Figure P22.73 illustrates the cycle ABCA for a 2.00-mol sample of an ideal diatomic gas, where the process CA is a reversible isothermal expansion. What is a. the net work done by the gas during one cycle? b. How much energy is added to the gas by heat during one cycle? c. How much energy is exhausted from the gas by heat during one cycle? d. What is the efficiency of the cycle? e. What would be the efficiency of a Carnot engine operated between the temperatures at points A and B during each cycle?
(a)
The net work done by the gas during one cycle.
Answer to Problem 73PQ
The net work done by the gas during one cycle is
Explanation of Solution
The work done by the gas along AB is
Write the expression to calculate the work done by the gas along BC.
Here,
Write the expression to calculate the work done by the gas along CA.
Here,
Write the expression to calculate the total work done by the gas.
Here, W is the total work done by the gas.
Conclusion:
Substitute
Substitute
Substitute
Therefore, the net work done by the gas during one cycle is
(b)
The heat added in one cycle.
Answer to Problem 73PQ
The heat added in one cycle is
Explanation of Solution
For the diatomic gases, the specific heat capacity at constant volume is
The temperature at A and C of the gas is same.
Write the expression to calculate the temperature of the gas at A and C.
Here, T is the temperature of the gas at A and C, P is the pressure at A, V is the pressure at A n is the number of moles and R is the universal gas constant.
Write the expression to calculate the temperature at B.
Here,
Write the expression to calculate the energy added to the system.
Here, Q is the energy added to the system,
Substitute
Conclusion:
Substitute
Substitute
Substitute
Therefore, the heat added in one cycle is
(c)
The energy exhausted from the gas.
Answer to Problem 73PQ
The energy exhausted from the gas is
Explanation of Solution
For the diatomic gases, the specific heat capacity at constant pressure is
Write the expression to calculate the heat exhausted from the gas.
Here,
Substitute
Conclusion:
Substitute
Therefore, the energy exhausted from the gas is
(d)
The efficiency of the cycle.
Answer to Problem 73PQ
The efficiency of the cycle is
Explanation of Solution
Write the expression to calculate the efficiency of one cycle.
Here, e is the efficiency.
Conclusion:
Substitute
Therefore, the efficiency of the cycle is
(e)
The efficiency of the Carnot engine.
Answer to Problem 73PQ
The efficiency of the Carnot engine is
Explanation of Solution
Write the expression to calculate the efficiency of the Carnot engine.
Here,
Conclusion:
Substitute
Therefore, the efficiency of the Carnot engine is
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Chapter 22 Solutions
Physics for Scientists and Engineers: Foundations and Connections
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