Calculate average kinetic energy from temperature, or find temperature, speed, or mass from kinetic energy for gases and moving objects.
Average Kinetic Energy Formula
The calculator uses three formulas depending on which mode you select.
Per-molecule (Temperature mode):
KE_avg = (3/2) * kB * T
Per-mole (also used in Energy to Temp mode):
KE_avg = (3/2) * R * T
RMS molecular speed (optional in Temperature mode):
v_rms = sqrt(3 * R * T / M)
Moving object (classical kinetic energy):
KE = (1/2) * m * v^2
- KE_avg = average translational kinetic energy
- kB = Boltzmann constant, 1.380649 × 10⁻²³ J/K
- R = molar gas constant, 8.31446 J/(mol·K)
- T = absolute temperature in kelvin
- M = molar mass in kg/mol (input in g/mol is converted)
- m = object mass in kg
- v = object speed in m/s
- v_rms = root-mean-square molecular speed in m/s
The gas formulas assume an ideal gas in thermal equilibrium and count only translational motion, not rotation or vibration. Temperature must be at or above absolute zero. The moving-object formula is non-relativistic and breaks down once speed approaches a noticeable fraction of the speed of light; the calculator flags results above 1% of c.
The three modes do the following:
- Temperature takes T and returns KE per molecule, per mole, and (if a gas is selected) the RMS speed.
- Energy to Temp inverts the gas formula. Give it KE in J/particle, eV/particle, J/mol, or kJ/mol and it returns T in K, °C, and °F.
- Moving Object uses KE = ½mv². Leave any one of energy, mass, or speed blank and the calculator solves for it.
Reference Tables
Average kinetic energy of an ideal-gas molecule at common temperatures.
| Temperature | KE per molecule (J) | KE per molecule (eV) | KE per mole (kJ/mol) |
|---|---|---|---|
| 100 K | 2.07 × 10⁻²¹ | 0.0129 | 1.247 |
| 200 K | 4.14 × 10⁻²¹ | 0.0259 | 2.494 |
| 273.15 K (0 °C) | 5.66 × 10⁻²¹ | 0.0353 | 3.406 |
| 298.15 K (25 °C) | 6.17 × 10⁻²¹ | 0.0385 | 3.718 |
| 500 K | 1.04 × 10⁻²⁰ | 0.0646 | 6.236 |
| 1000 K | 2.07 × 10⁻²⁰ | 0.1293 | 12.472 |
RMS speed of common gases at 25 °C (298.15 K).
| Gas | Molar mass (g/mol) | v_rms (m/s) |
|---|---|---|
| Hydrogen (H₂) | 2.016 | 1920 |
| Helium (He) | 4.003 | 1363 |
| Nitrogen (N₂) | 28.014 | 515 |
| Air | 28.97 | 506 |
| Oxygen (O₂) | 31.998 | 482 |
| Carbon dioxide (CO₂) | 44.01 | 411 |
Worked Examples and FAQ
Example 1: Air at room temperature. At T = 298.15 K, KE_avg = (3/2)(1.380649 × 10⁻²³)(298.15) = 6.17 × 10⁻²¹ J per molecule. Per mole that is (3/2)(8.314)(298.15) = 3718 J/mol, or about 3.72 kJ/mol.
Example 2: Solving for temperature. If the average kinetic energy of gas molecules is 8.00 × 10⁻²¹ J, then T = KE / ((3/2) kB) = 8.00 × 10⁻²¹ / (2.071 × 10⁻²³) = 386 K, which is 113 °C.
Example 3: Moving object. A 1500 kg car at 20 m/s has KE = 0.5 × 1500 × 20² = 300,000 J, or 300 kJ.
Why the factor 3/2? A monatomic ideal gas has three translational degrees of freedom. Each contributes (1/2) kB T to the average energy, giving (3/2) kB T total. Diatomic and polyatomic gases store more energy in rotation and vibration, but the translational part is still (3/2) kB T.
Does average kinetic energy depend on the type of gas? No. At the same temperature, every ideal gas has the same average translational kinetic energy per molecule. Lighter molecules just move faster to compensate, which is why hydrogen has a much higher v_rms than carbon dioxide at the same T.
What is the difference between average kinetic energy and total kinetic energy? Average kinetic energy is per molecule (or per mole). Total kinetic energy of a sample is the average multiplied by the number of molecules (or moles) present.
When does KE = ½mv² stop working? Once v exceeds roughly 1% of the speed of light, relativistic effects become noticeable and you need KE = (γ − 1)mc² instead. For everyday speeds the classical formula is accurate to many decimal places.
