Five moles of neon gas at 2.00 atm and 27.0 °C is adiabatically compressed to one-third its initial volume. Find the final pressure, final temperature, and external work done on the gas. For neon,
The final temperature, the final pressure, and the external work done on a neon gas if five moles of gas are adiabatically compressed to one-third of its initial volume.
Answer to Problem 48SP
Solution:
Explanation of Solution
Given data:
Five moles of neon gas are initially at
The neon gas is adiabatically compressed to one-third of its initial volume.
The specific heat of the neon gas at a constant volume is
The specific heat ratio of the neon gas is
The molar mass of the neon gas is
Formula used:
The formula for specific heat ratio is
Here,
The relation between specific heats and universal gas constant is written as
Here,
The ideal gas equation is written as
Here,
The number of moles are calculated by the formula:
Here,
The formula for conversion of temperature of gas from Celsius scale to Kelvin scale is
Here,
The formula for conversion of temperature of gas from Kelvin scale to Celsius scale is
The equation for an adiabatic process is written as
Here,
The heat transferred in an adiabatic process is zero. It is expressed as
Here,
The expression for the first law of thermodynamics for an adiabatic process is written as
Here,
The formula for change in internal energy of a gas is written as
Here,
Explanation:
Recall the formula for specific heat ratio for neon gas:
Substitute
Recall the expression relating specific heats and universal gas constant:
Substitute
Further solve as
Recall the expression for conversion of initial temperature of neon gas from Celsius to Kelvin scale:
Here,
Substitute
Recall the ideal gas equation for initial conditions of neon gas, that is, before compression:
The number of moles of neon gas are five. Therefore, substitute
Further solve as
According to the problem, the final volume of the gas after compression is one third of its initial volume. Therefore,
Substitute
Recall the equation for pressure and volume for an adiabatic process:
Substitute
The final pressure after adiabatic compression is
Recall the ideal gas equation for gas after adiabatic compression:
Here,
Substitute
The final temperature of the gas after adiabatic compression is
Recall the expression for the final temperature of the gas from Celsius to Kelvin scale:
Here,
Rearrange for
Substitute
Calculate the change in temperature of the gas:
Substitute
Recall the expression for number of moles of gas:
Substitute
Recall the formula for change in internal energy of the gas:
Substitute
Recall the expression for the first law of thermodynamics for adiabatic process:
Substitute
The negative sign indicates that the work is done on the system.
Therefore, the work done on the system is
Conclusion:
The final pressure is
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Chapter 20 Solutions
College Physics
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