Question

A container holds a sample of ideal gas in thermal equilibrium, as shown in the figure. (Figure 1) One end of the container is sealed with a piston whose head is perfectly free to move, unless it is locked in place. The walls of the container readily allow the transfer of energy via heat, unless the piston is insulated from its surroundings.

Refer to the pV diagram presented to answer the questions below. (Figure 2) In each case, the piston head is initially unlocked and the gas is in equilibrium at the pressure and volume indicated by point 0 on the diagram.

Part A

Starting from equilibrium at point 0, what point on the pV diagram will describe the ideal gas after the following process?

"Lock the piston head in place, and hold the container above a very hot flame."

Please Choosepoint 1 point 2 point 3 point 4 point 5 point 6 point 7 point 8

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Part B

Starting from equilibrium at point 0, what point on the pV diagram will describe the ideal gas after the following process?

"Immerse the container into a large water bath at the same temperature, and very slowly push the piston head further into the container."

Please Choosepoint 1 point 2 point 3 point 4 point 5 point 6 point 7 point 8

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Part C

Starting from equilibrium at point 0, what point on the pV diagram will describe the ideal gas after the following process?

"Lock the piston head in place and plunge the container into water that is colder than the gas."

Please Choosepoint 1 point 2 point 3 point 4 point 5 point 6 point 7 point 8

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Part D

Starting from equilibrium at point 0, what point on the pV diagram will describe the ideal gas after the following process?

"The piston is now insulated from its surroundings. Pull the piston head further out of the container."

Please Choosepoint 1 point 2 point 3 point 4 point 5 point 6 point 7 point 8

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Answer 1

Concepts and reason

The concept used to solve this problem is ideal gas equation, isothermal, isochoric and adiabatic processes.

According to the given situation, firstly find out what thermodynamic factor in the ideal gas equation changes. Later, find out the point on the PV graph corresponds to that change of thermodynamic factor.

Fundamentals

The equation of state for an ideal gas is given as follows:

$PV = nRT$

Here, P is the pressure, V is the volume, n is the number of moles, R is the universal gas constant and T is the temperature of ideal gas.

An isochoric process is a thermodynamic process in which the volume of a gas remains constant. In other words, it is given that pressure of an ideal gas is directly proportional to temperature under isochoric condition.

$P \propto T$

An adiabatic process is a thermodynamic process in which there is no transfer of heat occur between system and its surroundings. The equation of state for an adiabatic process is given as follows:

$P{V^\gamma } = {\rm{constant }}$

An isothermal process is a thermodynamic process in which the temperature of a gas remains constant. In other words, it is given that pressure of an ideal gas is inversely proportional to volume under isothermal condition.

$P \propto \frac{1}{V}$

(A)

The give graph represents the pressure and volume of the sample gas. If one of these quantities changes, the position of the point representing the gas sample must also change.

Since, from the ideal gas equation temperature (T) is proportional to the product of pressure and volume. Hence, temperature of the gas can also determine from the given graph.

Curve representing the constant temperature of gas are called as isotherms.

It is given that the we lock the piston head in place, and hold the container above a very hot flame. In this situation, the volume of the gas remains same.

In the graph, three isotherms are indicated and isotherm with temperature ${T_3}$ has the largest temperature.

On the isotherm with temperature ${T_3}$ , point 8, 1 and 2 are located. But since, the volume of the gas remains same and only point 1 represents the state of constant volume.

Hence, the point 1 represents the condition i.e. Lock the piston head in place and plunge the container into water that is colder than the gas."

(B)

In such case, the volume of the gas decreases slowly and the temperature of the gas remains the same. Here, the energy flows out of the container into the large water bath and since both the container and the bath are in thermal equilibrium with each other (temperature remains same), the curve which overlaps with the isotherm curve defined the situation significantly.

Hence, the point 7 and 3 on the isotherm 2 having temperature ${T_2}$ represents the above situation. But at point 3 volume is increasing and point 7 represents the state where the volume of gas decreases.

Thus, the point 7 represents the situation i.e. "Immerse the container into a large water bath at the same temperature, and very slowly push the piston head further into the container."

(C)

It is given that the we lock the piston head in place, and plug the container into water that is colder than the gas. In this situation, the volume of the gas also remains same.

In the graph, three isotherms are indicated and isotherm with temperature ${T_1}$ has the lowest temperature.

On the isotherm with temperature ${T_3}$ , point 6, 5 and 4 are located. But since, the volume of the gas remains same and only point 5 represents the state of constant volume.

Hence, the point 5 represents the condition i.e. "Lock the piston head in place and plunge the container into water that is colder than the gas."

(D)

It is given that "The piston is now insulated from its surroundings. Pull the piston head further out of the container. "Under this condition, the volume of the gas increases and no heat flows take place i.e. adiabatic process occurs.

Curve having points 8, 0 and 4 represents the curve of adiabatic process but as the situation demands that the volume of gas is increasing. Therefore, point 4 represents the given situation i.e. "The piston is now insulated from its surroundings. Pull the piston head further out of the container.

Ans: Part A

The point 1 represents the condition i.e. Lock the piston head in place and plunge the container into water that is colder than the gas."

Part B

The point 7 represents the situation i.e. "Immerse the container into a large water bath at the same temperature, and very slowly push the piston head further into the container."

Part C

The point 5 represents the situation i.e. "Lock the piston head in place and plunge the container into water that is colder than the gas."

Part D

The point 4 represents the situation i.e. "The piston is now insulated from its surroundings. Pull the piston head further out of the container.