4.2.3: Problems

Exercise \(\PageIndex{1}\): Increase \(N\), what happens?

In this animation \(N = nR\) (i.e., \(k_{B} = 1\)). This, then, gives the ideal gas law as \(PV = NT\). The average values shown, \(<\: >\), are calculated over intervals of five time units. Restart. When one particle is present, you see the temperature and average pressure. Add 24 more particles with the identical initial speed. What should the missing values in the table be? Explain.

Problem authored by Anne J. Cox.

Exercise \(\PageIndex{2}\): Find the warmer wall

The right and left walls of the box are at different temperatures. Restart. The graph shows the speed as a function of time for the blue particle (which is identical to the red particles). Which wall is at a higher temperature? Explain how you arrived at your answer.

Problem authored by Anne J. Cox.

Exercise \(\PageIndex{3}\): Rank mass of particles

In this animation \(N = nR\) (i.e., \(k_{B} = 1\)). This, then, gives the ideal gas law as \(PV = NT\). Restart.

Two different groups of particles, blue and another color (depending on the animation), are in a container with a piston between them. In each of the animations, the blue particles have the same mass, but the mass of the other particles is different. Rank the nonblue particles from most massive to least massive.

Illustration authored by Anne J. Cox.

Exercise \(\PageIndex{4}\): Kinetic energy of diatomic and monatmoic particles

The gas in the animation has \(20\) monatomic particles and \(10\) diatomic particles. Which of the lines on the graph corresponds to the total kinetic energy, the total kinetic energy of the monatomic particles, the total kinetic energy of the diatomic particles, and the translational and rotational kinetic energy of the diatomic particles? Explain. Restart.

Problem authored by Anne J. Cox.
Script authored by Wolfgang Christian modified by Anne J. Cox.
Applet authored by Ernesto Martin and modified by Wolfgang Christian.

Exercise \(\PageIndex{5}\): What's wrong with this compression?

If the gas inside the container being compressed is an ideal gas, what is wrong with this animation? The pressure is shown on the gauge (in atmospheres) and is caused by the black block sitting on the piston (position is given in centimeters). Restart.

Problem authored by Anne J. Cox and Mario Belloni.
Script authored by Mario Belloni and Anne J Cox.

Exercise \(\PageIndex{6}\): Balloon expanding as it rises in a liquid

A large balloon filled with an ideal gas is initially held at the bottom of a tank full of a liquid. It is released and floats to the top as shown in the animation. If you click on the graph, you can see the x and y positions of the cursor (position is given in meters). The top of the liquid (indicated by the blue line) is open to air at atmospheric pressure. Restart.

Find the density of the liquid (the liquid is at a constant temperature).

Note

You will need to recall what you learned about fluids (and the change in pressure as a function of depth in a fluid).

Problem authored by Anne J. Cox.

Exercise \(\PageIndex{7}\): Coefficient of expansion of an ideal gas

When heated, materials expand in all three dimensions (position is given in meters). The equation for the volume expansion is as follows:

\[\Delta V=V_{0}\beta\Delta T\nonumber\]

where the change in volume (\(\Delta V\)) is equal to the initial volume (\(V_{0}\)) multiplied by the coefficient of volume expansion, \(\beta\), and by the temperature increase. Note that this equation is similar to the equation for linear expansion (see Exploration 19.2), and that for solids the coefficient of expansion, \(\beta\), is approximately equal to \(3\alpha\). Restart. For a gas we need to be careful because a gas can expand even without a change in temperature (if the pressure decreases).

1. Assuming constant pressure, find the volume expansion coefficient of the gas at this initial temperature (\(100\text{ K}\)).
2. Assuming an ideal gas, use \(PV = nRT\) to find (i.e., derive an expression for) the volume expansion coefficient and see that it varies with temperature (is not a constant).
3. Pick a new initial temperature, use the animation, and verify that the results match the expression you derived.

Problem authored by Anne J. Cox.

Exercise \(\PageIndex{8}\): Find the work done in a compression

A mass is pushed downward on top of a container (the dimension of the container into the screen is \(30\text{ cm}\)) so that an ideal gas undergoes a temperature change as shown in the animation (pressure is given in atmospheres, position is given in centimeters, and time is given in seconds). Restart.

1. What is the work done on or by the gas?
2. Sketch a PV diagram of the process.
3. If the initial temperature of the gas is \(-73^{\circ}\text{C}\), what is the temperature scale (i.e., what are the divisions on the thermometer: \(10^{\circ}\text{C}\) units? \(25^{\circ}\text{C}\))?

Problem authored by Anne J. Cox.
Script authored by Anne J. Cox and Mario Belloni.

Exercise \(\PageIndex{9}\): Find work and heat input or output from a \(PV\) diagram

\(\color{red}{\text{There is a time delay—since the system must be in equilibrium—before the change of state occurs.}}\)

In this animation \(N = nR\) (i.e., \(k_{B} = 1\)). This, then, gives the ideal gas law as \(PV = NT\). Restart.

How much heat is added to or removed from the gas during the expansion of the ideal gas?

When you get a good-looking graph, right-click on it to clone the graph and resize it for a better view.

Problem authored by Anne J. Cox.

Exercise \(\PageIndex{10}\): Find work and heat input or output for a compression

A mass is pushed downward on top of a container with \(3\) moles of an ideal monatomic gas inside as shown in the animation (position is given in meters and time is given in seconds). The pressure remains constant as indicated by the round pressure gauge. Restart.

1. If the initial temperature is \(20^{\circ}\text{C}\), what is the work done on the gas?
2. How much heat is absorbed or released in the process?
3. Draw a \(PV\) diagram.

Problem authored by Anne J. Cox.
Script authored by Anne J Cox and Mario Belloni.

Exercise \(\PageIndex{11}\): Rank the expansions by work done and heat absorbed

\(\color{red}{\text{There is a time delay—since the system must be in equilibrium—before the change of state occurs.}}\)

In this animation \(N = nR\) (i.e., \(k_{B} = 1\)). This, then, gives the ideal gas law as \(PV = NT\). Restart.

1. Rank the expansions by the work done by the gas (from negative to positive).
2. Rank the expansions by change in internal energy (from negative to positive).
3. From these rankings, if possible, rank the expansions according to the heat required (again from negative to positive).

When you get a good-looking graph, right-click on it to clone the graph and resize it for a better view.

Problem authored by Anne J. Cox.

Exercise \(\PageIndex{12}\): Determine the adiabatic coefficient (\(\gamma\)) for the expanding gas

\(\color{red}{\text{There is a time delay—since the system must be in equilibrium—before the change of state occurs.}}\)

In this animation \(N = nR\) (i.e., \(k_{B} = 1\)). This, then, gives the ideal gas law as \(PV = NT\). The same amount of heat is added to an ideal gas in all three processes. Restart.

1. What is the value of \(\gamma\) (the ratio of \(C_{P}\) to \(C_{V}\)) for this ideal gas?
2. According to the equipartition of energy theorem, how many degrees of freedom do the particles that make up this gas have?
3. Is the gas monatomic? diatomic? polyatomic?

When you get a good-looking graph, right-click on it to clone the graph and resize it for a better view.

Problem authored by Anne J. Cox.

Physlets were developed at Davidson College and converted from Java to JavaScript using the SwingJS system developed at St. Olaf College.