2.1.3 Gases and the absolute scale of temperature

Learning objectives

2.1.3.1Qualitatively, in terms of particles• Describe qualitatively, in terms of particles, the effect on the pressure of a fixed mass of gas of: (a) a change of temperature at constant volume (b) a change of volume at constant temperature 2.1.3.2Convert temperatures between kelvin• Convert temperatures between kelvin and degrees Celsius; recall/use: T (in K) = θ (in °C) + 273 2.1.3.3Recall/use: pV = constant for a fixed• Recall/use: pV = constant for a fixed mass of gas at constant temperature, including a graphical representation of this relationship

For a fixed mass of gas, pressure changes when particle speed changes or when the same particles collide with the walls more or less frequently.

Controlled change	Particle account	Pressure effect
temperature increases at constant volume	particles gain average kinetic energy, move faster, and make more frequent and harder collisions with the same walls	pressure increases
temperature decreases at constant volume	particles move more slowly, so collisions are less frequent and less forceful	pressure decreases
volume decreases at constant temperature	average speed stays the same, but particles cross a shorter distance and hit each unit area of wall more frequently	pressure increases
volume increases at constant temperature	average speed stays the same, but collisions with each unit area of wall become less frequent	pressure decreases

Name the control before predicting the pressure. Constant volume isolates the effect of temperature on particle speed. Constant temperature isolates the effect of volume on collision frequency; the particles do not speed up merely because the gas is compressed slowly.

These comparisons require the same fixed mass of gas. If gas leaks in or out, or if temperature and volume both change, one simple comparison is not enough to predict the result without further information.

The kelvin and Celsius scales have equal-sized intervals but different zero points: 0 K corresponds to −273 °C.

$T\,(\mathrm{K}) = \theta\,(^{\circ}\mathrm{C}) + 273$

To convert °C to K, add 273. To convert K to °C, rearrange to $\theta = T - 273$ .

Examples: $25\,^{\circ}\mathrm{C} = 25 + 273 = 298\,\mathrm{K}$ . For nitrogen at $77\,\mathrm{K}$ , $\theta = 77 - 273 = -196\,^{\circ}\mathrm{C}$ .

Write K, not °K: kelvin has no degree sign. A temperature can be negative on the Celsius scale while remaining positive in kelvin; for example, −196 °C is 77 K.

For a fixed mass of gas at constant temperature, pressure and volume are inversely proportional: increasing one decreases the other so that their product stays constant.

$pV = \text{constant} \qquad \text{so} \qquad p_1V_1=p_2V_2$

A gas at $120\,\mathrm{kPa}$ occupies $50\,\mathrm{cm^3}$ and is compressed at constant temperature to $30\,\mathrm{cm^3}$ . Then $p_2=(120\times50)/30=200\,\mathrm{kPa}$ . The pressure rises because the volume falls.

On a graph of $p$ against $V$ , the relationship is a decreasing curved line: doubling $V$ halves $p$ , and halving $V$ doubles $p$ . The curve becomes less steep as volume increases and does not meet either axis. A graph of $p$ against $1/V$ is a straight line through the origin.

Use one consistent volume unit on both sides and one consistent pressure unit on both sides; matching units cancel in the ratio. Do not use $pV=\text{constant}$ if the gas mass or temperature changes.

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—2.2.1 Thermal expansion of solids, liquids and gasesRead

2.1.3 Gases and the absolute scale of temperature

Learning objectives

Predict gas pressure from temperature or volume

Convert temperatures between Celsius and kelvin

Use the inverse pressure–volume relationship

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