Class 11 Physics · NCERT notes for NEET

Kinetic Theory: NCERT Notes, Formulas and Tricks for NEET

Kinetic theory explains pressure, temperature and specific heats from the random motion of molecules. It is a short, formula-rich chapter; NEET usually has about one question from it, often on $v_{r m s}$ , degrees of freedom or $γ$ .

1. Ideal gas equation
2. Kinetic interpretation of pressure and temperature 🔒
3. Degrees of freedom and equipartition 🔒
4. Mean free path 🔒

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Topic 1

Ideal gas equation

A pressure cooker, a bicycle pump and a weather balloon all follow one line of physics: $p V = μ R T$ .

Gas molecules are far apart (about 10 times their size of 2 Å) and move freely; in solids and liquids they are close, so intermolecular forces (attraction at a few Å, repulsion closer) matter.
At low pressure and high temperature (far above liquefaction) every gas obeys $p V = N k_{B} T$ , with $k_{B} = 1.38 \times 1 0^{- 23} J K^{- 1}$ (Boltzmann constant), same for all gases.
Avogadro: equal volumes of all gases at the same $p$ and $T$ contain equal numbers of molecules. 1 mole $= N_{A} = 6.02 \times 1 0^{23}$ molecules and occupies 22.4 L at STP (273 K, 1 atm).
Ideal gas equation: $p V = μ R T$ , with $R = N_{A} k_{B} = 8.314 J mol^{- 1} K^{- 1}$ and $μ = N / N_{A} = M / M_{0}$ (mass / molar mass).
An ideal gas obeys $p V = μ R T$ exactly at all $p$ and $T$ . Real gases come close only at low $p$ and high $T$ ; all curves of $p V / μ T$ meet $R$ as $p \to 0$ .
Boyle's law: $T$ fixed, $p V$ = constant (p-V graph is a rectangular hyperbola). Charles' law: $p$ fixed, $V \propto T$ (T in kelvin).
Density form: $p = ρR T / M_{0}$ . At fixed $p$ and $T$ , density is proportional to molar mass.
Dalton's law: for non-reacting ideal gases in one vessel, $p = p_{1} + p_{2} + \dots = (μ_{1} + μ_{2} + \dots) R T / V$ .
Graham's law: rate of diffusion $\propto 1/ M$ ; this is how $^{235} U$ is separated from $^{238} U$ (as $UF_{6}$ ).

Boyle: hotter isotherms lie farther from the axes. Charles: extend the V-T line back and it hits zero volume at -273.15\,^circC, which is 0 K. — Boyle: hotter isotherms lie farther from the axes. Charles: extend the V-T line back and it hits zero volume at $- 273.15^{\circ} C$ , which is 0 K.

Real gases deviate from pV = mu RT, more so at high p and low T. As p to 0 every curve ends at R = 8.31, the ideal value. — Real gases deviate from $p V = μ R T$ , more so at high $p$ and low $T$ . As $p \to 0$ every curve ends at $R = 8.31$ , the ideal value.

Formulas

$p V = μ R T = N k_{B} T$
$μ$ = moles, $N$ = molecules, $T$ in kelvin
$R = N_{A} k_{B} = 8.314 J mol^{- 1} K^{- 1}$
Universal gas constant
$p = \frac{ρR T}{M _{0}}$
$ρ$ = density, $M_{0}$ = molar mass in kg/mol
$p = \frac{( μ _{1} + μ _{2} + \dots ) R T}{V}$
Dalton's law of partial pressures
$\frac{r _{1}}{r _{2}} = \frac{M _{2}}{M _{1}}$
Graham's law of diffusion

Tips, tricks and hacks

Ratio form, not numbersFor the same gas,

p V / T

is constant:

p_{1} V_{1} / T_{1} = p_{2} V_{2} / T_{2}

. Example:

T

doubles and

V

halves, so

p

becomes

2 \times 2 = 4

times. No need for

R

Count molecules fast

N = p V / (k_{B} T)

. At STP, 1

m^{3}

of any gas has about

2.7 \times 1 0^{25}

molecules (Loschmidt number). Use it to check orders of magnitude.

Mixture: add molesMixing gases in one vessel at the same

T

: total

p = (\sum μ_{i}) R T / V

. Example: 2 mol

O_{2}

+ 3 mol He in 10 L at 300 K behaves like 5 mol of one gas.

Density ratioAt the same

p

and

T

ρ \propto M_{0}

O_{2}

(32) is 16 times denser than

H_{2}

(2).

Common mistakes

✗ Putting temperature in $^{\circ}$ C in $p V = μ R T$ or in Charles' law.

✓ Always use kelvin; $V \propto T$ holds only on the absolute scale.

✗ Taking molar mass of $O_{2}$ as 32 in $p = ρR T / M_{0}$ with SI units.

✓ In SI, $M_{0} = 0.032 kg mol^{- 1}$ ; otherwise the answer is off by 1000.

✗ Real gases behave ideally at high pressure and low temperature.

✓ It is the reverse: low $p$ , high $T$ , where molecules are far apart and forces are negligible.

NEET traps

22.4 L per mole is true only at STP; at 300 K and 1 atm one mole occupies about 24.6 L.

" $k_{B}$ depends on the gas" is false: $k_{B}$ and $R$ are universal constants.

In a p-V graph two isotherms never cross; the one farther from the origin is at higher $T$ .

Quick check: A gas in a closed vessel has its absolute temperature doubled and its volume reduced to one-third. The pressure becomes

2/3 times
3/2 times
6 times
1/6 times

Answer: C. $p \propto T / V$ , so $p_{2} / p_{1} = 2/ (1/3) = 6$ .

Topic 2 · 🔒

Kinetic interpretation of pressure and temperature

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Topic 3 · 🔒

Degrees of freedom and equipartition

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Topic 4 · 🔒

Mean free path

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