Translation between representations (TBR)

Heating helium at constant pressure and at constant volume

Unit 9
12 points
About 28 minutes

You can use a calculator on this question, just like on exam day.

A multipart problem about one situation shown in several ways. You draw a diagram, derive equations, sketch or draw graphs, and then explain whether your answers agree with each other or use them to predict what happens when the situation changes. On the exam: Question 2 of 4. New for May 2027: the free-response section is 95 minutes, down from 100, for all 4 questions (50% of your score), one of each type in this order. Calculator and equation sheet allowed. The CED suggests 25–30 minutes. This question type started in May 2025, so Physics 2 free-response questions from 2024 and earlier are built differently.

The question

Cylinders 1 and 2 each contain 0.20 mol of helium, which can be treated as an ideal monatomic gas. Both samples start at pressure P₀, volume V₀ and temperature 300 K. Cylinder 1 has a piston that moves freely, so its gas is heated at constant pressure until its volume is 2V₀. Cylinder 2's piston is locked in place, so its gas is heated at constant volume until its pressure is 2P₀. The mass of a helium atom is 6.64 × 10⁻²⁷ kg; use R = 8.31 J/(mol·K) and $k_B$ = 1.38 × 10⁻²³ J/K.

Suggested time: 28 minutes

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Part (a)

2 points

On one set of axes of pressure versus volume, sketch both processes. You can't draw here, so describe the graph: the axes, the starting and ending points of each process in terms of P₀ and V₀, the shape of each process, and the final temperature of each sample.

Your answer to part (a)

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Part (b)

3 points

For each process, calculate (i) the change in internal energy of the gas, (ii) the work done on the gas, and (iii) the energy transferred to the gas by heating.

Your answer to part (b)

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Part (c)

4 points

Sketch, on one set of axes, the distribution of atomic speeds for the helium at 300 K and at 600 K. You can't draw here, so describe the graph: the axes, the shape of each curve and how they compare. Then calculate the speed of a helium atom that has the average kinetic energy, at 300 K and at 600 K.

Your answer to part (c)

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Part (d)

3 points

Using the motion of the atoms, explain (i) why the pressure in cylinder 2 doubles, and (ii) why the pressure in cylinder 1 stays the same even though its atoms also speed up. (iii) Explain how your explanations are consistent with the ideal gas law.

Your answer to part (d)

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