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Sam J

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Thread moved from the technical forums, so no HH Template is shown.

Two Thermally insulated cylinders, A and B, of equal volume, both equipped with pistons, are connected by a valve. Initially A has its piston fully withdrawn and contains a perfect monatomic gas at temperature T, while B has its piston fully inserted, and the valve is closed. Calculate the final temperature of the gas after the following operation. The thermal capacity of the cylinders is to be ignored.

Piston B is fully withdrawn and the valve is opened slightly; the gas is then driven as far as it will go into B by pushing home piston A at such a rate that the pressure in A remains constant: the cylinders are in thermal contact.

My attempt:

From intuition, were the piston in cylinder A to remain stationary, then we would have a Joule expansion in which no work is done on/by the system. Therefore, the depression of the piston in A means that work must be being done on the system.

Given that the system is thermally isolated from any surroundings, there can be no gain/loss of heat energy. Therefore the first law of thermo reduces to:

dU=dW

We know that U is a function of T, therefore any work done on the system will increase its temperature.

Let T be the initial temperature of the system. Let T′ be the final temperature. Let p be the pressure. Let V be the volume of each piston.

Given T increases and p remains constant, there must be a corresponding increase in the volume of the gas.

Let V' be the final volume of the gas in cylinder B.

I write down expressions for the ideal gas equation for the initial and final thermodynamic equilibria:

Initial:

pV=nRT

Final:

p(V+V′)=nRT′

Combining and rearranging I derive:

T′=T(1+V′/V)

I become stuck when I am required to quantify the ratio of V' and V. Clearly I need to be able to express V' in terms of V, yet I cannot think of an equation with which this can be done.

Piston B is fully withdrawn and the valve is opened slightly; the gas is then driven as far as it will go into B by pushing home piston A at such a rate that the pressure in A remains constant: the cylinders are in thermal contact.

My attempt:

From intuition, were the piston in cylinder A to remain stationary, then we would have a Joule expansion in which no work is done on/by the system. Therefore, the depression of the piston in A means that work must be being done on the system.

Given that the system is thermally isolated from any surroundings, there can be no gain/loss of heat energy. Therefore the first law of thermo reduces to:

dU=dW

We know that U is a function of T, therefore any work done on the system will increase its temperature.

Let T be the initial temperature of the system. Let T′ be the final temperature. Let p be the pressure. Let V be the volume of each piston.

Given T increases and p remains constant, there must be a corresponding increase in the volume of the gas.

Let V' be the final volume of the gas in cylinder B.

I write down expressions for the ideal gas equation for the initial and final thermodynamic equilibria:

Initial:

pV=nRT

Final:

p(V+V′)=nRT′

Combining and rearranging I derive:

T′=T(1+V′/V)

I become stuck when I am required to quantify the ratio of V' and V. Clearly I need to be able to express V' in terms of V, yet I cannot think of an equation with which this can be done.