Air (a diatomic ideal gas) at 27.0°C and atmospheric pressure is drawn into a bicycle pump that has a cylinder with an inner diameter of 2.50 cm and length 50.0 cm. The downstroke adiabatically compresses the air, which readies a gauge pressure of 8.00 × 105 Pa before entering the tire. We wish to investigate the temperature increase of the pump. (a) What is the initial volume of the air in the pump? (b) What is the number of moles of air in the pump? (c) What is the absolute pressure of the compressed air? (d) What is the volume of the compressed air? (c) What is the temperature of the compressed air? (f) What is the increase in internal energy of the gas during the compression? What If? The pump is made of steel that is 2.00 mm thick. Assume 4.00 cm of the cylinder’s length is allowed to come to thermal equilibrium with the air. (g) What is the volume of steel in this 4.00-cm length? (h) What is the mass of steel in this 4.00-cm length? (i) Assume the pump is compressed once. After the adiabatic expansion,
(a)
The initial volume of the air in the pump.
Answer to Problem 21.32P
The initial volume of the air in the pump is
Explanation of Solution
Initial temperature for diatomic gasis
Write the expression to calculate the radius of the pump.
Here,
Write the formula to calculate the initial volume of the air in the pump.
Here,
Substitute
Substitute
Thus, the initial volume of the air in the pump is
Conclusion:
Therefore, the initial volume of the air in the pump is
(b)
The number of moles of air in the pump.
Answer to Problem 21.32P
The number of moles of air in the pump is
Explanation of Solution
Initial temperature for diatomic gas is
Write the formula to calculate the number of moles of air in the pump.
Here,
The value of atmospheric pressure for diatomic gas is
Substitute
Thus, the number of moles of air in the pump is
Conclusion:
Therefore, the number of moles of air in the pump is
(c)
The absolute pressure of the compressed air.
Answer to Problem 21.32P
The absolute pressure of the compressed air is
Explanation of Solution
Initial temperature for diatomic gas is
Write the formula to calculate the absolute pressure of the compressed air.
Here,
Substitute
Thus, the absolute pressure of the compressed air is
Conclusion:
Therefore, the absolute pressure of the compressed air is
(d)
The volume of the compressed air.
Answer to Problem 21.32P
The volume of the compressed air is
Explanation of Solution
Initial temperature for diatomic gas is
Write the expression for the adiabatic compression.
Here,
Write the formula to calculate the volume of the compressed air.
Substitute
Thus, the volume of the compressed air is
Conclusion:
Therefore, the volume of the compressed air is
(e)
The temperature of the compressed air.
Answer to Problem 21.32P
The temperature of the compressed air is
Explanation of Solution
Initial temperature for diatomic gas is
Write the formula to calculate the temperature of the compressed air.
Here,
Substitute
Thus, the temperature of the compressed air is
Conclusion:
Therefore, the temperature of the compressed air is
(f)
The increase in internal energy of the gas during the compression.
Answer to Problem 21.32P
The increase in internal energy of the gas during the compression is
Explanation of Solution
Initial temperature for diatomic gas is
For adiabatic process, the work done on the gas is equal to the change in internal energyof the gas during the compression.
Here,
Write the expression for the change in internal energyof the gas during the compression.
Here,
Write the expression for specific heat at constant volume.
Here,
Equate the three expressions (7),(8) and (9)and re-arrange to get
Write the formula to calculate the change in temperature of a monatomic ideal gas.
Here,
Substitute
Thus, the change in temperature of a monatomic ideal gas is
Substitute
Thus, the increase in internal energy of the gas during the compression is
Conclusion:
Therefore, the increase in internal energy of the gas during the compression is
(g)
The volume of the steel in this
Answer to Problem 21.32P
The volume of the steel in this
Explanation of Solution
Initial temperature for diatomic gas is
Write the formula to calculate the volume of the steel in this
Here,
Write the expression to calculate the square radius of the pump for steel.
Here,
Write the formula to calculate the outer radius of the pump.
Here,
Substitute
Thus, the outer radius of the pump is
Substitute
Thus, the square radius of the pump for steel is
Substitute
Thus, the volume of the steel in this
Conclusion:
Therefore, the volume of the steel in this
(h)
The mass of the steel in this
Answer to Problem 21.32P
The mass of the steel in this
Explanation of Solution
Initial temperature for diatomic gas is
Write the formula to calculate the mass of the steel in this
Here,
The value of density of the steel is
Substitute
Thus, the mass of the steel in this
Conclusion:
Therefore, the mass of the steel in this
(i)
The increase in temperature of the steel after one compression.
Answer to Problem 21.32P
The increase in temperature of the steel after one compression is
Explanation of Solution
Initial temperature for diatomic gas is
After the adiabatic compression, conduction in the part (f) being shared between the gas and the
The work done on the gas is equal to the sum of change in internal energyof the gas during the compression and the heat supplied.
Here,
Write the expression for
Here,
The value of specific heat capacity is
Substitute
Substitute
Thus, the increase in temperature of the steel after one compression is
Conclusion:
Therefore, the increase in temperature of the steel after one compression is
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Chapter 21 Solutions
Physics for Scientists and Engineers, Technology Update (No access codes included)
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