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Q. 55

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Physics for Scientists and Engineers: A Strategic Approach with Modern Physics
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Short Answer

A heat engine using of a monatomic gas follows the cycle shown in FIGURE P. of heat energy is transferred to the gas during process .

a. Determine , and for each of the four processes in this cycle. Display your results in a table.

b. What is the thermal efficiency of this heat engine?

a)The table result is

b)As a result, the heat engine's efficiency is increased

See the step by step solution

Step by Step Solution

Step 1:Given data

The general expression for work done is the product of pressure and volume change.

Here, p is pressure and is change in volume.

Step 2: The monotonic gas

The monoatomic gas goes through the cyclic process depicted in the figure.

Processes and are isochoric and processes and are isobaric.

Step 3: First law of thermodynamics

The first law of thermodynamics states that the relationship between heat supplied or taken out of a gas, change in the internal energy of the gas, and work done is as follows:

Here, Q is the heat supplied or taken out, is the change in the internal energy of the gas and W denotes the work done by or on the gas.

The product of pressure and volume change is the general expression for work done.

Step 4: Volume of the gas

As, process Isochoric means that the volume of the gas remains constant or changes by zero. As a result, work done is also zero.

Use this in the equation

According to the graph, the heat given to the gas is . So, substitute for Q here

 Step 5:Find temperature of the gas at the end of the process

During an isochoric process, the heat transferred to the gas is,

Here, n is the number of moles in the gas, is the specific heat of the gas at constant temperature, is the initial temperature of the gas and is its temperature at the end of the process .

A monatomic gas's heat capacity at constant volume is,

Here, R is the gas constant.

Use this in the above equation to get,

Rearrange this for the gas temperature at the end of the process,

Step 6:  Find the volume of the gas

Substitute for for for $n$ and for in the above equation,

Find the volume of the gas :

ideal gas equation is

Rearrange this for the gas temperature at the end of the process,

For the case of process , write this as follows:

Here, is the pressure of the gas at the beginning of process .

Step 7: Solution

Substitute for for

Step 8: Pressure of the gas

Process :

According to the diagram, the volume of the gas remains constant at all points. and

That is, process isochoric.

As the process was isochoric, pressure of the gas is proportional to its temperature in this process,

Here, and are the pressure and temperature of the gas at the point 2 .

Rearrange this for the gas pressure at point 2. ,

Substitute for for and for .

Step 9: The heat transfer during an isobaric process 

From the given diagram, volume of the gas at the end of this process (or) at the point 3 is,

Substitute for ?

As process is an isobaric process, volume of the gas is proportional to its temperature,

Here, is the temperature at the point 3 .

Rearrange this for

Substitute for and for ,

The rate of heat transfer in an isobaric process is,

Substitute for for for and for ,

Step 10: Work done process 

The work done in this process is equal to the product of the gas pressure and the change in temperature.

Substitute for for and for

According to the first law of thermodynamics, the difference in the heat supplied and the work done equals the change in the internal energy of the gas..

Step 11: pressure is proportional to the temperature 

For the process

Keep in mind that the values used in this and the following processes were discovered earlier in the first two cases.

Pressure is proportional to temperature in this isochoric process.

Substitute for for and for ,

Step 12:Heat transferred in isochoric process 

Heat is transferred in the isochoric process by,

Substitute for for for and for

The work done in isochoric process is,

From the first law of thermodynamics, change in the internal energy of the gas is equal to the difference in the heat supplied and the work done,

Step 13:The pressure and change in volume (part a)

a)Process

This process is similar to the process , in reverse order. This is an isobaric process. Heat transferred is given by the equation,

Substitute for for for and for

Work done is equal to the product of pressure and volume change.

Substitute for for and for ,

From the first law of thermodynamics,

Substitute for and for ,

The below table shows the results of all processes. (All energies are in )

Step 14:Thermal efficiency of heat engine(part b)

b)A heat engine's thermal efficiency is equal to the ratio of total work done to total heat input.

Total amount of work completed is

Use the values in the preceding table to get,

The head absorbed is,

Use the values in the preceding table to get,

Step 15:Effiency of heat engine(part b)

b)Substitute for W and for Q in the equation for efficiency of the engine,

As a result, the heat engine's efficiency is increased.

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