Gas temperature and pressure

Internal Energy vs Temperature

• To understand how heating affects pressure, you must understand the difference between internal energy and temperature.
• Internal energy is the total sum of all the randomly distributed kinetic and potential energies of the particles in a system.
• In the GCSE ideal gas model, we assume there are no intermolecular forces between gas particles, meaning their potential energy is zero and internal energy is entirely made up of kinetic energy.
• Temperature, on the other hand, is a measure of the average kinetic energy of the particles in the gas.
• Therefore, heating a gas transfers energy to the kinetic energy stores of its particles, increasing both the internal energy and the temperature.

The Causal Mechanism: Why Pressure Increases

If you heat a fixed mass of gas inside a container with a constant volume (like a rigid metal cylinder), the pressure will increase. This happens through a specific sequence of events:

1. Thermal energy is transferred to the gas, increasing its temperature.
2. This means the average kinetic energy of the gas particles increases.
3. Because they have more kinetic energy, the particles move at a higher average speed.
4. These faster-moving particles hit the container walls with a higher collision frequency (more impacts per second).
5. When they hit the walls, they undergo a larger change in momentum, meaning each individual impact exerts a greater force.
6. With both more frequent and more forceful collisions, the total force on the container walls increases, leading to a higher gas pressure.

The Temperature-Pressure Relationship

• For a fixed mass of gas at a constant volume, the pressure is directly proportional to its absolute temperature.
• If you double the temperature, the pressure doubles — provided the temperature is measured on the Kelvin scale.
• This mathematical relationship (Gay-Lussac's Law) is written as:

$$\frac{P_1}{T_1} = \frac{P_2}{T_2}$$

Absolute Zero and the Kelvin Scale

• Since temperature is tied to kinetic energy, cooling a gas slows its particles down.
• At $-273.15^\circ\text{C}$, the particles reach their minimum possible internal energy and stop moving entirely — a point known as absolute zero.
• At absolute zero, there are no collisions with the container walls, so the gas exerts zero pressure.
• The Kelvin scale starts at absolute zero ($0\text{ K}$). To convert from Celsius to Kelvin, simply add $273$:

$$T(K) = T(^\circ\text{C}) + 273$$

Worked Example

A rigid metal gas canister at $20^\circ\text{C}$ has a pressure of $150\text{ kPa}$. The canister is left in the sun and its temperature rises to $85^\circ\text{C}$. Calculate the new pressure of the gas inside the canister. Assume the volume remains constant.

Step 1: Identify the values and convert temperatures to Kelvin.

• $P_1 = 150\text{ kPa}$
• $T_1 = 20 + 273 = 293\text{ K}$

Step 2: Rearrange the formula to find $P_2$.

• $\frac{P_1}{T_1} = \frac{P_2}{T_2}$

Step 3: Substitute and calculate.

• $P_2 = 150 \times \left(\frac{358}{293}\right)$

Doing Work on a Gas

• Heating is not the only way to increase the temperature of a gas; you can also do work on it.
• If you quickly compress a gas using a bicycle pump, you are applying a force over a distance to move the piston.
• This mechanical work transfers energy directly into the kinetic energy stores of the gas particles.
• As a result, the internal energy increases, the average kinetic energy rises, and the pump feels noticeably warmer.