Air (a diatomic ideal gas) at 30.0°C and atmospheric pressure is drawn into a bicycle pump that has a cylinder with an inner diameter of 2.50 cm and length 70 cm. The down stroke adiabatically compresses the air, which reaches a gauge pressure of 800 kPa before entering the tire. (a) Determine the volume of the compressed air.
m3
(b) Determine the temperature of the compressed air.
K
(c) The pump is made of steel and has an inner wall that is 1.80 mm thick. Assume that 14.70 cm of the cylinder's length is allowed to come to thermal equilibrium with the air. What will be the increase in wall temperature?
K
I tried using the pv=nrt formuala rearaanged but i am confused on how the thickness and length play a part?
To solve this problem, you can use the ideal gas law, which relates the pressure (P), volume (V), temperature (T), and number of moles (n) of a gas. The ideal gas law equation is:
PV = nRT
Where:
- P is the pressure of the gas
- V is the volume of the gas
- R is the ideal gas constant (8.314 J/(mol·K))
- T is the temperature of the gas
- n is the number of moles of the gas
(a) To determine the volume of the compressed air (V), we first need to find the initial volume of the air. We know the air is at atmospheric pressure, which is typically around 101.3 kPa. We can convert this to Pa for consistency:
Initial pressure (P1) = 101.3 kPa = 101,300 Pa
Final pressure (P2) = 800 kPa = 800,000 Pa
Since the compression process is adiabatic, we can use the adiabatic equation to relate the initial and final conditions:
P1 * V1^γ = P2 * V2^γ
Where γ is the heat capacity ratio for the diatomic ideal gas, which is approximately 1.4 for air. Rearranging the equation to solve for V2:
V2 = V1 * (P1 / P2)^(1/γ)
We know the diameter (d) and length (L) of the cylinder, so we can calculate the initial volume (V1) using the formula:
V1 = π * (d/2)^2 * L
Substituting known values in this formula, we can find V1. Then, substituting all the values in the equation above, we can calculate V2.
(b) To determine the temperature of the compressed air (T2), we can use the relationship:
T2 / T1 = (P2 / P1)^((γ - 1) / γ)
Where T1 is the initial temperature of the air. Rearranging the equation to solve for T2:
T2 = T1 * (P2 / P1)^((γ - 1) / γ)
We know the initial temperature (T1), and we can calculate T2 using the known values.
(c) To determine the increase in wall temperature, we can use the fact that the change in internal energy (ΔU) of an ideal gas is given by:
ΔU = Q - W
Where Q is the heat added to the gas and W is the work done on the gas. Since the process is adiabatic, there is no heat transfer (Q = 0). The work done on the gas (W) can be calculated using the equation:
W = P * ΔV
Where ΔV is the change in volume of the gas. By rearranging the ideal gas law, we can write:
ΔV = (nR / P) * ΔT
Where ΔT is the change in temperature of the gas. We can then substitute this expression for ΔV in the equation for W and calculate it. Finally, using the equation ΔU = Q - W, we can calculate the change in internal energy.
Note: This explanation provides the general approach to solve the problem. Make sure to substitute the correct values and units into the equations as needed.