Kinetic Theory

Kinetic TheoryNEET Physics · Class 11 · NCERT Chapter 12

Overview Notes NEET Questions Interactive Learning

7 interactive concept widgets for Kinetic Theory. Drag any slider, change any number, and watch the formula and the answer update live. Built so you understand how each NEET problem actually works, not just the final number.

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Ideal gas law solver (PV = nRT)

Gas pressure from molecular motion

RMS, average and most probable speed

Maxwell-Boltzmann distribution

Temperature to average KE per molecule

Equipartition of energy bar

Mean free path calculator

Ideal gas law and pressure

Solve PV = nRT for any unknown, and watch pressure emerge from molecular motion as P = (1/3) rho v_rms squared.

Ideal gas law

Ideal gas law solver (PV = nRT)

Pick the unknown, enter the rest, and see the answer in SI units. The same equation contains Boyle's, Charles's and Gay-Lussac's laws as special cases.

Pick the unknown. Enter the other three quantities. Use SI units (Pa, m³, mol, K).

P (Pa)

V (m³)

n (mol)

T (K)

Pressure P (Pa)

1.013e+5

P V = n R T, R = 8.314 J/ (mol \cdot K)

Try this

STP (T = 273 K, P = 101325 Pa): one mole of any ideal gas occupies 22.4 L = 0.0224 m³.
Same gas, double the temperature at constant V: pressure doubles. Gay-Lussac's law.
Same gas, double the volume at constant T: pressure halves. Boyle's law.
Same gas, isobaric: V is proportional to T (in kelvin). Charles's law.

Pressure of gas

Gas pressure from molecular motion

Pressure is one third of density times the square of v_rms. Tweak rho and v_rms to see how each affects pressure.

Each molecule slamming the wall transfers momentum. Add up all the molecules in a unit volume and the pressure comes out as one third of rho times v_rms squared.

Density rho: 1.29 kg/m³

v_rms: 485 m/s

Pressure

1.011e+5 Pa

0.998 atm

P = \frac{1}{3} ρ v_{rms}^{2}

Try this

Air at STP: rho around 1.29 kg/m³ and v_rms around 485 m/s give roughly 1 atm.
Doubling v_rms quadruples pressure (the v² factor).
At fixed density, raising temperature raises v_rms and so raises pressure.

Molecular speeds and distribution

How RMS, average and most-probable speed differ, and the actual Maxwell-Boltzmann distribution at any T.

Speeds

RMS, average and most probable speed

Three different averages of molecular speeds. NEET asks for the ratios and the formulas. Compare them across gases and temperatures.

Pick a gas, set a temperature, and compare the three speed measures. Notice that v_rms is always the largest.

Temperature T: 300 K

Selected: N₂ (M = 28 g/mol)

Most probable v_p

422 m/s

√(2RT/M)

Average v_avg

476 m/s

√(8RT/πM)

RMS v_rms

517 m/s

√(3RT/M)

v_{p} : v_{a v g} : v_{r m s} = 2 : 8/ π : 3 \approx 1 : 1.128 : 1.224

Try this

At fixed T, lighter gases (H₂, He) move faster than heavier gases (CO₂). v scales as 1/√M.
Quadruple T: speeds double (v scales as √T).
v_rms is always larger than v_avg, which is always larger than v_p. Memorise the ratios.

Maxwell distribution

Maxwell-Boltzmann speed distribution

The actual distribution of molecular speeds in a gas. Drag temperature and molar mass to see the curve shift.

Maxwell-Boltzmann distribution: most molecules cluster near v_p with a long tail of fast ones. Heat the gas, and the curve flattens and shifts to the right.

Temperature T: 300 K

Molar mass M: 28 g/mol

Most probable speed v_p

422 m/s

v_{p} = \frac{2 R T}{M}

Try this

Higher T flattens and broadens the distribution; the peak moves to a higher speed.
Lighter gases (small M) have a wider distribution at the same T.
The area under the whole curve always equals 1 (normalisation).

Kinetic temperature

Temperature to average KE

Average translational kinetic energy per molecule equals (3/2) k_B T. Move the temperature slider and watch the energy scale.

Temperature is a direct measure of the average translational kinetic energy of the gas molecules. The same formula works for any ideal gas, no matter the molecular mass.

Temperature T: 300 K (27 °C)

Avg translational KE per molecule

6.213e-21 J

⟨ K E ⟩ = \frac{3}{2} k_{B} T

Per mole

3.74 kJ/mol

⟨ K E ⟩_{mole} = \frac{3}{2} R T

Try this

At room T (300 K), avg translational KE per molecule is about 6.2e-21 J.
Double T: KE doubles. Halve T: KE halves. Linear in T (in kelvin).
At T = 0 K: avg KE = 0 (classical view). Real molecules still have zero-point energy.
Same T means same average KE for ALL gases. Heavier gases just move slower (since (1/2) m v² is fixed).

Equipartition and mean free path

How energy is shared among degrees of freedom, and the average distance a molecule travels between collisions.

Equipartition

Equipartition of energy

Each quadratic degree of freedom averages half k_B T of energy. The bar shows how much energy goes into translation vs rotation for each gas type.

Each quadratic degree of freedom carries half k_B T of energy on average. Pick a gas and a temperature.

Temperature T: 300 K

Diatomic (room T), examples: H₂, O₂, N₂. f = 5.

Energy share per molecule

Trans 60%

Rot 40%

Translational (3 dof)

6.21e-21 J

Rotational (2 dof)

4.14e-21 J

Total per molecule (f/2 k_B T)

1.04e-20 J

E_{molecule} = \frac{f}{2} k_{B} T, E_{mole} = \frac{f}{2} R T

Try this

Monoatomic gases have only 3 translational dof. No rotational energy share.
Diatomic gases at room T add 2 rotational dof. Vibration kicks in only at very high T.
Doubling T doubles the energy share per dof; the percentages stay the same.

Mean free path

Mean free path calculator

Average distance between collisions of gas molecules. Depends inversely on molecular cross-section and number density.

Mean free path lambda is the average distance a molecule travels between collisions. It depends on the molecular size and how packed the gas is.

Molecular diameter d: 0.37 nm

Number density n: 2.50e+25 m⁻³

Mean free path

65.8 nm

λ = \frac{1}{2 π d ^{2} n}

Try this

Air at STP (n ≈ 2.5e25 /m³, d ≈ 0.37 nm): lambda is around 70 nm, much larger than a molecule.
Halving the diameter quadruples the mean free path (d² in the denominator).
Reducing pressure (lower n) drastically increases lambda. In good vacuum (1e15 /m³), lambda is metres.
Mean free path also equals v_avg / collision rate. Faster molecules hit more often per second.

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