A vertical cylindrical tank contains 1.80 mol of an ideal gas under a pressure of 1.00 atm at 20.0oC. The round part of the tank has a radius of 10.0 cm, and the gas is supporting a piston that can move up and down in the cylinder without friction.

(a) What is the mass of this piston?
(b) How tall is the column of gas that is supporting the piston?

To answer these questions, we need to apply the ideal gas law and the concept of hydrostatic pressure.

(a) To find the mass of the piston, we need to calculate the force exerted by the gas on the piston. This force can be determined using the equation:

Force = Pressure x Area

First, we need to find the area of the piston. The area of a circular disk can be calculated using the formula:

Area = π x radius^2

Given that the radius of the piston is 10.0 cm, we convert it to meters by dividing by 100 since 1 meter is equal to 100 centimeters:

Radius = 10.0 cm / 100 = 0.1 m

Area = π x (0.1 m)^2

Next, we can calculate the force using the equation:

Force = Pressure x Area

Given that the pressure is 1.00 atm and 1 atm is equal to 101325 Pa, we convert the pressure:

Pressure = 1.00 atm x 101325 Pa/1 atm = 101325 Pa

Substituting the values into the equation:

Force = 101325 Pa x Area

Now we need to determine the mass of the piston. The force acting on the piston is equal to the weight of the piston, which can be calculated using the equation:

Force = mass x gravity

In this case, we assume the acceleration due to gravity is approximately 9.8 m/s^2.

Therefore:

mass x gravity = 101325 Pa x Area

Simplifying the equation to solve for mass:

mass = (101325 Pa x Area) / gravity

Substituting the values into the equation:

mass = (101325 Pa x Area) / 9.8 m/s^2

Calculate the area from above, then substitute and solve to find the mass of the piston.

(b) To determine the height of the gas column supporting the piston, we can use the concept of hydrostatic pressure. The hydrostatic pressure is calculated using the equation:

Pressure = density x gravity x height

Rearranging the equation to solve for height:

height = Pressure / (density x gravity)

To calculate the density of the gas, we can use the ideal gas law equation:

PV = nRT

Where:
P = pressure (in Pa)
V = volume (in m^3)
n = number of moles
R = ideal gas constant (8.314 J/(mol·K))
T = temperature (in K)

We have the pressure, number of moles, and temperature, but we need to calculate the volume of the gas. The volume of a cylinder can be calculated using the formula:

Volume = π x radius^2 x height

Given that the radius of the cylinder is 10.0 cm, we convert it to meters:

Radius = 10.0 cm / 100 = 0.1 m

Substituting the values into the equation:

Volume = π x (0.1 m)^2 x height

Now we can substitute the volume, pressure, number of moles, and temperature into the ideal gas law equation and solve for the density.

Once we know the density, we can substitute it into the hydrostatic pressure equation to calculate the height.