Kinetic Theory — Revision Notes

⚡ 30-Second Revision

KTG Postulates: — Point masses, random motion, no intermolecular forces, elastic collisions, negligible collision time.
Pressure: — $P = \frac{1}{3} \frac{Nm\overline{v^2}}{V}$
Avg. Kinetic Energy per molecule: — $E_{avg} = \frac{3}{2} k_B T$
RMS Speed: — $v_{rms} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3k_B T}{m}}$
Avg. Speed: — $v_{avg} = \sqrt{\frac{8RT}{\pi M}}$
Most Probable Speed: — $v_{mp} = \sqrt{\frac{2RT}{M}}$
Speed Ratio: — $v_{mp} : v_{avg} : v_{rms} = \sqrt{2} : \sqrt{8/\pi} : \sqrt{3} \approx 1.414 : 1.596 : 1.732$
Degrees of Freedom ($f$): — Monoatomic=3, Diatomic=5 (at moderate T), Polyatomic (non-linear)=6.
Equipartition Law: — Energy per degree of freedom = $\frac{1}{2} k_B T$
Internal Energy (1 mole): — $U = \frac{f}{2} RT$
Molar Specific Heat at constant V ($C_V$): — $C_V = \frac{f}{2} R$
Molar Specific Heat at constant P ($C_P$): — $C_P = (\frac{f}{2} + 1) R$
Mayer's Relation: — $C_P - C_V = R$
Ratio of Specific Heats ($gamma$): — $\gamma = \frac{C_P}{C_V} = 1 + \frac{2}{f}$
Mean Free Path: — $\lambda = \frac{k_B T}{\sqrt{2} \pi d^2 P}$

2-Minute Revision

Kinetic Theory of Gases (KTG) explains gas behavior by modeling molecules as tiny, randomly moving particles with elastic collisions. The core idea is that temperature is a direct measure of the average translational kinetic energy of these molecules, given by $E_{avg} = \frac{3}{2} k_B T$. This means if temperature doubles, average kinetic energy doubles. Pressure arises from molecular collisions with container walls, expressed as $P = \frac{1}{3} \frac{Nm\overline{v^2}}{V}$.

Gas molecules don't all move at the same speed; we use statistical speeds like **RMS speed ($v_{rms} = \sqrt{\frac{3RT}{M}}$)**, which is crucial for energy calculations. Remember $v_{rms} \propto \sqrt{T/M}$.

**Degrees of freedom ($f$) quantify how a molecule stores energy: 3 for monoatomic, 5 for diatomic (at moderate T), 6 for non-linear polyatomic. The Law of Equipartition of Energy** states each degree of freedom gets $\frac{1}{2} k_B T$ energy.

This leads to molar specific heats: $C_V = \frac{f}{2} R$ and $C_P = (\frac{f}{2} + 1) R$, with their ratio $\gamma = 1 + \frac{2}{f}$. Finally, the **mean free path ($\lambda$)** is the average distance between collisions, inversely proportional to pressure and directly proportional to temperature ($\lambda \propto T/P$).

Always convert temperatures to Kelvin for calculations.

5-Minute Revision

The Kinetic Theory of Gases (KTG) is a powerful model that bridges the microscopic world of molecules with the macroscopic properties of gases. It's built on the idea of an ideal gas, characterized by point-mass molecules in constant random motion, negligible intermolecular forces, and perfectly elastic collisions.

The most profound outcome is the kinetic interpretation of temperature: the absolute temperature ($T$) of a gas is directly proportional to the average translational kinetic energy of its molecules, $E_{avg} = \frac{3}{2} k_B T$.

This means $T$ is a measure of molecular 'jiggle'.

Pressure ($P$) is the result of these molecules colliding with the container walls. The fundamental relation is $P = \frac{1}{3} \frac{Nm\overline{v^2}}{V}$, where $\overline{v^2}$ is the mean square speed.

From this, we define the Root Mean Square (RMS) speed, $v_{rms} = \sqrt{\frac{3RT}{M}}$, which is the most relevant speed for energy calculations. Remember that $v_{rms}$ is always greater than the average speed ($v_{avg} = \sqrt{\frac{8RT}{\pi M}}$) and most probable speed ($v_{mp} = \sqrt{\frac{2RT}{M}}$).

**Degrees of freedom ($f$)** are crucial for understanding how a molecule stores energy. A monoatomic gas has $f=3$ (translational). A diatomic gas has $f=5$ (3 translational + 2 rotational) at moderate temperatures.

A non-linear polyatomic gas has $f=6$ (3 translational + 3 rotational). The Law of Equipartition of Energy states that each degree of freedom contributes $\frac{1}{2} k_B T$ to the average energy per molecule.

This directly leads to the internal energy of one mole of gas, $U = \frac{f}{2} RT$.

This also defines molar specific heats: $C_V = \frac{f}{2} R$ (at constant volume) and $C_P = (\frac{f}{2} + 1) R$ (at constant pressure). Their difference is Mayer's relation: $C_P - C_V = R$. The ratio of specific heats, $\gamma = \frac{C_P}{C_V} = 1 + \frac{2}{f}$, is a key parameter in adiabatic processes. For example, for a diatomic gas, $f=5$, so $C_V = \frac{5}{2}R$, $C_P = \frac{7}{2}R$, and $\gamma = 1.4$.

Finally, the **mean free path ($\lambda$)** is the average distance a molecule travels between collisions. It's inversely proportional to pressure and directly proportional to temperature: $\lambda = \frac{k_B T}{\sqrt{2} \pi d^2 P}$. Always convert temperatures to Kelvin for all calculations. Be mindful of units (e.g., molar mass in kg/mol). Practice problems involving gas mixtures and comparing properties under different conditions.

Prelims Revision Notes

1
Ideal Gas Postulates: — Molecules are point masses, in random motion, no intermolecular forces (except elastic collisions), negligible collision time. Temperature is proportional to average translational KE.
2
Pressure Formula: — $P = \frac{1}{3} \frac{Nm\overline{v^2}}{V}$. Pressure is due to molecular collisions with walls.
3
Kinetic Energy and Temperature: — Average translational kinetic energy per molecule $E_{avg} = \frac{3}{2} k_B T$. Total translational KE for $n$ moles is $n \cdot N_A \cdot \frac{3}{2} k_B T = \frac{3}{2} nRT$. This is a direct proportionality: $E_{avg} \propto T$.
4
Molecular Speeds:

* RMS speed: $v_{rms} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3k_B T}{m}}$. ($M$ is molar mass in kg/mol, $m$ is mass of one molecule in kg). * Average speed: $v_{avg} = \sqrt{\frac{8RT}{\pi M}}$. * Most probable speed: $v_{mp} = \sqrt{\frac{2RT}{M}}$. * Order: $v_{mp} < v_{avg} < v_{rms}$. * Ratio: $v_{mp} : v_{avg} : v_{rms} = \sqrt{2} : \sqrt{8/\pi} : \sqrt{3}$.

1
Degrees of Freedom ($f$): — Number of independent ways a molecule can store energy.

* Monoatomic (He, Ne, Ar): $f=3$ (3 translational). * Diatomic (O$_2$, N$_2$, H$_2$): $f=5$ (3 translational + 2 rotational) at moderate T. At high T, $f=7$ (vibrational modes active). * Polyatomic (non-linear, H$_2$O, NH$_3$): $f=6$ (3 translational + 3 rotational). * Polyatomic (linear, CO$_2$): $f=5$ (3 translational + 2 rotational).

1
Law of Equipartition of Energy: — Each active degree of freedom contributes $\frac{1}{2} k_B T$ per molecule (or $\frac{1}{2} RT$ per mole) to the internal energy.
2
Internal Energy ($U$): — For $n$ moles, $U = n \frac{f}{2} RT$.
3
Molar Specific Heats:

* At constant volume: $C_V = \frac{f}{2} R$. * At constant pressure: $C_P = (\frac{f}{2} + 1) R$. * Mayer's Relation: $C_P - C_V = R$.

1
Ratio of Specific Heats ($gamma$): — $\gamma = \frac{C_P}{C_V} = 1 + \frac{2}{f}$.

* Monoatomic: $\gamma = 5/3 \approx 1.67$. * Diatomic: $\gamma = 7/5 = 1.4$. * Polyatomic (non-linear): $\gamma = 4/3 \approx 1.33$.

1
Mean Free Path ($lambda$): — Average distance between collisions. $\lambda = \frac{k_B T}{\sqrt{2} \pi d^2 P}$. $\lambda \propto T/P$ and $\lambda \propto 1/d^2$.
2
Unit Conversion: — Always convert temperature to Kelvin ($T_K = T_C + 273.15$). Molar mass $M$ in kg/mol. $R = 8.314 \text{ J mol}^{-1} \text{ K}^{-1}$. $k_B = 1.38 \times 10^{-23} \text{ J K}^{-1}$.

Vyyuha Quick Recall

To remember the order of molecular speeds: Most Average RMS. Think of it as 'MAR' for the increasing order of speeds: Most Probable < Average < RMS. For degrees of freedom: Mono Di Poly (non-linear) is 3-5-6 (at moderate temperatures).