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?

14

To solve this problem, we need to consider the adiabatic compression of the air in the bicycle pump. Adiabatic compression means that there is no heat exchange between the air and its surroundings during the compression process.

Let's first tackle part (a) - determining the volume of the compressed air.
We can use the adiabatic formula for an ideal gas: PV^γ = constant, where P is the pressure, V is the volume, and γ (gamma) is the heat capacity ratio or the adiabatic index, which is equal to 1.4 for diatomic gases like air.

The initial pressure of the air is the atmospheric pressure, which we'll assume to be approximately equal to 101.3 kPa. The final pressure is given as 800 kPa. We can convert these pressures to absolute values by adding atmospheric pressure (101.3 kPa) to both.

Using the formula mentioned above, we have:
P1V1^γ = P2V2^γ

P1 = 101.3 kPa (initial pressure)
V1 = ? (initial volume)
P2 = 101.3 kPa + 800 kPa = 901.3 kPa (final pressure)
V2 = ? (final volume)

Since the compression process is adiabatic, the equation can be simplified as:
V1^γ = V2^γ

Rearranging the equation to solve for V2, we get:
V2 = V1 * (P1/P2)^(1/γ)

The initial volume V1 can be calculated using the cylinder's dimensions. The inner diameter is given as 2.50 cm, so the radius (r) is half of that: 2.50 cm / 2 = 1.25 cm = 0.0125 m. The length (L) of the cylinder is given as 70 cm = 0.7 m.

The volume V1 is calculated as the area of the cross-section (πr^2) multiplied by the length (L):
V1 = π * (0.0125 m)^2 * 0.7 m

Now, substitute these values into the equation to calculate V2:
V2 = V1 * (P1/P2)^(1/γ)

This gives you the volume of the compressed air in cubic meters, which is the answer to part (a).

Moving on to part (b) - determining the temperature of the compressed air.
To find the temperature, we can use the ideal gas law equation: PV = nRT, where n is the number of moles of gas and R is the ideal gas constant (8.31 J/(mol·K)).

Since we're dealing with one mole of diatomic air, the equation becomes: PV = RT.
We know the final pressure (P2 = 901.3 kPa) and the volume (V2, which we calculated in part (a)). Rearrange the equation to solve for the final temperature (T2):

T2 = (P2 * V2) / R

This gives you the temperature of the compressed air in Kelvin, the answer to part (b).

Finally, for part (c) - the increase in wall temperature, we need to consider the thermal equilibrium between the air and the wall. The heat transfer between the air and the wall can be calculated using the formula: Q = mcΔT, where Q is the heat transferred, m is the mass, c is the specific heat capacity, and ΔT is the change in temperature.

In this case, we are interested in finding ΔT, but first, we need to calculate the heat transferred (Q).
The mass (m) can be determined using the ideal gas law equation, where PV = nRT. Rearrange to solve for the mass:
m = (P2 * V2) / (RT2)

The specific heat capacity (c) for steel is approximately 500 J/(kg·K). The change in temperature (ΔT) can be calculated using the relationship between the heat transferred (Q) and the mass and specific heat capacity:

Q = mcΔT
ΔT = Q / (mc)

Substitute the values into the equation to find ΔT, which will give you the increase in wall temperature in Kelvin, the answer to part (c).

By following these steps, you should be able to obtain the answers for all three parts of the question.