Kinetic Theory of Gases & Statistical Distributions: Master Notes

Chapter Overview

Kinetic Theory of Gases (KTG) provides a microscopic statistical interpretation of macroscopic thermodynamic variables. In IIT-JEE (Advanced), top problems focus on molecular speed ratios ($v_{\text{rms}},v_{\text{avg}},v_{\text{mp}}$), mixture degrees of freedom & ${\gamma }_{\text{mix}}$, mean free path scaling ($\lambda \propto T/P$), and Van der Waals critical states.

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1. Microscopic Pressure Derivation & Molecular Speeds

$$P=\frac{1}{3}\rho v_{\text{rms}}^2=\frac{1}{3}\frac{Nm}{V}{v}_{\text{rms}}^2⟹PV=\frac{2}{3}N\left(\frac{1}{2}m{v}_{\text{rms}}^2\right)=\frac{2}{3}{E}_{\text{internal}}$$

1.1 The Three Characteristic Molecular Speeds

Speed Type	Formula	Numerical Multiplier
Most Probable Speed ($v_{\text{mp}}$)	$\sqrt{\frac{2RT}{M}}=\sqrt{\frac{2k_{B}T}{m}}$	$1.414\sqrt{\frac{RT}{M}}$ (Peak of distribution)
Average / Mean Speed ($⟨v⟩$)	$\sqrt{\frac{8RT}{\pi M}}=\sqrt{\frac{8k_{B}T}{\pi m}}$	$1.596\sqrt{\frac{RT}{M}}$
Root-Mean-Square Speed ($v_{\text{rms}}$)	$\sqrt{\frac{3RT}{M}}=\sqrt{\frac{3k_{B}T}{m}}$	$1.732\sqrt{\frac{RT}{M}}$

$$\boxed{{\displaystyle v_{\text{mp}}<⟨v⟩<v_{\text{rms}}\text{Ratio: }\sqrt{2}:\sqrt{\frac{8}{\pi }}:\sqrt{3}\approx 1:1.128:1.225}}$$

Multi-Mode DiagramKinetic Theory: Maxwell-Boltzmann Speed Distribution

Option 1: Publication-Grade Scientific Vector SVG

Molecular speed distribution comparing v_most_probable, v_average, and v_rms.

RMS Speed: $v_{\text{rms}} = \sqrt{\frac{3RT}{M}}$

Speed Ordering: $v_{\text{rms}} > v_{\text{avg}} > v_{\text{mp}} = \sqrt{\frac{2RT}{M}}$

Option 2: Interactive 60 FPS Canvas Physics Engine

Gas particle kinetic collision container simulator.

Temperature T: 300 K

v_rms ($v_{\text{rms}}$):517 m/s

Option 3: Interactive 3D Spatial Viewport

3D molecular gas container.

+X

+Y

+Z

👆 Click and drag anywhere to rotate in 3D space

Azimuth Angle (ψ):35°

Elevation Angle (θ):-22°

Kinetic Pressure:$P = \frac{1}{3}\rho v_{\text{rms}}^2$

Option 5: Reactive Parametric Physics with Live Calculus

Speed distribution parameter solver.

Temperature T: 300 K

: 45

v_rms ($v_{\text{rms}}$):$484 m/s$

Active Mode:Option 1 (Print & Derivation Vector SVG)Switch modes using the tabs above to explore different visualizations.

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2. Degrees of Freedom ($f$) & Gas Mixtures

$$C_v=\frac{f}{2}R,C_p=C_v+R=(\frac{f}{2}+1)R,\gamma =\frac{C_p}{C_v}=1+\frac{2}{f}$$

Gas Atomicity	Degrees of Freedom $f$	$C_v$	$C_p$	$\gamma =C_p/C_v$
Monoatomic (He, Ne, Ar)	$3$ (Translational only)	$\frac{3}{2}R$	$\frac{5}{2}R$	$\frac{5}{3}\approx 1.67$
Diatomic (Rigid: $O_2,N_2$)	$5$ ($3$ Trans + $2$ Rot)	$\frac{5}{2}R$	$\frac{7}{2}R$	$\frac{7}{5}=1.40$
Non-Linear Polyatomic ($H_{2}O,N{H}_3$)	$6$ ($3$ Trans + $3$ Rot)	$3R$	$4R$	$\frac{4}{3}\approx 1.33$

2.1 Gas Mixture Formulas

For a mixture of $n_1$ moles of gas 1 and $n_2$ moles of gas 2:

$$C_{v,\text{mix}}=\frac{n_1{C}_{v1}+n_2{C}_{v2}}{n_1+n_2},\boxed{{\displaystyle {\gamma }_{\text{mix}}=\frac{n_1{C}_{p1}+n_2{C}_{p2}}{n_1{C}_{v1}+n_2{C}_{v2}}=\frac{n_1\frac{{\gamma }_1}{{\gamma }_1-1}+n_2\frac{{\gamma }_2}{{\gamma }_2-1}}{\frac{n_1}{{\gamma }_1-1}+\frac{n_2}{{\gamma }_2-1}}}}$$

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3. Mean Free Path & Van der Waals Real Gas

• Mean Free Path ($\lambda$): Average distance between successive molecular collisions:

  $$\boxed{{\displaystyle \lambda =\frac{1}{\sqrt{2}\pi n{d}^2}=\frac{k_{B}T}{\sqrt{2}\pi d^{2}P}}}$$

  (where $d$ is molecular diameter, $n=N/V$ is number density).

• Van der Waals Equation of State:

  $$(P+\frac{a{n}^2}{V^2})(V-nb)=nRT$$
  • Critical Temperature: $T_c=\frac{8a}{27Rb}$
  • Critical Pressure: $P_c=\frac{a}{27b^2}$
  • Critical Volume: $V_c=3b$

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4. Authentic Previous Years Questions (PYQs)

PYQ 1: JEE Advanced 2022 (Paper 2) — Gas Mixture Adiabatic Index

Question:
A vessel contains a mixture of $n_1=1\text{ mole}$ of helium (${\gamma }_1=5/3$) and $n_2=2\text{ moles}$ of oxygen (${\gamma }_2=7/5$). Find the adiabatic exponent ${\gamma }_{\text{mix}}$ of the mixture.

Step-by-Step Solution:

1. $C_{v1}=\frac{3}{2}R$ (Helium is monoatomic).
2. $C_{v2}=\frac{5}{2}R$ (Oxygen is diatomic).
3. Molar heat capacity of mixture at constant volume:$$C_{v,\text{mix}}=\frac{n_1{C}_{v1}+n_2{C}_{v2}}{n_1+n_2}=\frac{1\times \frac{3}{2}R+2\times \frac{5}{2}R}{1+2}=\frac{\frac{3}{2}R+5R}{3}=\frac{\frac{13}{2}R}{3}=\frac{13}{6}R$$
4. Molar heat capacity at constant pressure:$$C_{p,\text{mix}}=C_{v,\text{mix}}+R=\frac{13}{6}R+R=\frac{19}{6}R$$
5. Ratio:$${\gamma }_{\text{mix}}=\frac{C_{p,\text{mix}}}{C_{v,\text{mix}}}=\frac{\frac{19}{6}R}{\frac{13}{6}R}=\boxed{{\displaystyle \frac{19}{13}\approx 1.46}}$$

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PYQ 2: JEE Advanced 2019 — Mean Free Path Scaling

Question:
An ideal gas in a closed container is heated such that its absolute temperature increases by a factor of 4 while the volume remains constant. How does the mean free path $\lambda$ and the collision frequency $f_{\text{coll}}$ change?

Step-by-Step Solution:

1. Mean free path depends only on number density $n=N/V$:$$\lambda =\frac{1}{\sqrt{2}\pi n{d}^2}$$Since volume $V$ and particle number $N$ are constant, $\lambda$ remains strictly unchanged (${\lambda }_f={\lambda }_i$).
2. Collision frequency is $f_{\text{coll}}=\frac{v_{\text{avg}}}{\lambda }\propto \sqrt{T}$. Since $T\to 4T⟹v_{\text{avg}}\to 2v_{\text{avg}}$:$$\boxed{{\displaystyle f_{\text{coll, final}}=2f_{\text{coll, initial}}}}$$

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5. High-Yield Formula Sheet

Quantity	Formula	Notes
RMS Speed	$v_{\text{rms}}=\sqrt{\frac{3RT}{M}}$	$M$ in kg/mol
Internal Energy	$U=\frac{f}{2}nRT$	Ideal gas
Mean Free Path	$\lambda \propto \frac{T}{P}\propto \frac{1}{n}$	Density dependent
Critical Temp	$T_c=\frac{8a}{27Rb}$	Liquefaction threshold