AP® Physics 2: Algebra-Based · Unit 9 · Topic 9.1
Temperature and Average Kinetic Energy
Temperature is a measure of the average translational kinetic energy of a huge number of particles, given by K = 3/2 kT, not a property any single atom can have on its own.
A single atom flying alone through space has a speed and a kinetic energy, but no temperature. Temperature only describes a crowd. It is a statistical statement about the average kinetic energy of trillions of atoms bouncing around, colliding, speeding up, slowing down. Any one atom's speed is constantly changing from collisions, but the average over the whole swarm settles into a stable number, and that number is what the thermometer reads.
The link is the equation K = 3/2 kT, where K is the average translational kinetic energy per atom, k is Boltzmann's constant (1.38 x 10−23 J/K), and T is absolute temperature in kelvin. Raise T and you raise the average energy each atom carries, in a straight line, no exceptions. Because kinetic energy also equals 1/2 m v2, you can solve for a typical speed: setting 1/2 m v_rms2 equal to 3/2 kT gives v_rms = √(3kT/m), the root-mean-square speed, a kind of average speed for the swarm.
This sits underneath everything else in thermodynamics. Pressure, internal energy, and the ideal gas law all trace back to this same picture of atoms in motion, temperature just measures how energetically they are moving. Keep T in kelvin always, since the whole relationship is built on absolute zero meaning zero average kinetic energy.
Key terms
- Boltzmann's constant (k)
- A fixed number, 1.38 x 10−23 J/K, that links temperature to energy at the particle level.
- Average translational kinetic energy (K)
- The mean kinetic energy per particle in a gas due to its motion through space, equal to 3/2 kT.
- RMS speed (v_rms)
- The square root of the average of the squared particle speeds, a representative speed for the whole gas.
- Absolute temperature (T)
- Temperature measured in kelvin starting from absolute zero, required for K = 3/2 kT to hold.
Watch it explained
- How it worksHow does the average kinetic energy of trillions of atoms turn into one number on a thermometer?
- Worked exampleFind the average kinetic energy of a helium atom at 300 K using K = 3/2 kT.
- Worked exampleFind the rms speed of a nitrogen molecule at 300 K if its mass is 4.7 x 10−26 kg.
- Common mistakeWhy doesn't doubling the Celsius reading double the average kinetic energy of the atoms?
Check yourself
End of topic.