Average Kinetic Energy of Gas Molecules and Temperature
A common question in Class 11 is: “What is the kinetic energy of a gas molecule, and how does it depend on temperature?” The answer is short and surprisingly strong. At a given temperature, the average kinetic energy of the molecules of every ideal gas is the same, whether the gas is hydrogen, oxygen or carbon dioxide. Temperature is, in a way, the measure of this average energy.
The formulas
Average kinetic energy per mole: KE = (3/2) R T
Root mean square speed: vrms = √(3RT / M)
Relation: k = R / NA
k is the Boltzmann constant, 1.381 × 10−23 J/K. R = 8.314 J K−1 mol−1. T must be in kelvin. M is the molar mass in kilograms per mole when you want vrms in m/s. These formulas apply to translational energy, which is the energy of the molecule moving through space. They come from the kinetic theory of an ideal gas.
What the formula says
- KE is proportional to the absolute temperature. Double T in kelvin and the average energy doubles.
- KE does not depend on the type of gas. A hydrogen molecule and an oxygen molecule have the same average KE at the same T.
- KE does not depend on pressure or volume directly. Pressure changes because the molecules hit the walls more often, not because each one moves faster.
- Molecules of a lighter gas move faster at the same temperature, since the same KE = ½ m v2 is shared by a smaller mass.
Worked example 1: energy at 300 K
Find the average translational kinetic energy per mole and per molecule of an ideal gas at 300 K.
Per mole: (3/2) × 8.314 × 300 = 3741 J = 3.74 kJ/mol.
Per molecule: (3/2) × 1.381 × 10−23 × 300 = 6.21 × 10−21 J.
Check: 6.21 × 10−21 × 6.022 × 1023 = 3740 J/mol. Both routes agree.
Worked example 2: oxygen and hydrogen at the same temperature
Compare the average KE and the rms speed of O2 (M = 0.0320 kg/mol) and H2 (M = 0.00202 kg/mol) at 300 K.
Average KE: the same for both, 3.74 kJ/mol.
vrms of O2 = √(3 × 8.314 × 300 / 0.0320) = √(233 800) = 484 m/s.
vrms of H2 = √(3 × 8.314 × 300 / 0.00202) = √(3 704 000) = 1925 m/s.
The ratio is 1925 / 484 = 3.98, about 4, which equals √(32 / 2.02) = 3.98. Hydrogen is about four times faster but carries the same average energy, since its mass is about sixteen times smaller (the speed has a square root, so the speed ratio is 4).
Worked example 3: doubling the energy
A gas is at 27 °C. To what temperature must it be heated to double the average kinetic energy of its molecules?
T1 = 27 + 273 = 300 K. KE is proportional to T, so T2 = 2 × 300 = 600 K = 327 °C.
Many students write 2 × 27 = 54 °C. That is wrong, because the proportionality holds only for the kelvin scale.
Check: (3/2) × 8.314 × 600 = 7483 J = 2 × 3741 J.
Worked example 4: speed when T is four times larger
By what factor does vrms change if the temperature is raised from 300 K to 1200 K?
vrms is proportional to √T. The ratio = √(1200 / 300) = √4 = 2. The speed doubles, while the average kinetic energy becomes four times larger. Remember that energy goes as T and speed goes as √T.
Where the 3/2 comes from
A molecule can move in three independent directions: x, y and z. Each direction contributes (1/2) kT of average energy, so three directions give (3/2) kT. This is also why a monatomic gas such as helium, which has only these three translational motions, has an internal energy U = (3/2) RT per mole. For molecules like N2, rotation and vibration can store extra energy, but the translational part is still (3/2) RT.
Common mistakes
- Using °C instead of K. The energy at 0 °C is not zero. It is (3/2) R × 273.15 = 3.41 kJ/mol.
- Thinking a heavier gas has more KE at the same T. The average KE is the same for all ideal gases. Only the speed differs.
- Using M in g/mol in the vrms formula with R = 8.314. M must be in kg/mol, otherwise the speed is wrong by a factor of √1000.
- Mixing up the three types of speed. The root mean square speed is not the same as the average speed or the most probable speed; the three differ by small factors.
- Saying that each molecule has the average energy. Individual molecules have a spread of speeds; only the average is fixed by T.
Exam relevance
| Type of question | Key idea |
|---|---|
| Compare two gases at the same T | Equal average KE; lighter gas has the higher speed |
| Temperature doubled or tripled | KE scales with T, speed scales with √T |
| Energy per molecule | (3/2) kT, with k = R / NA |
| Speed of molecules | vrms = √(3RT / M) with M in kg/mol |
Some textbooks give R as 8.314 J K−1 mol−1 and others as 8.31; both are fine if used consistently. Check which value your question or book gives.
The Ideal Gas calculator helps you work with temperature in kelvin in gas problems.
Open the Ideal Gas Law Calculator →Turn this concept into marks: solve previous-year CSIR-NET questions and take a timed quiz.
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