A cylinder that has a 40.0 cm radius and is 50.0 cm deep is filled with air at 20.0°C and 1.00 atm shown in figure (a). A 25.0 kg piston is now lowered into the cylinder, compressing the air trapped inside shown in figure (b). Finally, a 71.0 kg man stands on the piston, further compressing the air, which remains at 20°C shown in figure (c).
To calculate the pressure of the air after each step, we'll use the ideal gas law equation:
PV = nRT
where:
P = pressure
V = volume
n = number of moles
R = gas constant
T = temperature
Step 1: Initial state
In figure (a), the cylinder is filled with air at 20.0°C and 1.00 atm pressure. The volume of the cylinder can be calculated using the formula for the volume of a cylinder:
V = πr^2h
Given that the radius of the cylinder is 40.0 cm (or 0.4 m) and the depth is 50.0 cm (or 0.5 m), we can substitute these values into the formula to find the volume V.
V = π * (0.4)^2 * 0.5 = π * 0.16 * 0.5 ≈ 0.251m^3
Now, we can calculate the number of moles of air using the ideal gas law equation:
n = PV / RT
Given that the pressure P is 1.00 atm, the gas constant R is 8.31 J/(mol K), and the temperature T is 20.0°C (or 293.15 K), we can substitute these values into the equation to find the number of moles n.
n = (1.00 atm * 0.251m^3) / (8.31 J/(mol K) * 293.15 K) ≈ 0.0101 mol
So, initially, we have approximately 0.0101 moles of air in the cylinder.
Step 2: Piston is lowered
In figure (b), a 25.0 kg piston is lowered into the cylinder, compressing the air trapped inside. Since the piston is lowering, the volume of the cylinder decreases. The initial volume V from step 1 will be the final volume after the piston is lowered.
Now, we can calculate the new pressure P after the piston is lowered using the ideal gas law equation:
P = nRT / V
Substituting the known values, we have:
P = (0.0101 mol * 8.31 J/(mol K) * 293.15 K) / 0.251m^3 ≈ 10.4 atm
So, after the piston is lowered, the air pressure inside the cylinder is approximately 10.4 atm.
Step 3: Man stands on the piston
In figure (c), a 71.0 kg man stands on the piston, further compressing the air. This additional compression will reduce the volume further, leading to an increase in pressure.
To calculate the new pressure P, we need to find the final volume V. Since the cylinder's volume is further reduced due to the man standing on the piston, this new volume can be calculated by adding the additional weight to the piston.
Given that the mass of the man is 71.0 kg and the acceleration due to gravity is 9.8 m/s^2, we can calculate the additional force exerted on the piston using Newton's second law:
Force = mass * acceleration = 71.0 kg * 9.8 m/s^2 = 696.8 N
Now, we can convert this force into pressure using the formula:
Pressure = Force / Area
The area of the piston can be calculated using the formula for the area of a circle:
Area = πr^2
Substituting the known value for the radius r (40.0 cm or 0.4 m), we have:
Area = π * (0.4)^2 ≈ 0.5027 m^2
Now, we can calculate the additional pressure added by the man standing on the piston:
Pressure = 696.8 N / 0.5027 m^2 ≈ 1384.9 Pa
To add this additional pressure to the pressure calculated in step 2, we convert it to atm:
Additional pressure = 1384.9 Pa / (101325 Pa/atm) ≈ 0.0137 atm
Adding the additional pressure to the pressure calculated in step 2, we find:
New pressure = 10.4 atm + 0.0137 atm ≈ 10.4 atm
So, after the man stands on the piston, the air pressure inside the cylinder remains approximately 10.4 atm at 20.0°C.