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?

a) For the adiabatic compression, use the formula P V^1.4 = constant to get the final volume

b) Once you know the final volume V2, use the perfect gas law to get T2, before heat is lost to the cylinder.

c) Use the heat capacity of the steel to determine how much the gas temperature drops at equilibrium

im sorry i tried using PV^1.4=22.3l but this was not the right answer could you explain further the equation for a.

To solve this problem, we can use the ideal gas law equation along with the adiabatic compression equation. Let's break down each part of the problem step by step:

(a) Determine the volume of the compressed air:

To find the volume of the compressed air, we can use the ideal gas law equation: PV = nRT.

Given:
P = gauge pressure + atmospheric pressure = 800 kPa + atmospheric pressure
V = (π/4)(d^2)(h) = (π/4)(0.025 m)^2(0.70 m) (where d is the diameter and h is the length of the cylinder)
n = number of moles of air (we will assume it's constant throughout the process)
R = gas constant = 8.314 J/(mol·K)
T = initial temperature of air in Kelvin

Since the process is adiabatic, there is no heat exchange with the surroundings, and therefore, no change in the number of moles of air. Hence, we can assume n is constant throughout the process.

To find the volume of the compressed air, we need to solve for V. Rewrite the ideal gas law equation as:
V = (nRT)/(P + atmospheric pressure)

Now, substitute the known values into the equation and calculate the volume of the compressed air.

(b) Determine the temperature of the compressed air:

To find the final temperature of the compressed air, we can use the adiabatic compression equation: P1V1^γ = P2V2^γ.

Given:
P1 = initial pressure = gauge pressure + atmospheric pressure = 800 kPa + atmospheric pressure
V1 = initial volume of the air (which we already calculated in part (a))
γ = ratio of specific heats for diatomic ideal gases = 7/5 (for air)
P2 = final pressure = gauge pressure + atmospheric pressure = 800 kPa + atmospheric pressure
V2 = volume of the compressed air (which we calculated in part (a))

Rearrange the equation to solve for the final temperature (T2) of the compressed air:
T2 = T1 * (P2/P1)^((γ-1)/γ)

Now substitute the known values into the equation and calculate the temperature of the compressed air.

(c) The pump is made of steel, and we are asked to find the increase in wall temperature. To calculate this, we need to consider the heat transfer from the compressed air to the pump.

Given:
Inner wall thickness of the cylinder = 1.80 mm = 0.0018 m
Length allowed to come to thermal equilibrium with the air = 14.70 cm = 0.147 m

We will assume that the heat transfer occurs only from the inner wall of the cylinder to the compressed air. Hence, we can use the formula for conduction heat transfer through a cylinder:

Q = (k * A * ΔT * t) / L

Where:
Q = heat transfer
k = thermal conductivity of the material (for steel, k is approximately 50 W/(m·K))
A = surface area of the inner wall of the cylinder (which can be calculated using the inner diameter and length of the cylinder)
ΔT = increase in temperature (final temperature - initial temperature) of the inner wall
t = time for the heat transfer (assumed to be equal to the time taken for the compression process)
L = length of the portion of the cylinder in thermal equilibrium with the air

Calculate the surface area 'A' using the inner diameter and length of the cylinder. Then, substitute the known values into the formula and calculate the increase in wall temperature (ΔT).