A certain ideal gas has a molar specific heat of Cv =
(a)
The final pressure, final volume, final temperature, change in the internal energy, the energy added to the gas by heat and the work done on the gas.
Answer to Problem 21.51AP
The final pressure is
Explanation of Solution
Given info: The molar specific heat of the gas at constant volume is
The value of universal gas constant is
The gas is heated at constant pressure.
Write the expression for molar specific heat of the gas at constant pressure.
Here,
Substitute
The gas is heated at constant pressure, therefore the final pressure of the gas is equal to the initial pressure of the gas.
The expression for the final pressure of the gas is,
Here,
Substitute
The final pressure of the gas is
The final temperature is the temperature to which the gas is heated.
The value of final temperature of the gas is,
Write the expression for the change in the temperature of the gas.
Here,
Substitute
The ideal gas equation is,
Here,
Rearrange the above equation for the value of
Substitute
Substitute
Thus, the final volume of the gas is
The expression for the change in the internal energy of the gas is,
Substitute
Substitute
Thus the change in the internal energy of the gas is
The expression for the energy added to the gas by heat is,
Substitute
Substitute
Thus the energy added to the gas by heat is
The formula for the work done on the gas is,
Substitute
Thus the work done on the gas is
Conclusion:
Therefore, for the gas, the final pressure is
(b)
The final pressure, final volume, final temperature, change in the internal energy, the energy added to the gas by heat and the work done on the gas.
Answer to Problem 21.51AP
For the gas, the final pressure is
Explanation of Solution
Given info: The molar specific heat of the gas at constant volume is
The value of universal gas constant is
The gas is heated at constant volume.
The gas is heated at constant volume, therefore the final volume of the gas is equal to the initial pressure of the gas.
The formula or ideal gas is,
Rearrange the above equation for the value of
Substitute
Write the expression for the final volume of the gas.
Substitute
Substitute
Thus, the final volume of the gas is
The final temperature is the temperature to which the gas is heated.
The value of final temperature of the gas is,
Write the expression for the change in the temperature of the gas.
Here,
Substitute
Write the expression for ideal gas.
Rearrange the above equation for the value of
Substitute
Substitute
Divide equation (2) from equation (1).
Substitute
The expression for the change in the internal energy of the gas is,
Substitute
Substitute
Thus the change in the internal energy of the gas is
The formula for the energy added to the gas by heat for constant volume is,
Substitute
Substitute
Substitute
Thus the energy added to the gas by heat is
The forrmula for the work done on the gas is,
Substitute
Thus the work done on the gas is
Conclusion:
Therefore, for the gas, the final pressure is
(c)
The final pressure, final volume, final temperature, change in the internal energy, the energy added to the gas by heat and the work done on the gas.
Answer to Problem 21.51AP
For the gas, the final pressure is
Explanation of Solution
Given info: The molar specific heat of the gas at constant volume is
The value of universal gas constant is
The gas is compressed at constant temperature.
The gas is heated at constant volume, therefore the final volume of the gas is equal to the initial pressure of the gas.
The formula for ideal gas is,
Rearrange the above equation for the value of
Substitute
Substitute
Thus, the initial volume of the gas is
Substitute
Substitute
Thus, the final volume of the gas is
The expression for the final temperature of the gas is,
Substitute
The final temperature of the gas is
The expression for the change in the temperature of the gas is,
Substitute
The final pressure of the gas is the pressure at which the gas is compressed.
The value of final; pressure of the gas is,
The expression for the change in the internal energy of the gas is,
Substitute
Substitute
Thus the change in the internal energy of the gas is
The expression for the work done on the gas is,
Divide and multiply the above equation by
Substitute
Substitute
Thus, the work done on the system is
The expression for the energy added to the gas by heat for constant volume is,
Substitute
Thus the energy added to the gas by heat is
Conclusion:
Therefore, for the gas, the final pressure is
(d)
The final pressure, final volume, final temperature, change in the internal energy, the energy added to the gas by heat and the work done on the gas.
Answer to Problem 21.51AP
For the gas, the final pressure is
Explanation of Solution
Given info: The molar specific heat of the gas at constant volume is
The value of universal gas constant is
The gas is compressed adiabatically to the final pressure.
The value of the final pressure is,
Thus, the value of the final pressure is
The expression for the ratio of the specific heats is,
Substitute
The expression for an adiabatic process for the initial condition of gas is,
Here,
The expression for an adiabatic process for the final condition of gas is,
Divide equation (4) by equation (3).
Rearrange the above equation for the value of
Substitute
Thus, the final value of the volume is
The expression for ideal gas is,
Rearrange the above equation .
Substitute
Substitute
Divide equation (7) by equation (6).
Rearrange the above expression for the value of
Substitute
The expression for the change in the temperature of the gas is,
Here,
Substitute
Thus the change in the temperature is
The expression for the change in the internal energy of the gas is,
Substitute
Substitute
Thus the change in the internal energy of the gas is
The adiabatic process is insulated to heat supplied externally.
The expression for the energy added to the gas by heat is,
Thus the energy added to the gas by heat is
The expression for the work done on the gas is,
Substitute
Thus the work done on the gas is
Conclusion:
Therefore, for the gas, the final pressure is
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Chapter 21 Solutions
EBK PHYSICS FOR SCIENTISTS AND ENGINEER
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