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# Ideal Gases

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Age 16 to 18

Challenge Level

This problem gives a sequence of questions concerning ideal gases. Each part might require standard data not provided in the question and may require certain assumptions which should be clearly considered. If estimates are needed, try to give definite bounds between which you are confident that the estimated quantity lies.

The assumptions of an ideal gas may be stated as:

A) Particles are in a state of constant motion.

B) The volume of the gas molecules is negligible compared to the volume of the gas.

C) The attractive forces between the particles are negligible.

D) The collisions between the particles of gas are perfectly elastic.

E) Molecules of gas travel in straight lines in the absence of collisions.

The scenario: Imagine a rigid box of side $1$m contains oxygen at standard temperature and pressure.

0. How much oxygen is in the box?

1. What is the average distance between nearest-neighbour molecules in the box?

2. What is the average speed of the molecules in the box? Is it possible to give an estimate of a speed below which $95\%$ of the molecules will be moving at any one time?

3. In the absence of any collisions, estimate how long it takes a particular molecule to cross the box.

4. Imagine that a gas molecule has just directly struck the wall and rebounded at right angles. Do you expect it to strike the other side of the box before a collision occurs with another molecule? Think carefully about any factors and assumptions which enter into any calculations you perform.

5. In the absence of collisions, each molecule is assumed to travel in a straight line. What effect would gravity have on the crossing time of the box? Is it correct to ignore gravity? Would gravity have any effect on $P$, $V$ or $T$ for the box?

6. Bullet point 'E' in the above list of ideal gas assumptions is not usually listed explicitly as an assumption. Are there any other 'hidden' assumptions underlying the behaviour of ideal gases that you can think of? How might they affect $P$, $V$ or $T$ for the box?

7. Imagine that the box is gradually reduced in volume, whilst keeping the same amount of gas trapped inside. In what order would you expect the ideal gas assumptions to break down? Try to justify your thinking quantitatively.

8. The box is placed in a vacuum and one side opened for a nanosecond and then closed. Estimate the volume of gas that will remain in the box. What effect will this have on $T$ and $P$?