Kinetic Theory — NEET notes

Ideal Gas & Gas Laws

An ideal gas obeys PV = nRT exactly — no intermolecular forces, no molecular volume. Boyle's law ( PV = const at constant T ), Charles's law ( V/T = const at constant P ), and Gay-Lussac's law ( P/T = const at constant V ) are all special cases. Avogadro's number N_A = 6.022 10^ 23 connects moles to molecules. NEET frequently tests ideal gas equation numericals and gas law applications.

PV = nRT = Nk_BT

The Ideal Gas

An ideal gas is a theoretical gas where molecules are point particles with no mutual attraction. Real gases behave like ideal gases at low pressure and high temperature.

An ideal gas is a simplified model where: (1) molecules are point particles with negligible volume compared to the container, (2) they exert no forces on each other except during elastic collisions, and (3) collisions with container walls are perfectly elastic.

• Ideal gas assumptions: point-sized molecules, no intermolecular forces, perfectly elastic collisions
• Real gases approach ideal behavior at low pressure and high temperature
• At high pressure or low temperature, intermolecular forces and molecular volume matter — the gas becomes 'real'
• All gas laws (Boyle, Charles, Gay-Lussac, Avogadro) assume ideal behavior

Boyle's, Charles's & Gay-Lussac's Laws

Each gas law holds one variable constant and relates the other two. They are all special cases of PV = nRT.

These three laws are the historical building blocks of the ideal gas equation. Each keeps one state variable constant:

Boyle's Law: PV = constant (T constant, isothermal)

Charles's Law: V T = constant (P constant, isobaric)

Gay-Lussac's Law: P T = constant (V constant, isochoric)

• Boyle: P_1V_1 = P_2V_2 (constant T ) — pressure and volume are inversely proportional
• Charles: V_1 T_1 = V_2 T_2 (constant P ) — volume and temperature are directly proportional
• Gay-Lussac: P_1 T_1 = P_2 T_2 (constant V ) — pressure and temperature are directly proportional
• Temperature MUST be in kelvin in all gas law equations — using Celsius gives wrong answers

The Ideal Gas Equation

The ideal gas equation combines all gas laws into one. It connects macroscopic variables (P, V, T) to the amount of gas (n moles or N molecules).

PV = nRT = Nk_BT

Here n = number of moles, R = 8.314 J mol ^ -1 K ^ -1 (universal gas constant), N = number of molecules, and k_B = R N_A = 1.38 10^ -23 J K ^ -1 (Boltzmann constant). The two forms are equivalent: nR = Nk_B .

• R = 8.314 J mol ^ -1 K ^ -1 — universal gas constant (same for ALL ideal gases)
• k_B = 1.38 10^ -23 J K ^ -1 — Boltzmann constant (gas constant per molecule)
• N_A = 6.022 10^ 23 mol ^ -1 — Avogadro's number
• Relation: R = N_A k_B , so k_B = R / N_A
• At STP ( 0° C, 1 atm): 1 mole of any ideal gas occupies 22.4 L

Quick recall

A gas at 27° C and 1 atm occupies 10 L. What is its volume at 127° C and 2 atm?

Using P_1V_1 T_1 = P_2V_2 T_2 : T_1 = 300 K, T_2 = 400 K. So V_2 = 1 10 400 300 2 = 6.67 L.

Dalton's Law of Partial Pressures

In a mixture of non-reacting ideal gases, the total pressure equals the sum of partial pressures each gas would exert alone in the same volume.

P_ total = P_1 + P_2 + P_3 + = _i n_i RT V

Each gas in a mixture behaves as if the other gases aren't there. The partial pressure P_i of gas i is the pressure it would exert if it alone occupied the entire volume. Since ideal gas molecules don't interact, their pressures simply add up.

• Partial pressure of gas i : P_i = x_i P_ total , where x_i = n_i / n_ total is the mole fraction
• Valid only for non-reacting gas mixtures
• For the mixture as a whole: PV = (n_1 + n_2 + )RT = n_ total RT
• Each gas in the mixture independently satisfies P_iV = n_iRT

Kinetic Theory of Ideal Gas

Gas pressure arises from billions of molecular collisions with container walls. Starting from Newtonian mechanics, the kinetic theory derives P = 1 3 v_ rms ^2 and shows that temperature is a measure of average kinetic energy: 1 2 mv_ rms ^2 = 3 2 k_BT . This connects the microscopic world (molecular speeds) to macroscopic observables (pressure, temperature). NEET frequently tests the KE-temperature relation and the pressure formula.

P = 1 3 v_ rms ^2 | KE_ avg = 3 2 k_BT

Assumptions of Kinetic Theory

The kinetic theory builds on a few simple assumptions about molecular motion to explain macroscopic gas behavior.

The kinetic theory models a gas as a large number of tiny molecules in constant, random motion. The key assumptions are:

• A gas consists of a large number of identical molecules (of the order 10^ 23 )
• Molecules are point particles — their size is negligible compared to the average intermolecular separation
• Molecules are in constant, random motion in all directions with all speeds
• Collisions between molecules and with walls are perfectly elastic — total KE is conserved
• Between collisions, molecules move in straight lines (no forces except during collision)
• The time of collision is negligible compared to the time between collisions

Pressure from Molecular Collisions

By calculating the momentum transferred by molecules hitting a wall, we derive the fundamental relation between gas pressure and molecular speed.

Consider a gas of N molecules, each of mass m , in a cubic container of side L . A molecule moving with velocity component v_x toward a wall bounces back with -v_x (elastic collision). The change in momentum per collision is p = 2mv_x .

The molecule travels a distance 2L (to the opposite wall and back) between consecutive hits on the same wall. Time between hits = 2L/v_x . So force by one molecule on the wall = p t = 2mv_x 2L/v_x = mv_x^2 L .

Summing over all N molecules and dividing by the wall area A = L^2 :

P = 1 3 Nm v^2 V = 1 3 v^2 = 1 3 v_ rms ^2

The factor 1 3 appears because molecules move randomly in 3D: on average, v_x^2 = v_y^2 = v_z^2 = 1 3 v^2 . Here v^2 is the mean of the square of molecular speeds, and v_ rms = v^2 .

• Pressure is proportional to v_ rms ^2 — denser gas or faster molecules means higher pressure
• The 1 3 factor comes from the 3D nature of molecular motion
• = Nm/V is the gas density
• This derivation assumes elastic collisions and ignores gravity

Kinetic Energy & Temperature

The average translational kinetic energy of a gas molecule is directly proportional to the absolute temperature. This is the deepest insight of kinetic theory.

Combining PV = nRT with P = 1 3 Nm v^2 V gives:

1 2 m v^2 = 3 2 k_BT

This is one of the most important results in all of physics. It says: temperature is a direct measure of the average translational kinetic energy of molecules. Double the absolute temperature → double the average KE.

Total KE of n moles = 3 2 nRT = 3 2 Nk_BT

• Average KE per molecule = 3 2 k_BT — depends ONLY on temperature, not on the gas type
• At the same temperature, a heavy molecule ( N_2 ) and a light molecule ( H_2 ) have the SAME average KE
• But the lighter molecule moves FASTER (since 1 2 mv^2 is the same, smaller m means larger v )
• At absolute zero ( T = 0 K), molecular motion stops (classically) — this is the lowest possible temperature
• KE is proportional to T in kelvin, NOT in ° C

Quick recall

Two gases — helium (mass 4 u) and argon (mass 40 u) — are at the same temperature. Compare their average KE and RMS speeds.

Average KE is the same (depends only on T ). RMS speed of He is 40/4 = 10 3.16 times that of Ar (lighter molecules move faster).

Molecular Speeds

Not all molecules in a gas move at the same speed — they follow the Maxwell-Boltzmann distribution. Three characteristic speeds matter: RMS speed v_ rms = 3k_BT/m (from energy), average speed v_ avg = 8k_BT/ m (arithmetic mean), and most probable speed v_ mp = 2k_BT/m (peak of distribution). Their ratio v_ mp : v_ avg : v_ rms = 1 : 1.128 : 1.224 is a NEET staple.

v_ rms = 3k_BT m = 3RT M

RMS Speed

The root-mean-square speed is obtained from the average kinetic energy. It is the most commonly used molecular speed in kinetic theory.

The RMS (root-mean-square) speed is defined as the square root of the mean of the squares of molecular speeds. It connects directly to kinetic energy via 1 2 mv_ rms ^2 = 3 2 k_BT .

v_ rms = v^2 = 3k_BT m = 3RT M

Here m = mass of one molecule, M = molar mass (in kg/mol), T = absolute temperature (K). The two forms are equivalent since k_B/m = R/M .

• v_ rms T — double the temperature (in K) and speed increases by 2
• v_ rms 1/ M — lighter molecules move faster at the same temperature
• The RMS speed is always greater than the average speed (because squaring gives more weight to fast molecules)
• Use M in kg/mol (not g/mol) when calculating in SI units

Average & Most Probable Speeds

The average speed is the arithmetic mean, the most probable speed is where the Maxwell distribution peaks, and the RMS speed comes from the energy. They are always in the order v_mp < v_avg < v_rms.

v_ mp = 2k_BT m = 2RT M

v_ avg = 8k_BT m = 8RT M

v_ rms = 3k_BT m = 3RT M

The three speeds differ only in the numerical factor under the square root: 2 (most probable), 8/π ≈ 2.55 (average), 3 (RMS). Since 2 < 8/ < 3 , the ordering is always:

v_ mp < v_ avg < v_ rms

• Ratio v_ mp : v_ avg : v_ rms = 2 : 8/ : 3 = 1 : 1.128 : 1.224
• v_ mp = speed possessed by the maximum number of molecules (peak of Maxwell curve)
• v_ avg = arithmetic mean of all molecular speeds
• v_ rms = square root of the mean of squared speeds (linked to KE)
• All three speeds are proportional to T/M

Quick recall

If the RMS speed of oxygen molecules at a temperature is v , what is the most probable speed?

v_ mp = v_ rms 2/3 = v 2/3 0.816v . Since v_ rms = 3RT/M and v_ mp = 2RT/M .

Maxwell Speed Distribution

The Maxwell-Boltzmann distribution describes the fraction of molecules at each speed. It is asymmetric — more molecules have speeds below the average than above it.

In a gas at temperature T , molecules don't all move at the same speed. The Maxwell-Boltzmann speed distribution f(v) gives the fraction of molecules with speeds between v and v + dv . The curve starts at zero, rises to a peak at v_ mp , then falls off gradually — it has a long tail toward high speeds.

• The curve is asymmetric — it has a longer tail on the high-speed side
• As T increases: peak shifts right (higher v_ mp ), peak height decreases, curve broadens
• The area under the entire curve = 1 (accounts for all molecules)
• At any temperature, a small fraction of molecules have very high speeds — this is why some molecules can escape from a liquid surface (evaporation)
• The distribution depends on T and m — heavier molecules have a narrower, taller peak

Degrees of Freedom & Equipartition

A degree of freedom (DOF) is an independent way a molecule can store energy. Monoatomic gases have 3 (translational), diatomic have 5 (3 translational + 2 rotational), and triatomic have 6-7. The law of equipartition of energy assigns 1 2 k_BT of energy to each DOF. Total internal energy U = f 2 nRT , where f = number of DOFs. This is the bridge between molecular structure and thermodynamic quantities like C_v and C_p .

Energy per DOF = 1 2 k_BT | U = f 2 nRT

Degrees of Freedom

The number of independent ways a molecule can possess kinetic energy. Depends on whether the molecule is monoatomic, diatomic, or polyatomic.

A degree of freedom (DOF) is an independent mode of motion that contributes to the molecule's energy. There are three types: translational (moving through space), rotational (spinning about an axis), and vibrational (atoms oscillating within the molecule).

• Monoatomic (He, Ne, Ar): 3 DOF — only translational ( x , y , z directions). Single atom can't rotate or vibrate meaningfully
• Diatomic (H _2 , O _2 , N _2 ): 5 DOF — 3 translational + 2 rotational (about axes perpendicular to the bond). Rotation about the bond axis has negligible moment of inertia
• Triatomic/Polyatomic (H _2 O, CO _2 ): 6 DOF (nonlinear) or 7 DOF (linear) — 3 translational + 2 or 3 rotational + vibrational modes
• Vibrational DOFs become significant only at high temperatures ( > 1000 K for diatomic); at room temperature, they are 'frozen out'

Law of Equipartition of Energy

In thermal equilibrium, each degree of freedom gets an equal share of the total energy. This elegant principle determines the internal energy and heat capacities of ideal gases.

The law of equipartition of energy states: in thermal equilibrium, each degree of freedom contributes 1 2 k_BT to the average energy of a molecule. For a molecule with f degrees of freedom:

Average energy per molecule = f 2 k_BT

Total internal energy of n moles: U = f 2 nRT

For translational DOF: each of the 3 directions ( x , y , z ) gets 1 2 k_BT , giving total translational KE = 3 2 k_BT . For rotational DOF: each rotation axis gets 1 2 k_BT . For vibrational DOF: each mode gets k_BT ( 1 2 k_BT kinetic + 1 2 k_BT potential).

• Each translational DOF: 1 2 k_BT (kinetic only)
• Each rotational DOF: 1 2 k_BT (kinetic only)
• Each vibrational DOF: k_BT ( 1 2 k_BT kinetic + 1 2 k_BT potential)
• Monoatomic: U = 3 2 nRT , Diatomic (room T ): U = 5 2 nRT
• Energy is shared equally among ALL available DOF — no mode is 'preferred'

Quick recall

What is the total internal energy of 2 moles of a diatomic gas at 300 K? (Ignore vibrations.)

U = f 2 nRT = 5 2 2 8.314 300 = 12471 J 12.5 kJ.

Specific Heat of Gases

Gases have two specific heats: C_v (at constant volume) and C_p (at constant pressure). From equipartition, C_v = f 2 R and C_p = C_v + R (Mayer's relation). The ratio = C_p/C_v = 1 + 2/f determines how the gas behaves in adiabatic processes. For monoatomic gases = 5/3 , diatomic = 7/5 , and polyatomic = 4/3 . These values are heavily tested in NEET.

C_p - C_v = R | = C_p C_v = 1 + 2 f

C_v, C_p & Mayer's Relation

At constant volume, all heat goes into increasing internal energy. At constant pressure, the gas must also do work to expand — so more heat is needed for the same temperature rise.

When you heat a gas at constant volume, no work is done ( W = P V = 0 since V = 0 ). ALL the heat goes into increasing molecular kinetic energy (internal energy). So C_v measures the heat needed to raise the temperature of 1 mole by 1 K at constant volume.

When you heat a gas at constant pressure, the gas expands and does work on its surroundings. You need to supply extra heat to compensate for this work. So C_p > C_v always.

C_p - C_v = R (Mayer's relation — for 1 mole of an ideal gas)

The difference is exactly R because for 1 mole expanding at constant pressure, the work done is P V = nR T = R 1 K = R .

• Mayer's relation: C_p - C_v = R = 8.314 J mol ^ -1 K ^ -1 (for ideal gases)
• C_p > C_v ALWAYS for any gas
• The extra heat in C_p goes into the P V work of expansion
• For solids and liquids, C_p C_v because they barely expand

The Ratio γ = C_p/C_v

The adiabatic exponent gamma depends only on the degrees of freedom of the gas molecule. It governs adiabatic processes, speed of sound, and much more.

From equipartition: C_v = f 2 R (energy stored in f degrees of freedom). Using Mayer's relation: C_p = C_v + R = f 2 R + R = f+2 2 R . Their ratio:

= C_p C_v = f + 2 f = 1 + 2 f

• depends ONLY on the molecular structure (number of DOF), not on temperature or pressure
• More DOF → smaller (monoatomic has highest , polyatomic has lowest)
• governs adiabatic processes: PV^ = constant
• Speed of sound in a gas: v_s = RT/M — also depends on
• At very high temperatures, vibrational DOF activate → f increases → decreases

Quick recall

A gas has = 1.4 . What is the gas — monoatomic, diatomic, or polyatomic? Find C_v and C_p .

= 1.4 = 7/5 → diatomic ( f = 5 ). C_v = R - 1 = R 0.4 = 5 2 R . C_p = C_v = 7 2 R .

Specific Heat of Gas Mixtures

For a mixture of ideal gases, the effective Cv and Cp are mole-fraction weighted averages of the individual values. The mixture gamma is then Cp_mix / Cv_mix.

When two or more ideal gases are mixed, the total internal energy is the sum of individual internal energies. For a mixture of n_1 moles of gas 1 (with C_ v1 ) and n_2 moles of gas 2 (with C_ v2 ):

C_ v,mix = n_1 C_ v1 + n_2 C_ v2 n_1 + n_2

C_ p,mix = n_1 C_ p1 + n_2 C_ p2 n_1 + n_2

_ mix = C_ p,mix C_ v,mix

• Mayer's relation still holds for the mixture: C_ p,mix - C_ v,mix = R
• _ mix lies between _1 and _2 — it's a weighted average
• Equal moles of two gases: _ mix is the simple average only if you use the n/( - 1) formula
• The mixture formula works only for non-reacting ideal gases

Quick recall

1 mole of He ( = 5/3 ) is mixed with 1 mole of H _2 ( = 7/5 ). Find _ mix .

2 _ mix - 1 = 1 5/3 - 1 + 1 7/5 - 1 = 1 2/3 + 1 2/5 = 3 2 + 5 2 = 4 . So _ mix - 1 = 1/2 , giving _ mix = 3/2 = 1.5 .

Mean Free Path

The mean free path l is the average distance a molecule travels between successive collisions: l = 1 2 n d^2 = k_BT 2 d^2 P . It depends on molecular diameter d , number density n , temperature, and pressure. Brownian motion — the random zigzag of pollen grains in water — provides visible evidence of molecular collisions. NEET tests the formula and the factors affecting mean free path.

l = 1 2 n d^2 = k_BT 2 d^2 P

Mean Free Path

A molecule constantly collides with others. The average straight-line distance between successive collisions is the mean free path, and it depends on how crowded and how big the molecules are.

A molecule moving through a gas is like running through a crowded room — you bump into people. The mean free path l is the average distance you run between bumps. It depends on two things: how crowded the room is (number density n ) and how big you are (molecular diameter d ).

l = 1 2 n d^2

Using n = P/k_BT (from ideal gas law PV = Nk_BT , so n = N/V = P/k_BT ):

l = k_BT 2 d^2 P

• l T — higher temperature (at constant P ) means molecules spread out, fewer collisions, longer mean free path
• l 1/P — higher pressure (at constant T ) means more molecules per volume, more collisions, shorter mean free path
• l 1/d^2 — bigger molecules have a larger 'collision cross-section', so they collide more often
• The 2 factor accounts for the relative motion of molecules (they're ALL moving, not just one)

Brownian Motion

The random zigzag motion of small suspended particles (like pollen grains in water) provides direct experimental evidence for the existence of molecules and their ceaseless random motion.

In 1827, botanist Robert Brown observed pollen grains suspended in water under a microscope. The grains moved in a random, erratic zigzag pattern — never settling down. This is Brownian motion, and it's caused by the unbalanced bombardment of the grain by water molecules from all sides.

• Brownian motion is the random, ceaseless, zigzag movement of small suspended particles in a fluid
• Caused by unequal bombardment by fluid molecules from different sides
• Provides direct evidence for the molecular nature of matter and random molecular motion
• Increases with temperature (faster molecules → stronger kicks)
• Decreases with particle size (larger particles feel more balanced bombardment)
• Decreases with viscosity of the medium (thicker fluid dampens the kicks)