056 – The relationship between a gas’ temperature and its volume

Heat a gas and it spreads out. That single fact keeps a hot-air balloon in the sky, drives the convection that circulates air in a room, makes bread rise, and is the reason a car’s tyre pressure warning light comes on during the first cold morning of the winter. Put quantitatively, it is Charles’s law: for a fixed amount of gas held at constant pressure, the volume is directly proportional to the absolute temperature, V / T = constant. The word absolute is doing the work — the proportionality only holds when temperature is counted from the point where molecular motion would cease, not from the freezing point of water.

The reason is the same kinetic picture that explains Boyle’s law, run the other way. Temperature sets how fast the molecules move. Warm the gas and each molecule arrives at the wall harder and more often, so if it were confined the pressure would rise. Here it is not confined: the gas is free to push its seal along the tube until it has spread out enough that the collisions once again balance the pressure of the room outside. The volume it settles at is proportional to the absolute temperature.

In this laboratory you will trap a short column of air in a glass capillary tube of 0.5 mm internal radius, sealed at the top by a drop of olive oil that slides freely and leaks nothing. The tube is clamped beside a ruler and immersed in a beaker of iced water on a stirred hot plate. Warming the bath in steps of about ten degrees and reading the height of the bottom of the drop at each plateau gives ten matched pairs of temperature and length. Because the bore is uniform, the height is the volume up to a constant factor, so a plot of height against temperature is a plot of Charles’s law — and the point where that straight line reaches zero height is absolute zero, located from a bench in a warm room.

Educational Goals

Familiarization with the laboratory environment

• Identify the capillary tube, the oil seal, the water bath, the magnetic stirrer and the clamped ruler, and state the role of each.
• Explain why the trapped column must be completely below the water line and why neither the tube nor the thermometer may touch the beaker wall.

Preparing a sealed sample

• Heat the capillary evenly and draw an oil plug of 5 mm to 1 cm into the open end as the tube cools, and explain why cooling air draws the oil in.
• Handle hot glassware with thermal gloves and shut down the Bunsen burner before moving on to Part B.

Measurement and data recording

• Read the water temperature and the height of the bottom of the oil drop only after the reading has stabilised, and say why the stirrer matters.
• Record ten matched pairs across the widest temperature range the bath allows, rather than two points at the extremes.

Quantitative treatment of Charles’s law

• Convert each Celsius reading to kelvin and test whether h / T is constant.
• Convert a height to a volume with V = πr²h and show that the choice of units does not affect the test.
• Plot height against Celsius temperature, fit a straight line and extrapolate it to zero height to locate absolute zero.

Critical evaluation of the measurement

• Estimate the pressure the oil plug itself adds and show that it is negligible beside atmospheric pressure.
• Explain why the full temperature range must be used for the extrapolation.

Protocol

PART A: The preparation of the capillary tube

1. Above the heating plate; attach a universal clamp to each of the two supports.
2. Using the dropper; take some olive oil and place two drops of oil on the watch glass.
3. Light the Bunsen burner.
4. Using thermal gloves; heat the capillary tube along its entire length by exposing it to the flame while making back-and-forth movements for about 20 sec.
5. Place the open end (the transparent end) of the very hot tube onto the oil drops prepared in step 1. The oil should rise on its own into the capillary tube.
6. Return the tube to an upright position (open end facing upwards) and wait a few seconds for it to cool.
7. Attach the capillary tube to the universal clamp of the right support; positioning the open (transparent) end upwards.
8. Turn off the Bunsen burner.

• The drop of oil in the capillary tube should have a thickness between 5 mm and 1 cm.

PART B: Measurement of the volume-temperature relationship

9. Place the 250 mL beaker on the hot plate (do not turn on the hot plate).
10. Place the magnetic stirrer in the beaker.
11. Attach the thermometer to the universal clamp of the left stand and place it vertically in the 250 mL beaker. Neither the capillary tube nor the thermometer should touch the walls of the beaker.
12. Secure the ruler behind the capillary tube to measure the height of the oil drop.
13. Take the beaker containing ice and add 250 mL of cold tap water to it.
14. Then pour the cold water and ice into the beaker containing the capillary tube. The water level must exceed that of the oil drop.
15. Observe the water temperature and the height of the bottom of the oil drop once the temperature has stabilized.
16. Start the stirrer on the hot plate.
17. Start the stopwatch.
18. Turn on the heating plate at low intensity (20°C) and wait for the temperature to rise by about ten degrees.
19. In the results table; observe the water temperature and the height of the bottom of the oil drop once the temperature has stabilized. You will notice the change in the height of the drop based on changes in the water temperature.
20. Repeat steps 18 and 19 while increasing the temperature of the plate by an additional 10 degrees.
21. Turn off the hot plate and wait for everything to cool down.

• Note: The inside of the capillary tube has a radius of 0.5 mm.

Anticipated Outcomes

Expected results

The bore of the capillary is uniform, so the trapped air occupies a cylinder of cross-section πr² = π × (0.5 mm)² = 0.785 mm². Its volume is therefore V = 0.785 × h, with h in millimetres and V in mm³, and since 1 mm³ = 1 µL the height in millimetres converts to a volume in microlitres by that one factor: 47.3 mm → 37.1 µL, 62.1 mm → 48.8 µL. Because the factor is the same at every temperature, testing V ∝ T and testing h ∝ T are the same test, and the ruler alone is enough.

What the numbers say

Charles’s law states that V₁ / T₁ = V₂ / T₂ for a fixed amount of gas at constant pressure, with T in kelvin. Taking the two ends of the series: 47.3 mm / 277.2 K = 0.1706 mm/K and 62.1 mm / 364.2 K = 0.1705 mm/K. Every intermediate reading gives the same figure to within 0.03 %. Put the other way round, the temperature rises by a factor of 364.2 / 277.2 = 1.314 across the run and the column lengthens by 62.1 / 47.3 = 1.313 — the two ratios agree to one part in a thousand, which is the whole content of the law.

Finding absolute zero without going anywhere near it

This is the reason the experiment has survived for two centuries. Plot the drop height against the Celsius temperature and the points fall on a straight line, h = 0.1705 t + 46.6, where the slope is in mm per °C and the intercept is the height the column would have at 0 °C. Extend that line backwards to h = 0 and it reaches t = −46.6 / 0.1705 = −273 °C. Nothing in the apparatus goes below 4 °C; the zero of the absolute scale is found by extrapolation from a beaker of warm water, and the fact that the same intercept comes out whatever gas is used is what makes it a property of temperature rather than of air.

Why the pressure really is constant

The oil drop is free to slide, so the trapped air is always at the pressure of the room plus two small corrections. The weight of the plug itself contributes ρgL: for olive oil at about 910 kg/m³ and a 7 mm plug that is 910 × 9.81 × 0.007 = 62 Pa, six hundredths of one per cent of atmospheric pressure. Surface tension contributes 2γ/r at each meniscus, about 2 × 0.032 / 0.0005 = 128 Pa, but the two menisci curve in opposite senses and very nearly cancel; only their difference survives. Both corrections are far below anything a millimetre ruler could reveal, which is why the constant-pressure condition can be treated as exactly met.

The amount of gas, and the kinetic picture

From n = PV / RT at the first reading, n = (101 325 Pa × 3.71 × 10⁻⁸ m³) / (8.314 J·mol⁻¹·K⁻¹ × 277.2 K) = 1.63 × 10⁻⁶ mol — about 47 µg of air, roughly 9.8 × 10¹⁷ molecules. That number is fixed by the oil seal and does not change. Warming the sample from 277 K to 364 K raises the root-mean-square molecular speed by only √(364/277) = 1.15, fifteen per cent, while the volume grows by the full 31 %: the molecules speed up as the square root of temperature but the gas must expand in direct proportion to it, because pressure depends on both how hard and how often the molecules strike, and the gas is spreading out precisely so that the rate of arrival per unit area stays matched to the room outside.

Summary of Assignment by Grade Range

Grade 9–10

Focus. Observation and vocabulary. Students see that warming a gas makes it take up more room, and learn that the seal must slide freely for the pressure to stay constant.

• Prepare the capillary, capture the oil drop, and describe what happens to it as the bath warms.
• Record five matched readings of water temperature and drop height and enter them in a table.
• Plot height against temperature by hand and describe the result as a straight line rather than a curve.
• Answer in words why the drop moves up rather than the pressure rising.

Grade 11

Focus. Quantitative treatment. Students convert to kelvin, test the proportionality numerically and make the extrapolation that gives absolute zero.

• Take all ten readings, convert each to kelvin, and add a column of h / T; state its mean and its spread.
• Convert two heights to volumes with V = πr²h and confirm that V / T gives the same constancy as h / T.
• Plot height against Celsius temperature, fit a straight line, and read off both the slope and the intercept.
• Extrapolate the line to zero height and quote the temperature it reaches, with a comment on how far outside the data that point lies.

Grade 12 / College Level

Focus. Derivation, error analysis and independent interpretation.

• Derive V ∝ T from PV = nRT and identify exactly which quantities the apparatus holds fixed and by what mechanism.
• Compute the amount of trapped air in moles, micrograms and molecules, and show it is unchanged across the run.
• Estimate the hydrostatic and surface-tension contributions of the oil plug and show that the constant-pressure assumption holds to better than 0.1 %.
• Explain why the root-mean-square molecular speed rises as √T while the volume rises as T, and reconcile the two using P = (1/3)(N/V)m〈v²〉.
• Discuss what the experiment can and cannot say about absolute zero, given that air liquefies near 80 K.

Laboratory essentials

Instruments

• Beaker (250 mL)
• Bunsen burner
• Capillary tube
• Dropper
• Hot plate
• Lab Stand & Clamps
• Magnetic stirrer
• Ruler
• Thermometers
• Timer
• Watch glass
• Thermal gloves

Products

• Olive oil
• Ice
• Cold tap water