Temperature feels like a single number, but underneath it lies a whole crowd of particles moving at different speeds. What temperature actually measures is their average kinetic energy.
📘 What you need to know
The molecules in a gas don’t all move at the same speed — they have a range of speeds
For an ideal gas, the average kinetic energy of the molecules is given by Ēk = 3⁄2 kBT
kB = 1.38 × 10⁻²³ J K⁻¹ is the Boltzmann constant
The absolute temperature of a gas is directly proportional to the average kinetic energy of its molecules
Temperature must be in kelvin whenever this equation is used
Average Kinetic Energy of Gas Particles
In any real gas, individual particles are moving at all sorts of different speeds — some fast, some slow, most somewhere in between. Rather than tracking every particle individually, the kinetic theory of matter deals with their average kinetic energy, which links directly to a quantity you can actually measure: temperature.
Average kinetic energy of an ideal gasĒk = 3⁄2 kBT
Where Ēk is the average kinetic energy of the molecules in joules, kB is the Boltzmann constant, and T is the absolute temperature of the gas in kelvin.
Why This Proportionality Matters
Because Ēk is directly proportional to T, doubling the absolute temperature of a gas doubles the average kinetic energy of its particles too. This is what makes temperature such a useful quantity — it’s a direct, measurable stand-in for something happening at the microscopic scale that you could never observe particle by particle.
A straight line through the origin confirms that absolute temperature and average kinetic energy are directly proportional
🧭 Recipe: Finding Average Particle Speed From Temperature
Write down the average kinetic energy equation — Ēk = 3⁄2 kBT
Write down the kinetic energy equation — Ek = ½mv², using the mass of a single particle
Equate the two — 3⁄2 kBT = ½mv²
Rearrange for v and substitute in the known values
Quick recap: Ē_k = 3⁄2 k_B T. Absolute temperature and average particle kinetic energy rise and fall together — always use kelvin, never Celsius, in this equation.
WE 1
Calculate the average kinetic energy of gas particles in a sample of air at room temperature, 293 K.
The core of a small red dwarf star has a temperature of 4200 K and consists mainly of hydrogen atoms. Calculate the average speed of the hydrogen atoms in the core, in km s⁻¹.
Step 1 — List the known quantities
T = 4200 K, mass of hydrogen atom m_p = 1.673 × 10⁻²⁷ kg, k_B = 1.38 × 10⁻²³ J K⁻¹
Step 2 — Equate the two kinetic energy expressions3⁄2 k_B T = ½ m_p v²Step 3 — Rearrange and calculatev = √(3k_B T ÷ m_p) = √[(3 × 1.38 × 10⁻²³ × 4200) ÷ (1.673 × 10⁻²⁷)]≈ 10 200 m s⁻¹ ≈ 10.2 km s⁻¹
💡 Top tips
Always use the mass of a single particle when equating Ē_k with ½mv², not the mass of a whole mole
T in this equation must always be in kelvin — using Celsius will give a nonsense answer
Remember this equation gives an average — real particles in the sample will still cover a wide range of individual speeds
kB is a fixed constant given in the data booklet — you don’t need to memorise its value
⚠ Common mistakes
Substituting a Celsius temperature directly into Ēk = 3⁄2 kBT without converting to kelvin first
Confusing the Boltzmann constant, kB, with the spring constant, k, or the gas constant, R
Using molar mass instead of the mass of one particle when solving for speed
Treating the calculated speed as the speed of every particle, rather than an average across the whole sample
Up next: Internal Energy — where kinetic energy and potential energy come together to explain what’s really changing inside a substance.
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