Work Done by an Expanding Gas



Work Done by an Expanding Gas

Learning Goal: To derive the expression for the work done by an expanding gas, [pic], and to understand how it follows from the expression [pic]for mechanical work.

Especially from the historically important perspective of making engines to convert heat energy into work, the work in thermodynamics is defined as the work done by the system on the exterior world, and not vice versa as is done in the rest of classical mechanics. In classical mechanics, one always considers the work done on a system by the outside world. Rarely does one think about the work done by the system. Suppose you push a large block with a certain force of magnitude [pic]over some distance. You have done work on the block; hence the energy of the block should increase. According to Newton's 3rd law, the block exerts the same magnitude of force [pic], but in the opposite direction (i.e., directed back at you). Hence, the work done by the block (on you) is negative, since the direction of motion opposes the direction of the force. In summary, you have to be careful about the sign of the work: the same situation gives opposite signs of the work depending on whether our perspective is classical mechanics or thermodynamics.

In thermodynamics, one often deals with liquids and gases that exert forces on their containers (i.e., the fluids exert pressure over an area). If the container changes volume, then this force acts through a distance and hence does work.

For a steam engine, the example pictured here, the "container" is a cylinder whose volume changes as the piston slides in or out. Suppose a gas is confined within the cylinder. The pressure of the gas is [pic], and the area of the cylinder is [pic]. Consider the work done as the gas expands, pushing the piston to the right. Call the infinitesimal distance the piston moves [pic].

A. What force [pic]does the gas exert on the piston? (Note that the positive x axis is to the right in the figure.)

Express the force in terms of [pic], [pic], and any constants,

|[pic]|p*A | | |

|= | | | |

B. If the piston moves a distance [pic], what is [pic], the work done by the gas?

Express the work done by the gas in terms of given quantities.

|[pic]= |p*A*dx | | |

| | | | |

C. What is [pic], the increase in volume of the gas?

Express the differential increase in terms of [pic]and other given quantities.

|[pic]=|A*dx | | |

| | | | |

D. [pic]Now find the work done by the gas in terms of the thermodynamic variables.

Express the differential work [pic]in terms of thermodynamic variables such as the gas's pressure [pic], temperature [pic], volume [pic], and its change in volume [pic].

|[pic]= |p*dV | | |

| | | | |

E. Suppose that the gas expands from [pic]to [pic]at constant pressure [pic]. How much work [pic]is done by the gas?

Express the work in terms of [pic], [pic], and [pic].

|[pic]=|p_0*(V_1-V_0) | | |

| | | | |

F. [pic]Is the work you just computed positive or negative?

|[pic]positive | | |

|[pic]negative | | |

G. Assume now that the diameter of the piston is reduced by a factor of 2. What is the amount of work [pic]done by a gas of pressure [pic]in expanding from [pic]to [pic]?

Express your answer in terms of [pic],[pic], [pic], and simple numerical factors.

|[pic]= |p_0*(V_1-V_0) | | |

| | | | |

[pic] [ Print ] 

Work Integral in the pV Plane

The diagram shows the pressure and volume of an ideal gas during one cycle of an engine. As the gas proceeds from state 1 to state 2, it is heated at constant pressure. It is then cooled at constant volume, until it reaches state 3. The gas is then cooled at constant pressure to state 4. Finally, the gas is heated at constant volume until it returns to state 1.

A. Find [pic], the work done by the gas as it expands from state 1 to state 2.

Express the work done in terms of [pic]and [pic].

|[pic]= |9*p_0*V_0 | | |

| | | | |

B. Find [pic], the work done by the gas as it cools from state 2 to state 3.

Express your answer in terms of [pic]and [pic].

|[pic]= |0 | | |

| | | | |

C. Find [pic], the work done by the gas as it is compressed from state 3 to state 4.

Express your answer in terms of [pic]and [pic].

|[pic]= |-3*p_0*V_0 | | |

| | | | |

D. Find [pic], the work done by the gas as it is heated from state 4 to state 1.

Express your answer in terms of [pic]and [pic].

|[pic]= |0 | | |

| | | | |

E. What is [pic], the total work done by the gas during one cycle?

Express your answer in terms of [pic]and [pic].

|[pic]= |6*p_0*V_0 | | |

| | | | |

F. When the gas is in state 1, its temperature is [pic]. Find the temperature [pic]of the gas when it is in state 3. (Remember, this is an ideal gas.)

Express [pic]in terms of [pic].

|[pic]|(4/3)*T_1 | | |

|= | | | |

[pic] [ Print ] 

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[pic]Various Gas Expansions: pV Plots and Work

An ideal monatomic gas is contained in a cylinder with a movable piston so that the gas can do work on the outside world, and heat can be added or removed as necessary. The figure[pic] shows various paths that the gas might take in expanding from an initial state whose pressure, volume, and temperature are [pic], [pic], and [pic]respectively. The gas expands to a state with final volume [pic]. For some answers it will be convenient to generalize your results by using the variable [pic], which is the ratio of final to initial volumes (equal to 4 for the expansions shown in the figure.)

The figure shows several possible paths of the system in the pV plane. Although there are an infinite number of paths possible, several of those shown are special because one of their state variables remains constant during the expansion. These have the following names:

• Adiabiatic: No heat is added or removed during the expansion.

• Isobaric: The pressure remains constant during the expansion.

• Isothermal: The temperature remains constant during the expansion.

A. Which of the curves in the figure represents an isobaric process?

|[pic]A [pic]B [pic]C [pic]D | | |

B. What happens to the temperature of the gas during an isobaric expansion?

|[pic]Temperature increases. | | |

|[pic]Temperature remains constant. | | |

|[pic]Temperature decreases. | | |

C. Which of the curves in the figure represents an isothermal process?

|[pic]A [pic]B [pic]C [pic]D | | |

D. Graphically, the work along any path in the pV plot ____________.

|[pic]is the area to the left of the curve from [pic]to [pic] | | |

|[pic]is the area under the curve from [pic]to [pic] | | |

|[pic]requires knowledge of the temperature [pic] | | |

E. Calculate [pic], the work done by the gas as it expands along path A from [pic]to [pic].

Express [pic]in terms of [pic], [pic], and [pic].

|[pic]= |p_0* V_0 * (R_v - 1) | | |

| | | | |

F. [pic]Calculate the work [pic]done by the gas during the isothermal expansion.

Express [pic]in terms of [pic], [pic], and [pic].

|[pic]= |p_0*V_0*ln(R_v) | | |

| | | | |

G. Which of the curves shown represents an adiabatic expansion?

|[pic]A [pic]B [pic]C [pic]D | | |

[pic] [ Print ] 

Fast vs. Slow Tire Pumping

Imagine the following design for a simple tire pump. The pump is filled with a volume [pic]of air at atmospheric pressure [pic]and ambient temperature [pic]. When you push the pump handle, the air is compressed to a new (smaller) volume [pic], raising its pressure. A valve is then opened, allowing air to flow from the pump into the tire until the remaining air in the pump reaches the pressure of the air in the tire, [pic].

In this problem, you will consider whether you can get more air into the tire per pump cycle by pushing the pump handle quickly or slowly.

We will make the following simplifying assumptions:

• The pressure of the air in the tire, [pic], does not change significantly as air flows into the tire from the pump.

• The temperature of the air in the pump does not change significantly while air is flowing into the tire (i.e., this is an isothermal process).

• The air is mainly composed of diatomic molecules with [pic].

A. First imagine that you push the pump handle quickly, so that the compression of air in the pump occurs adiabatically. Find the absolute temperature [pic]of the air inside the pump after a rapid compression from volume [pic]to volume [pic], assuming an ambient temperature of [pic].

Express your answer in terms of [pic], [pic], [pic], and [pic]

|[pic]=|T_a*(V_i/V_f)^(gamma-1) | | |

| |or | | |

| |T_a*(V_i/V_f)^(0.40) | | |

| | | | |

B. [pic]Once the tire and pump pressures have equilibrated at [pic], what fraction [pic]of the gas particles initially in the pump will have ended up in the tire?

Express the fraction in terms of [pic], [pic], [pic], [pic], and [pic]. The temperatures [pic]and [pic]should not appear in your answer.

|[pic|1-(p_t/p_a)*(V_f/V_i)^gamma | | |

|]= |or | | |

| |1-(p_t/p_a)*(V_f/V_i)^(1.40) | | |

| | | | |

C. Now imagine that you push the pump handle slowly, so that the compression of air in the pump occurs isothermally. Find the absolute temperature [pic]of the air inside the pump assuming an ambient temperature of [pic].

|[pic]=|T_a | | |

| | | | |

D. [pic]Once the tire and pump pressures have equilibrated at [pic], what fraction [pic]of the gas particles initially in the pump will have ended up in the tire?

Express the fraction in terms of [pic], [pic], [pic], [pic], and [pic]. The temperatures [pic]and [pic]should not appear in your answer.

|[pic|1-(p_t/p_a)*(V_f/V_i) | | |

|]= | | | |

E. Assume that [pic]and [pic]. Which method, fast pumping (adiabatic process) or slow pumping (isothermal process) will put a larger amount of air into the tire per pump cycle?

|[pic]fast pumping | | |

|[pic]slow pumping | | |

Carnot Cycle

After Count Rumford (Benjamin Thompson) and James Prescott Joule had shown the equivalence of mechanical energy and heat, it was natural that engineers believed it possible to make a "heat engine" (e.g., a steam engine) that would convert heat completely into mechanical energy. Sadi Carnot considered a hypothetical piston engine that contained [pic]moles of an ideal gas, showing first that it was reversible, and most importantly that--regardless of the specific heat of the gas--it had limited efficiency, defined as [pic], where [pic]is the net work done by the engine and [pic]is the quantity of heat put into the engine at a (high) temperature [pic]. Furthermore, he showed that the engine must necessarily put an amount of heat [pic]back into a heat reservoir at a lower temperature [pic].

The cycle associated with a Carnot engine is known as a Carnot cycle. A pV plot of the Carnot cycle is shown in the figure. The working gas first expands isothermally from state A to state B, absorbing heat [pic]from a reservoir at temperature [pic]. The gas then expands adiabatically until it reaches a temperature [pic], in state C. The gas is compressed isothermally to state D, giving off heat [pic]. Finally, the gas is adiabatically compressed to state A, its original state.

A. Indicate whether the following statements are true or false:

1. For the gas to do positive work, the cycle must be traversed in a clockwise manner.

2. Positive heat is added to the gas as it proceeds from state C to state D.

3. The net work done by the gas is proportional to the area inside the closed curve.

4. The heat transferred as the gas proceeds from state B to state C is greater than the heat transferred as the gas proceeds from state D to state A.

Use t for true or f for false. Separate your answers with commas.

|t | | |

|, | | |

|f | | |

|, | | |

|t | | |

|, | | |

|f | | |

| | | |

B. Find the total work [pic]done by the gas after it completes a single Carnot cycle.

Express the work in terms of any or all of the quantities [pic], [pic], [pic], and [pic].

|[pic]=|Q_h-Q_c | | |

| |or | | |

| |Q_h*(1-T_c/T_h) | | |

| | | | |

C. Suppose there are [pic]moles of the ideal gas, and the volumes of the gas in states A and B are, respectively, [pic]and [pic]. Find [pic], the heat absorbed by the gas as it expands from state A to state B.

Express the heat absorbed by the gas in terms of [pic], [pic], [pic], the termperature of the hot reservoir [pic], and the gas constant [pic].

|[pic]=|n*R*T_h*ln(V_B/V_A) | | |

| | | | |

D. The volume of the gas in state C is [pic], and its volume in state D is [pic]. Find [pic], the magnitude of the heat that flows out of the gas as it proceeds from state C to state D.

Express your answer in terms of [pic], [pic], [pic], [pic](the temperature of the cold reservoir), and [pic].

|[pic]=|n*R*T_c*ln(V_C/V_D) | | |

| | | | |

E. Now, by considering the adiabatic processes (from B to C and from D to A), find the ratio [pic]in terms of [pic]and [pic].

|[pic]= |V_B/V_A | | |

| | | | |

F. Using your expressions for [pic]and [pic](found in Parts C and D), and your result from Part E, find a simplified expression for [pic].

No volume variables should appear in your expression, nor should any constants (e.g., [pic]or [pic]).

|[pic]= |T_c/T_h | | |

| | | | |

G. The efficiency of any engine is, by definition, [pic]. Carnot proved that no engine can have an efficiency greater than that of a Carnot engine. Find the efficiency [pic]of a Carnot engine.

Express the efficiency in terms of [pic]and [pic].

|[pic]= |1-T_c/T_h | | |

| | | | |

[pic] [ Print ] 

Isothermal Engine

Consider a thermal engine filled with [pic]molecules of a monatomic ideal gas. The gas undergoes a cyclic transformation, as shown.[pic] Processes AB and CD are isothermal (constant temperature), and processes BC and DA are isochoric (constant volume).

The quantities [pic], [pic], [pic], and [pic]are defined in the figure. For example, when the gas is in state A, it has pressure [pic]and volume [pic]; when the gas is in state B, it has volume [pic], but the pressure is not given.

Work done by the gas

A. Find [pic], the work done by the gas during process AB.

Express the work in terms of [pic], [pic], [pic], and/or [pic].

|[pic]= |p_2*V_1*ln(V_2/V_1) | | |

| | | | |

B. Find [pic], the work done by the gas during process BC.

Express your answer in terms of [pic], [pic], [pic], and/or [pic].

|[pic]= |0 | | |

| | | | |

C. Find [pic], the work done by the gas during process CD.

Express your answer in terms of [pic], [pic], [pic], and/or [pic].

|[pic]= |-p_1*V_2*ln(V_2/V_1) | | |

| | | | |

D. Find [pic], the work done by the gas during process DA.

Express your answer in terms of [pic], [pic], [pic], and/or [pic].

|[pic]= |0 | | |

| | | | |

The internal energy

Now, you will investigate the change in the gas's internal energy during each of the four segments of its cycle. You may find it helpful to recall that, for a monatomic ideal gas, [pic], where [pic]is the heat capacity of the gas at constant volume.

A. What is the change in internal energy [pic]during the process AB?

Express the change in internal energy in terms of [pic], [pic], [pic], and/or [pic].

|[pic]= |0 | | |

| | | | |

B. What is the change in internal energy [pic]during the process BC?

Express the change in internal energy in terms of [pic], [pic], [pic], and/or [pic].

|[pic]= |(3/2)*(p_1*V_2-p_2*V_1) | | |

| | | | |

C. What is the change in internal energy [pic]during the process CD?

Express your answer in terms of [pic], [pic], [pic], and/or [pic].

|[pic]= |0 | | |

| | | | |

D. What is the change in internal energy [pic]during the process DA?

Express your answer in terms of [pic], [pic], [pic], and/or [pic].

|[pic]= |-(3/2)*(p_1*V_2-p_2*V_1) | | |

| | | | |

E. What is the total change in the gas's internal energy after it has completed a cycle?

|[pic]= |0 | | |

| | | | |

Heat

Finally, consider the heat absorbed by the gas during its cycle. Define [pic]to be heat absorbed by the gas in process AB, etc. If a process is exothermic (i.e., heat is given off by the gas during the process), then the quantity of heat absorbed will be negative.

A. Indicate the processes in this cycle during which heat is flowing into the gas. That is, indicate the segments where [pic]is positive.

|[pic]DA only | | |

|[pic]CD and DA | | |

|[pic]DA and AB | | |

|[pic]AB and BC | | |

[pic] [ Print ] 

Heat Pumps and Refrigerators

Learning Goal: To understand that a heat engine run backward is a heat pump that can be used as a refrigerator.

By now you should be familiar with heat engines--devices, theoretical or actual, designed to convert heat into work. You should understand the following:

1. Heat engines must be cyclical; that is, they must return to their original state some time after having absorbed some heat and done some work).

2. Heat engines cannot convert heat into work without generating some waste heat in the process.

The second characteristic is a rigorous result even for a perfect engine and follows from thermodynamics. A perfect heat engine is reversible, another result of the laws of thermodynamics.

If a heat engine is run backward (i.e., with every input and output reversed), it becomes a heat pump (as pictured schematically[pic]). Work [pic]must be put into a heat pump, and it then pumps heat from a colder temperature [pic]to a hotter temperature [pic], that is, against the usual direction of heat flow (which explains why it is called a "heat pump"). The big surprise is that the laws of thermodynamics seem to allow more than 100% efficiency: The heat coming out the hot side [pic]of a heat pump or the heat going in to the cold side [pic]of a refrigerator is more than the work put in; in fact it can be many times as large. For this reason, the ratio of the heat to the work in heat pumps and refrigerators is called the coefficient of performance. In a refrigerator, this is the ratio of heat removed from the cold side [pic]to work put in:

[pic].

In a heat pump the coefficient of performance is the ratio of heat exiting the hot side [pic]to the work put in:

[pic].

A. What is the relationship of [pic]to the usual thermodynamic definition of work [pic]?

Express [pic]in terms of [pic]and other quantities given in the introduction.

|[pic]= |-W | | |

| | | | |

B. Find [pic], the heat pumped out by the ideal heat pump.

Express [pic]in terms of [pic]and [pic].

|[pic]=|W_in + Q_c | | |

| | | | |

C. A heat pump is used to heat a house in winter; the inside radiators are at [pic]and the outside heat exchanger is at [pic]. If it is a perfect (e.g., Carnot cycle) heat pump, what is [pic], its coefficient of performance?

Give your answer in terms of [pic]and [pic].

|[pic]= |T_h/(T_h - T_c) | | |

| | | | |

D. The heat pump is designed to move heat. This is only possible if certain relationships between the heats and temperatures at the hot and cold sides hold true. Indicate the statement that must apply for the heat pump to work.

|[pic][pic]and [pic]. | | |

|[pic][pic]and [pic]. | | |

|[pic][pic]and [pic]. | | |

|[pic][pic]and [pic]. | | |

E. Assume that you heat your home with a heat pump whose heat exchanger is at [pic], and which maintains the baseboard radiators at [pic]. If it would cost $1000 to heat the house for one winter with ideal electric heaters (which have a coefficient of performance of 1), how much would it cost if the actual coefficient of performance of the heat pump were 75% of that allowed by thermodynamics?

Express the cost in dollars.

|Cost = |187.5 |dollars | |

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