Molecular Speeds — Revision Notes

⚡ 30-Second Revision

Most Probable Speed ($v_p$) — $v_p = \sqrt{\frac{2RT}{M}}$
Average Speed ($v_{avg}$) — $v_{avg} = \sqrt{\frac{8RT}{\pi M}}$
Root Mean Square Speed ($v_{rms}$) — $v_{rms} = \sqrt{\frac{3RT}{M}}$
Order of Speeds — $v_p < v_{avg} < v_{rms}$
Ratio of Speeds — $v_p : v_{avg} : v_{rms} = \sqrt{2} : \sqrt{8/\pi} : \sqrt{3} \approx 1.414 : 1.596 : 1.732$
Dependence on Temperature — $v \propto \sqrt{T}$ (T in Kelvin)
Dependence on Molar Mass — $v \propto \frac{1}{\sqrt{M}}$ (M in kg/mol)
Units — $T$ in Kelvin, $M$ in $\text{kg mol}^{-1}$, $R = 8.314,\text{J mol}^{-1}\text{K}^{-1}$.

2-Minute Revision

Molecular speeds describe the distribution of velocities among gas particles, not a single speed. The Maxwell-Boltzmann distribution illustrates this range. We define three key characteristic speeds: most probable speed ($v_p$), average speed ($v_{avg}$), and root mean square speed ($v_{rms}$).

$v_p$ is the speed of the largest fraction of molecules, $v_{avg}$ is the simple arithmetic mean, and $v_{rms}$ is crucial as it directly relates to the average kinetic energy and absolute temperature of the gas.

All three speeds increase with the square root of absolute temperature ($v \propto \sqrt{T}$) and decrease with the square root of molar mass ($v \propto 1/\sqrt{M}$). Their magnitudes are always in the order $v_p < v_{avg} < v_{rms}$, with a fixed ratio of $\sqrt{2} : \sqrt{8/\pi} : \sqrt{3}$.

Remember to use Kelvin for temperature and kilograms per mole for molar mass in calculations. These concepts are vital for understanding gas behavior, diffusion, and effusion.

5-Minute Revision

The kinetic theory of gases posits that gas molecules are in continuous, random motion, leading to a distribution of speeds rather than a single value. This distribution is described by the Maxwell-Boltzmann curve. For NEET, three specific speeds are critical:

1
Most Probable Speed ($v_p$) — The speed at which the maximum number of molecules are moving. It's the peak of the distribution curve. Formula: $v_p = \sqrt{\frac{2RT}{M}}$.
2
Average Speed ($v_{avg}$) — The arithmetic mean of all molecular speeds. Formula: $v_{avg} = \sqrt{\frac{8RT}{\pi M}}$.
3
Root Mean Square Speed ($v_{rms}$) — The square root of the average of the squares of the speeds. This is directly linked to the average translational kinetic energy ($E_k = \frac{1}{2}mv_{rms}^2 = \frac{3}{2}k_BT$) and thus to the absolute temperature. Formula: $v_{rms} = \sqrt{\frac{3RT}{M}}$.

Key Relationships:

Order — $v_p < v_{avg} < v_{rms}$. This is because the squaring in $v_{rms}$ gives more weight to higher speeds.
Ratio — $v_p : v_{avg} : v_{rms} = \sqrt{2} : \sqrt{8/\pi} : \sqrt{3} \approx 1.414 : 1.596 : 1.732$.
Temperature Dependence — All speeds are proportional to $\sqrt{T}$ (absolute temperature in Kelvin). Increasing temperature increases molecular speeds.
Molar Mass Dependence — All speeds are inversely proportional to $\sqrt{M}$ (molar mass in kg/mol). Lighter gases move faster.

Example: If $v_{rms}$ of a gas at $300,\text{K}$ is $500,\text{m/s}$, what is its $v_{rms}$ at $1200,\text{K}$? Since $v_{rms} \propto \sqrt{T}$, we have $\frac{v_{rms,2}}{v_{rms,1}} = \sqrt{\frac{T_2}{T_1}}$. $\frac{v_{rms,2}}{500} = \sqrt{\frac{1200}{300}} = \sqrt{4} = 2$. So, $v_{rms,2} = 2 \times 500 = 1000,\text{m/s}$.

Always ensure temperature is in Kelvin and molar mass in kg/mol for calculations. These concepts are fundamental for solving problems related to gas laws, diffusion, and kinetic energy.

Prelims Revision Notes

1
Molecular Speeds — Gas molecules do not have a single speed; they have a distribution of speeds (Maxwell-Boltzmann distribution).
2
Three Characteristic Speeds

* **Most Probable Speed ($v_p$)**: Speed of maximum number of molecules. $v_p = \sqrt{\frac{2RT}{M}}$. * **Average Speed ($v_{avg}$)**: Arithmetic mean of all speeds. $v_{avg} = \sqrt{\frac{8RT}{\pi M}}$. * **Root Mean Square Speed ($v_{rms}$)**: Related to average kinetic energy. $v_{rms} = \sqrt{\frac{3RT}{M}}$.

1
Units — $T$ must be in Kelvin ($T_K = T_{^circ C} + 273$). $M$ must be in $\text{kg mol}^{-1}$ (convert $ext{g mol}^{-1}$ to $ext{kg mol}^{-1}$ by dividing by 1000). $R = 8.314,\text{J mol}^{-1}\text{K}^{-1}$.
2
Order of Magnitudes — $v_p < v_{avg} < v_{rms}$.
3
Ratio of Speeds — $v_p : v_{avg} : v_{rms} = \sqrt{2} : \sqrt{8/\pi} : \sqrt{3}$. (Approximate values: $1.414 : 1.596 : 1.732$).
4
Effect of Temperature ($T$) — All speeds are directly proportional to $\sqrt{T}$. If $T$ increases, speeds increase. $\frac{v_2}{v_1} = \sqrt{\frac{T_2}{T_1}}$.
5
Effect of Molar Mass ($M$) — All speeds are inversely proportional to $\sqrt{M}$. If $M$ increases, speeds decrease. $\frac{v_2}{v_1} = \sqrt{\frac{M_1}{M_2}}$.
6
Kinetic Energy Relation — Average translational kinetic energy per molecule $E_k = \frac{1}{2}mv_{rms}^2 = \frac{3}{2}k_BT$. Total kinetic energy for $n$ moles is $U = n \frac{3}{2}RT$.
7
Maxwell-Boltzmann Curve — At higher temperatures, the curve broadens, flattens, and shifts to the right (higher speeds). For lighter gases, the curve also shifts to higher speeds and broadens.
8
Common Traps — Incorrect unit conversions (especially $T$ to Kelvin, $M$ to kg/mol), mixing up formulas, or misinterpreting the proportionality relationships.

Vyyuha Quick Recall

To remember the order of speeds ($v_p < v_{avg} < v_{rms}$): People Always Remember: Probable, Average, Root-mean-square. (P is smallest, R is largest).

To remember the constants in the formulas for $v_p, v_{avg}, v_{rms}$ (2, $8/\pi$, 3): 2 People 8 Apples 3 Roots. $v_p \propto \sqrt{2}$, $v_{avg} \propto \sqrt{8/\pi}$, $v_{rms} \propto \sqrt{3}$.