a gas confined in a 0.47m diameter cylinder by a piston, on which rests a weight. The mass of the piston and weight together is 150kg. The local acceleration of gravity is 9.813meters per second squared and atmospheric pressure is 101.57kPa
a. what is the force in newtons exerted on the gas by the atmosphere, the piston, and the weight, assuming no friction between the piston and the cylinder?
b. what is the pressure of the gas in kPa?
c. if the gas is heated, it expands, pushing the piston and weight upward. if the piston and weight are raised 0.83m, what is the work done by the gas in kJ? what is the change in potential energy of the piston and the weight?
a. To find the force exerted on the gas by the atmosphere, piston, and weight, we need to consider the forces acting on the piston. These forces are the atmospheric pressure, the force exerted by the weight, and the force exerted by the gas.
1. Force exerted by the atmosphere:
The force exerted by the atmosphere is equal to the atmospheric pressure multiplied by the area of the piston. The formula to calculate force is:
Force = Pressure × Area
Area = πr^2, where r is the radius of the piston.
Given that the diameter of the cylinder, and consequently, the piston is 0.47m, we can find the radius:
Radius = diameter / 2 = 0.47m / 2 = 0.235m
Now, we can calculate the force:
Force by atmosphere = Atmospheric pressure × Area
= 101.57kPa × π × (0.235m)^2
2. Force exerted by the weight:
The force exerted by the weight is equal to the mass of the piston and weight together multiplied by the acceleration due to gravity.
Force by weight = Mass × Gravity
= 150kg × 9.813m/s^2
b. To find the pressure of the gas, we can use the ideal gas law:
PV = nRT
Here, P is the pressure, V is the volume, n is the number of moles of the gas, R is the ideal gas constant, and T is the temperature in Kelvin.
Since we don't have the number of moles or temperature, but we are given the diameter of the cylinder, we can use the fact that the cylinder is a right circular cylinder, so the volume is given by:
V = πr^2h
Where h is the height of the cylinder.
Given that the diameter of the cylinder is 0.47m, we can calculate the height using the formula:
h = 4 × radius
= 4 × 0.235m
Now, we can substitute the values into the equation:
V = π × (0.235m)^2 × (4 × 0.235m)
To find the pressure of the gas:
Pressure = Force exerted by the atmosphere / Area
c. To find the work done by the gas, we can use the formula:
Work = Force × Distance
Given that the piston and weight are raised 0.83m, the distance is given. We already calculated the force exerted by the atmosphere and weight. Multiply the force exerted by the gas by the distance to find the work done by the gas.
To calculate the change in potential energy of the piston and weight, we can use the formula:
Potential energy = Mass × gravity × height
Given that the height is 0.83m, and the mass is 150kg (mass of piston + weight), we can calculate the change in potential energy.