Temperature and pressure calculations

When a gas is trapped inside a container which has a fixed size (its volume cannot change) and the gas is heated, the particles will gain kinetic energy which will make them move faster.

The temperature of the gas is proportional to the average kinetic energy of its molecules. Faster moving particles will collide with the container walls more frequently and with greater force. This causes the force on the walls of the container to increase and so the pressure increases.

If the temperature of the gas is measured on the Kelvin scale, the pressure is proportional to the temperature.

From this we can derive the equation

\[\frac{P_{1}}{T_{1}} = \frac{P_{2}}{T_{2}}\]

where:

P1 is the initial pressure

T1 is the initial temperature

P2 is the final pressure

T2 is the final temperature

All the temperatures are measured on the Kelvin scale of temperature.

curriculum-key-fact
This equation is true as long as the volume and mass of the gas are constant
Question

A car tyre contains air at 1.25 × 105 Pa when at a temperature of 27°C. Once the car has been running for a while, the temperature of the air in the tyre rises to 42°C. If the volume of the tyre does not change, what is the new pressure of the air in the tyre?

First convert the temperatures into kelvin.

T1 = 27 + 273 = 300 K

T2 = 42 + 273 = 315 K

Now calculate the new pressure

T1 = 300 K

T2 = 315 K

P1 = 1.25 × 105 Pa

\[\frac{P_{1}}{T_{1}} = \frac{P_{2}}{T_{2}}\]

Therefore

\[P_{2} = \frac{P_{1}T_{2}}{T_{1}}\]

\[P_{2} = \frac{(1.25 \times 10^5 \times 315)}{300}\]

The new pressure is 1.31 × 105 Pa.

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