(a)
All unknown pressures, volumes, and temperatures in the table given in the question.
(a)
Answer to Problem 32P
The table completed with unknown pressures, volumes, and temperatures given beow.
state | P(kPa) | V (L) | T (K) |
A | 1400 | 10.0 | 720 |
B | 875 | 16.0 | 720 |
C | 445 | 24.0 | 549 |
D | 712 | 15.0 | 549 |
Explanation of Solution
Consider the adiabatic process
Write the adiabatic condition for the monoatomic ideal gas.
Here,
Rearrange above equation to get
Write ideal gas equation for ideal gas at point
Here,
Rearrange above equation to get
Write ideal gas equation for ideal gas at point
Rearrange above equation to get
Use equation (III) in (IV) in equation (I) to get
Consider the isothermal process
Write the condition for the isothermal process.
Write the equation for an isothermal process.
Here,
Apply isothermal equation to process
Rearrange above equation to get
Substitute
Consider the adiabatic process
Write the adiabatic condition for the monoatomic ideal gas.
Here,
Apply isothermal equation to process
Rearrange to get
Use equation (XI) and (IX) in equation (X) to get
Conclusion:
Substitute
Substitute
Substitute
Substitute
Substitute
The table completed with unknown pressures, volumes, and temperatures given below.
state | P(kPa) | V (L) | T (K) |
A | 1400 | 10.0 | 720 |
B | 875 | 16.0 | 720 |
C | 445 | 24.0 | 549 |
D | 712 | 15.0 | 549 |
(b)
The energy added by heat, work done by the engine, and the change in internal energy for each of the steps
(b)
Answer to Problem 32P
The energy added by heat, work done by the engine, and the change in internal energy for each of the steps
Process | Q(kJ) | W(kJ) | |
+6.58 | 0 | ||
0 | |||
0 | |||
0 |
Explanation of Solution
Consider the isothermal process
Write the expression for the change in internal energy.
Here,
Write the expression for the first law of
Here,
Write the expression for the work done in the isothermal process.
Consider the adiabatic process
Therefore,
Write the expression for the change in internal energy.
Here,
Consider the isothermal process
Write the expression for the work done in the isothermal process.
Consider the adiabatic process
Therefore,
Write the expression for the change in internal energy.
Here,
Conclusion:
For the isothermal process
For isothermal process
Substitute
Substitute
Substitute
For the adiabatic process
Substitute
Substitute
Consider the isothermal process
Substitute
Substitute
Substitute
For the adiabatic process
Substitute
Substitute
Therefore,
Process | Q(kJ) | W(kJ) | |
+6.58 | 0 | ||
0 | |||
0 | |||
0 |
(c)
The efficiency
(c)
Answer to Problem 32P
The efficiency
Explanation of Solution
Write the expression for the efficiency.
Here,
The total work done by the engine is the negative of the work input.
Write the expression for the net work done by the engine.
Here,
Write the expression for the
Here,
Write the expression for
Use equation (XXII) and (XXIII) in equation (XX) to get
Conclusion:
Substitute
Substitute
Therefore, the efficiency
(d)
To show that the efficiency is equal to Carnot efficiency.
(d)
Answer to Problem 32P
It is showed that the efficiency of the engine calculated in part(c) is equal to Carnot efficiency.
Explanation of Solution
Write the expression for the Carnot efficiency.
Here,
Conclusion:
Substitute
The above calculated efficiency is equal to that obtained in part(c). Thus, efficiency of the engine is equal to Carnot efficiency.
Therefore, it is showed that the efficiency of the engine calculated in part(c) is equal to Carnot efficiency.
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Chapter 22 Solutions
Physics For Scientists And Engineers With Modern Physics, 9th Edition, The Ohio State University
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