At point A in a Carnot cycle, 2.34 mol of a monatomic ideal gas has a pressure of 1 4000 kPa, a volume of 10.0 L, and a temperature of 720 K. The gas expands isothermally to point B and then expands adiabatically to point C, where its volume is 24.0 L. An isothermal compression brings it to point D, where its volume is 15.0 L. An adiabatic process returns the gas to point A. (a) Determine all the unknown pressures, volumes, and temperatures as you f ill in the following table:
(b) Find the energy added by heat, the work done by the engine, and the change in internal energy for each of the steps A → B, B → C, C → D, and D → A (c) Calculate the efficiency Wnet/|Qk|. (d) Show that the efficiency is equal to 1 - TC/TA, the Carnot efficiency.
(a)
The unknown pressures, volumes and the temperature in the table.
Answer to Problem 22.32P
The values of unknown pressures, volumes and the temperature in the table are,
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Explanation of Solution
Given: The number of moles of a mono atomic ideal gas is
Write the equation of adiabatic process
Here,
The value of
Substitute
Thus, the pressure of the gas at point
Write the ideal gas equation.
Here,
The value of gas constant is
Substitute
Thus, the temperature of the gas at point
In isothermal process, the temperature is constant.
For isothermal process
The temperature of the gas at point
Thus, the temperature of the gas at point
Write the ideal gas equation.
Here,
Substitute
Thus, the pressure of the gas at point
Write the equation of adiabatic process
Here,
Substitute
Thus, the volume of the gas at point
In isothermal process, the temperature is constant.
For isothermal process
The temperature of the gas at point
Thus, the temperature of the gas at point
Write the ideal gas equation.
Here,
Substitute
Thus, the pressure of the gas at point
Form a table and show the unknown value of pressures, volumes and temperatures.
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Conclusion:
Therefore, the values of unknown pressures, volumes and the temperature in the table are,
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(b)
The energy added by heat, work done by the engine and the change in internal energy for each of the steps
Answer to Problem 22.32P
The values of energy added by heat, work done by the engine and the change in internal energy for each of the steps in the table are,
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Explanation of Solution
Given: The number of moles of a mono atomic ideal gas is
The process
Write the equation of change in temperature for process
Here,
Substitute
Thus, the change in internal energy for process
Write the equation of work done by the engine for process
Substitute
Thus, the work done by the engine for process
Write the equation of isothermal process
Substitute
Thus, the energy added by heat for process
Write the equation of change in temperature for process
Here,
The value of
Substitute
Substitute
Thus, the change in internal energy for process
The process
Thus, the energy added by heat for process
Write the equation of change in internal energy for process
Substitute
Thus, the work done by the engine for process
The process
Write the equation of change in temperature for process
Substitute
Thus, the change in internal energy for process
Write the equation of work done by the engine for process
Substitute
Thus, the work done by the engine for process
Write the equation of isothermal process
Substitute
Thus, the energy added by heat for process
Write the equation of change in temperature for process
Substitute
Substitute
Thus, the change in internal energy for process
The process
Thus, the energy added by heat for process
Write the equation of change in internal energy for process
Substitute
Thus, the work done by the engine for process
Form a table and show the value of energy added by heat, work done by the engine and the change in internal energy.
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Conclusion:
Therefore, the values of energy added by heat, work done by the engine and the change in internal energy for each of the steps in the table are,
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(c)
The value of efficiency
Answer to Problem 22.32P
The value of efficiency
Explanation of Solution
Given: The number of moles of a mono atomic ideal gas is
Calculate the net work done from the table is,
Write the equation for efficiency.
Here,
Substitute
The value of efficiency
Conclusion:
Therefore, the value of efficiency
(d)
To show: The efficiency is equal to the Carnot efficiency
Answer to Problem 22.32P
The efficiency is equal to the Carnot efficiency
Explanation of Solution
Given: The number of moles of a mono atomic ideal gas is
Write the equation for Carnot efficiency.
Here,
The value of
Substitute
Thus, the Carnot efficiency is
Write the equation for efficiency.
Substitute
The value of efficiency is
Conclusion:
Therefore, the efficiency is equal to the Carnot efficiency
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Chapter 22 Solutions
Physics for Scientists and Engineers, Technology Update (No access codes included)