“Root three, root eight by pi, root two”
The three molecular speeds
vrms uses 3, average uses 8/π, most probable uses 2, all over M. The numbers fall in that order, and so do the speeds.
Class 11 Physics · Gujarat Board, CBSE, GUJCET and JEE
Kinetic Theory of Gases formulas
Kinetic theory explains pressure and temperature by counting molecular collisions. It is the bridge between mechanics and thermodynamics, and its three speeds, with the order between them, are asked nearly every year.
12 formulas, 3 memory hooks, and the 5 mistakes that cost the most marks.
The gas laws
| What | Formula | Symbols and notes |
|---|---|---|
| Ideal gas equation | pV = nRT = NkT | R = 8.314 J mol−1 K−1, k = 1.38 × 10−23 J K−1 |
| Boltzmann constant | k = R/NA | the gas constant per molecule |
| Pressure from kinetic theory | p = ⅓ρvrms² = ⅓(mN/V)vrms² | ρ is the gas density |
Molecular speeds
| What | Formula | Symbols and notes |
|---|---|---|
| Root mean square speed | vrms = √(3RT/M) = √(3kT/m) | M is the molar mass in kg per mole |
| Average speed | vav = √(8RT/πM) | slightly smaller than vrms |
| Most probable speed | vmp = √(2RT/M) | the peak of the Maxwell distribution |
| Order of the three | vrms > vav > vmp | in the ratio √3 : √(8/π) : √2 |
Energy and degrees of freedom
| What | Formula | Symbols and notes |
|---|---|---|
| Kinetic energy per molecule | KE = (3/2)kT | depends only on temperature, not on the gas |
| Kinetic energy per mole | KE = (3/2)RT | for a monatomic ideal gas |
| Law of equipartition | each degree of freedom gets ½kT per molecule | ½RT per mole |
| Degrees of freedom | monatomic 3, diatomic 5, triatomic 6 | at ordinary temperatures |
| Mean free path | λ = 1/(√2 nπd²) | n is molecules per unit volume, d the molecular diameter |
Taught by Darshit Bhatt, founder and lead faculty at Darshit Sir Institute. Instrumentation and Control engineer, with a master's degree and a chartered degree, teaching in Paldi, Ahmedabad since 2006.
Know the formula but not when to use it?
Darshit Sir has taught this in Paldi since 2006, in the classroom and live online.
Memory hooks
How to remember this chapter.
“Temperature is just kinetic energy”
What temperature means
Average translational kinetic energy is (3/2)kT for every gas. At the same temperature hydrogen and oxygen molecules carry the same energy, so the lighter one moves faster.
“Half kT per freedom”
Equipartition
Every degree of freedom, translational or rotational, gets ½kT per molecule. Count the freedoms and you have the internal energy.
Where marks go
The mistakes that cost the most.
- Using grams per mole for M in the speed formulas. They need kilograms per mole, so 32 g per mole is 0.032.
- Thinking a heavier gas is hotter because it has more energy. At the same temperature every gas has the same average kinetic energy per molecule.
- Getting the speed order backwards. Root mean square is the largest of the three, most probable the smallest.
- Using pV = nRT with temperature in °C. Every gas law needs kelvin.
- Forgetting that an ideal gas has no intermolecular force, so its internal energy is entirely kinetic.
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