Temperature & Kinetic Theory

You’ll be able to

Describe the postulates of kinetic molecular theory
Relate absolute temperature to the average kinetic energy of molecules
Convert between Celsius and Kelvin and reason on the absolute scale

What temperature really measures

A gas is a swarm of molecules in ceaseless, random motion — flying, colliding, and rebounding billions of times a second. Kinetic molecular theory (KMT) models this crowd with a few clean assumptions: molecules are tiny compared with the space between them, they move in random directions, they collide elastically (no kinetic energy is lost), and between collisions they feel no forces. What we sense as temperature is nothing more than the average kinetic energy of that molecular motion. Hotter means faster.

Average translational kinetic energy

KE_avg = (3/2) k_B · T

k_B = 1.38 × 10⁻²³ J·K⁻¹ is Boltzmann’s constant. T must be in kelvin — this relationship only holds on the absolute scale.

Why we need an absolute scale

The Celsius zero is arbitrary — it just marks where water freezes. But kinetic energy cannot be negative, so there must be a temperature where molecular motion is minimized: absolute zero, 0 K = −273 °C. The Kelvin scale starts there, which is why every gas and energy equation uses kelvin. A kelvin is the same size as a Celsius degree, so you convert by shifting, not scaling: T(K) = T(°C) + 273. Doubling a Kelvin temperature genuinely doubles the average kinetic energy; doubling a Celsius temperature means nothing physical.

Root-mean-square speed

v_rms = √(3 k_B · T / m)

The typical molecular speed grows with √T, not T. Heavier molecules (larger m) move slower at the same temperature.

Worked example

Find the average translational kinetic energy of a gas molecule at room temperature, 27 °C.

1.Convert to kelvin first: T = 27 + 273 = 300 K.
2.Apply KE_avg = (3/2) k_B T = (3/2)(1.38 × 10⁻²³ J·K⁻¹)(300 K).
3.Multiply: (3/2)(1.38 × 10⁻²³)(300) = 1.5 × 414 × 10⁻²³.

Answer: 6.21 × 10⁻²¹ J per molecule

Watch out

Never plug Celsius into a kinetic-energy or gas equation. Using 27 instead of 300 K here would understate the energy more than tenfold. Convert to kelvin before you compute anything.

Checkpoint

A thermometer reads 27 °C. What is this temperature on the absolute (Kelvin) scale?

Checkpoint

A gas is heated from 27 °C to 327 °C. By what factor does the average translational kinetic energy of its molecules increase?

On the exam

Two proportionalities to lock in: KE_avg ∝ T (linear in absolute temperature), but v_rms ∝ √T. When a problem doubles the temperature, energy doubles while speed rises only by √2.

Answer the 2 checkpoints as you read.