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, we can use the adiabatic compression formula for an ideal gas:
P1 * V1^γ = P2 * V2^γ
where P1 and V1 are the initial pressure and volume of the gas, P2 and V2 are the final pressure and volume of the gas, and γ is the adiabatic index (1.4 for diatomic gases like air).
(a) To determine the volume of the compressed air, we need to find V2. Given that the initial pressure (P1) = atmospheric pressure = 101.3 kPa, and the final pressure (P2) = 800 kPa, we can set up the equation:
101.3 Pa * V1^1.4 = 800,000 Pa * V2^1.4
Next, we need to convert the cylinder dimensions to SI units. The inner diameter of the cylinder is 2.50 cm, which corresponds to a radius of 1.25 cm or 0.0125 m. The length of the cylinder is 70 cm or 0.70 m.
The volume of the cylinder (V1) can be calculated using the formula for the volume of a cylinder:
V1 = π * r^2 * h
= π * (0.0125 m)^2 * 0.70 m
Plugging in the values and solving for V1:
V1 = 3.14 * (0.0125 m)^2 * 0.70 m
Now, we can substitute the values into the adiabatic compression equation:
101.3 Pa * (3.14 * (0.0125 m)^2 * 0.70 m)^1.4 = 800,000 Pa * V2^1.4
Solving for V2 gives us the volume of the compressed air.
(b) To determine the temperature of the compressed air, we can use the ideal gas law:
P2 * V2 = n * R * T2
where P2 and V2 are the final pressure and volume of the gas, n is the number of moles of the gas, R is the ideal gas constant (8.314 J/(mol·K)), and T2 is the final temperature of the gas.
We can rearrange the equation to solve for T2:
T2 = (P2 * V2) / (n * R)
Assuming air behaves ideally, we can use the ideal gas law to calculate the number of moles of air (n). We know that the pressure (P1) and volume (V1) initially are at atmospheric conditions, temperature (T1) is 30.0°C or 303.15 K, and the pressure and volume (P2, V2) correspond to the compressed air conditions. So:
n = (P1 * V1) / (R * T1)
Finally, we can substitute the values into the equation for T2 and calculate the temperature of the compressed air.
(c) To find the increase in wall temperature, we need to consider the heat transfer that occurs between the air and the cylinder wall during the adiabatic compression. The heat transfer is given by the equation:
Q = m * C * ΔT
where Q is the heat transferred, m is the mass of the cylinder wall, C is the specific heat capacity of the steel, and ΔT is the change in temperature.
The mass of the cylinder wall can be calculated using its volume and density. Given that the length of the cylinder that comes to thermal equilibrium is 14.70 cm or 0.147 m, and the thickness of the wall is 1.80 mm or 0.0018 m, we can calculate the volume of the cylinder wall as:
V_wall = π * ((0.0125 m + 0.0018 m)^2 - (0.0125 m)^2) * 0.147 m
To calculate the mass of the cylinder wall, we need the density of steel. Let's assume it is 7800 kg/m^3.
m_wall = V_wall * density
Now, we can solve for ΔT using the equation:
ΔT = Q / (m_wall * C)
Substituting the known values, we can find the increase in wall temperature.
To solve this problem, we can use the ideal gas law equation, as well as the adiabatic compression equation. Let's break it down step by step:
(a) Determining the volume of the compressed air:
1. First, we need to calculate the initial volume of the air in the pump cylinder.
The cylinder has an inner diameter of 2.50 cm, so the radius (r) is half of that, 1.25 cm or 0.0125 m.
The length (h) of the cylinder is given as 70 cm or 0.7 m.
Using the formula for the volume of a cylinder (V = π * r^2 * h), we can find the initial volume.
2. Next, we need to determine the final pressure. The atmospheric pressure (P1) is given in the problem, and the gauge pressure (P2) is given as 800 kPa. To get the absolute pressure, we need to add the atmospheric pressure to the gauge pressure.
3. To find the final volume (V2) of the compressed air, we can use the adiabatic compression equation:
P1 * V1^(γ) = P2 * V2^(γ),
where γ is the heat capacity ratio (1.4 for diatomic ideal gases).
(b) Determining the temperature of the compressed air:
We can use the ideal gas law equation, PV = nRT, where R is the gas constant. Rearrange the formula to solve for T:
T = P * V / (n * R).
(c) Determining the increase in wall temperature:
To calculate the increase in wall temperature, we can use the formula:
ΔT = (Q / (m * C)),
where Q is the heat transferred, m is the mass of the steel cylinder, and C is the specific heat capacity of steel.
As for how the thickness and length of the cylinder play a part, they determine the heat transfer between the compressed air and the cylinder walls. The given length of 14.70 cm indicates the portion of the cylinder that comes into thermal equilibrium with the air, allowing heat exchange to occur. The thickness of the cylinder wall affects the rate at which heat is transferred between the air and the cylinder material. By calculating the increase in wall temperature, we can assess the impact of this heat transfer.