat is added at constant pressure, more heat enters the gas than work leaves it so there is a net increase in the internal energy. If less than the isothermal amount of heat enters the gas while the gas expands, then the internal energy will decrease.