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Gases and temperature

Heat and matter · level 5, lesson 5 of 6

A question to start

A sealed metal box of air sits on the Mars surface at the Viking 1 site. In the afternoon it is at 242 K; before dawn, 184 K. How much does the pressure inside change?

The idea

Heat a gas and its particles move faster. What happens next depends on what is held fixed.

If the volume is fixed (a sealed rigid box), the pressure rises in step with the kelvin temperature. This is the pressure law:

\[\frac{p_1}{T_1} = \frac{p_2}{T_2}.\]

If the pressure is fixed (a balloon free to grow), the volume rises in step with the kelvin temperature. This is Charles's law:

\[\frac{V_1}{T_1} = \frac{V_2}{T_2}.\]

Both only work with temperatures in kelvin, measured from absolute zero.

0 Ktemperature (K)preal gas measured

Worked examples

1. The sealed box. It was sealed at 101 kPa and 242 K. Before dawn, at 184 K: \(p_2 = 101 \times 184/242 = 77\) kPa, a fall of nearly a quarter, every night.

Swing in a sealed box of gas, each sol Mars (Viking 1 site) Earth
Day to night 242 K to 184 K suppose 293 K to 278 K
Pressure change about 24% about 5%

2. A balloon at Charles's law. 3 litres at 300 K is warmed to 330 K at the same pressure: \(V_2 = 3 \times 330/300 = 3.3\) litres.

3. A car tyre. Its air is at 300 kPa (counting the outside air pressure too, not just what a tyre gauge shows) and 283 K (10 °C). On a hot motorway it warms to 313 K (40 °C): \(p_2 = 300 \times 313/283 = 332\) kPa.

Try it

  1. Why must temperatures be in kelvin for these laws?
  2. A sealed can of gas at 100 kPa and 300 K is heated to 450 K. Find its pressure.
  3. A balloon of 2 litres at 280 K warms to 350 K at the same pressure. Find its volume.
  4. What would happen to the pressure of a gas cooled towards 0 K, if it stayed a gas?
Answers
  1. The pressure and volume are in proportion to the temperature measured from absolute zero, not from 0 °C.
  2. \(100 \times 450/300 = 150\) kPa.
  3. \(2 \times 350/280 = 2.5\) litres.
  4. It would fall towards zero, since the particles would almost stop.

Common mistake

Using °C. Warming from 10 °C to 20 °C does not double the pressure. In kelvin it is 283 K to 293 K: a rise of about 3.5%.

Where it leads