A vertical cylindrical tank contains 1.80 mol of an ideal...

A vertical cylindrical tank contains 1.80 mol of an ideal gas under a pressure of 0.300 atm at 20.0$^\circ$C. 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. There is a vacuum above the piston. (a) What is the mass of this piston? (b) How tall is the column of gas that is supporting the piston?

Key Concepts

Ideal Gas Law

This concept describes the relationship between the pressure, volume, temperature, and number of moles of an ideal gas. It is expressed by the equation PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is the absolute temperature. This law allows one to predict changes in any one of these variables when the others are held constant.

Force Equilibrium on a Piston

Force equilibrium is a principle stating that when a system is in a steady state, the net forces acting on it sum to zero. In the context of a piston supporting a gas, the upward force produced by the gas pressure over the area of the piston must exactly balance the downward gravitational force (weight) of the piston, which is used to determine parameters like the piston's mass.

Cylindrical Geometry

Understanding the geometry of cylinders is crucial, especially the relationship between the cross-sectional area and the height of the cylinder. The volume of a cylinder is calculated by multiplying the area of its circular base by its height, which is important when translating a gas's volume into a measurable column height in a cylindrical container.

Unit Conversions and Pressure Units

Consistent units are key in physical calculations. Often, pressure is provided in atmospheres but must be converted to SI units, such as Pascals, to align with other quantities in formulas like the Ideal Gas Law. Ensuring proper unit conversion is essential for accurate computation of physical properties.

Transcript

00:01 So we need to find the mass of this piston and then the height of the heights of the gas column within this piston.

00:08 So we should write down our gibbons.

00:10 We have the number of moles of gas within this piston, 1 .80 moles.

00:15 And then we have a pressure of 0 .30 ,000 atmospheres.

00:22 This is going to be 30 ,397 .5 pascals, where one hour, one hour.

00:31 Atmosphere equals 100 ,1 ,325 pascal.

00:38 We have a temperature of 20 degrees celsius, and we're going to just add 273 to convert it to kelvin, so we'll have 293 kelvin.

00:49 And then we have a radius of 10 centimeters.

00:54 We should automatically convert this into meters.

01:01 And then we're going to say that pressure is defined as a force per unit area and then weight is defined as mass times gravity which can then say we can then say that this is going to be equal to the force so we can say that m g over a is going to be equal to the pressure and then we're going to solve for the mass is going to be equal to p a over g and then the area is simply going to be pi r squared so it'll be pi r squared p over at this point we can now simply plug in for our constants and solve for the mass so we'll have 30 ,397 .5 times 0 .1 squared xx2x5 all over the acceleration due to gravity 9 .8 and it's going to be 97 .4 kilograms.

02:19 So this is the mass of the piston.

02:24 So this is the mass of the piston and this is going to be the answer for part a.

02:31 And then it's saying what is the height of the gas column? well, we can first use the ideal gas law, pv equals nrt and solve for volume.

02:43 Is going to be equal to nrt over p.

02:48 And then we're going to say that volume is also equal to pi r squared h...

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