Relate the temperature of a gas to the average kinetic energy of the particles; recall and use the equation T (in K) = θ (in °C) + 273

2.1.2 Particle Model 📦

What is the Particle Model?

The particle model says that all matter is made of tiny particles that are always moving. In a gas, these particles are far apart and bounce around like a crowded dance floor. The faster they move, the higher the temperature of the gas. Think of the particles as energetic dancers: when the music (heat) gets louder, they dance faster and the room feels warmer.

Objective 🎯

Relate the temperature of a gas to the average kinetic energy of its particles and remember the simple conversion between Celsius and Kelvin:

\$T(\text{K}) = \theta(\degree\text{C}) + 273\$

Temperature & Kinetic Energy 🔥

In a gas, the average kinetic energy of the particles is directly proportional to the temperature:

\$\langle KE \rangle \propto T\$

If you double the temperature, the particles move roughly twice as fast on average. This is why a hot cup of coffee feels warmer than a cold one – the coffee’s molecules are dancing faster!

Converting Temperatures 🔢

°C (Celsius)K (Kelvin)
0273
25298
100373

How to Find Average Kinetic Energy 🧮

  1. Measure the temperature of the gas in °C.
  2. Convert to Kelvin using \$T = \theta + 273\$.
  3. Use the relation \$\langle KE \rangle = \frac{3}{2}kB T\$ (where \$kB\$ is the Boltzmann constant, \$1.38 \times 10^{-23}\,\text{J/K}\$).
  4. Plug in the Kelvin value to get \$\langle KE \rangle\$ in joules.

Quick Example 🏃‍♂️

Suppose a gas is at \$25\degree\text{C}\$.

1. Convert: \$T = 25 + 273 = 298\,\text{K}\$.

2. Calculate: \$\langle KE \rangle = \frac{3}{2} (1.38 \times 10^{-23}) (298) \approx 6.2 \times 10^{-21}\,\text{J}\$.

Remember the Key Point!

Temperature is a *measure* of how fast the particles are moving on average. The higher the temperature, the higher the average kinetic energy. And the simple conversion \$T(\text{K}) = \theta(\degree\text{C}) + 273\$ lets you switch between the two units in a snap. 🚀