Mean Free Path — Revision Notes

⚡ 30-Second Revision

Definition: — Average distance a molecule travels between collisions.
Formula 1 (with number density): — $\lambda = \frac{1}{\sqrt{2} n \pi d^2}$
Formula 2 (with P and T): — $\lambda = \frac{kT}{\sqrt{2} P \pi d^2}$
Proportionalities:

- $\lambda \propto 1/n$ - $\lambda \propto 1/P$ (at constant T) - $\lambda \propto T$ (at constant P) - $\lambda \propto 1/d^2$ - $\lambda$ is independent of $T$ at constant $V$.

Constants: — $k$ (Boltzmann's constant), $d$ (molecular diameter).

2-Minute Revision

The mean free path ($\lambda$) is a crucial concept in the kinetic theory of gases, representing the average distance a gas molecule travels before colliding with another molecule. It is not a fixed value but depends on the gas's properties and conditions.

The primary formulas are $\lambda = \frac{1}{\sqrt{2} n \pi d^2}$ (where $n$ is number density, $d$ is molecular diameter) and $\lambda = \frac{kT}{\sqrt{2} P \pi d^2}$ (where $k$ is Boltzmann's constant, $T$ is temperature, $P$ is pressure).

Key relationships to remember for NEET are: $\lambda$ is inversely proportional to number density ($n$) and pressure ($P$, at constant $T$). It is directly proportional to temperature ($T$, at constant $P$).

Crucially, $\lambda$ is inversely proportional to the square of the molecular diameter ($d^2$). Remember that at constant volume, $n$ is constant, so $\lambda$ is independent of temperature. These proportionalities are frequently tested in conceptual and ratio-based problems.

5-Minute Revision

The mean free path ($\lambda$) is the average distance a gas molecule traverses between successive collisions. This concept is fundamental to understanding the microscopic dynamics of gases and their macroscopic properties like diffusion and viscosity. The two key mathematical expressions for $\lambda$ are:

1
$\lambda = \frac{1}{\sqrt{2} n \pi d^2}$, where $n$ is the number density (molecules per unit volume) and $d$ is the molecular diameter. This form highlights the inverse relationship with the 'crowdedness' of the gas and the size of the molecules.
2
$\lambda = \frac{kT}{\sqrt{2} P \pi d^2}$, derived by substituting $n = P/(kT)$ from the ideal gas law. This form is particularly useful for analyzing the effects of temperature ($T$) and pressure ($P$).

Key Proportionalities for NEET:

Pressure: — $\lambda \propto 1/P$ (at constant temperature). Higher pressure means more molecules, more collisions, shorter $\lambda$.
Temperature: — $\lambda \propto T$ (at constant pressure). Higher temperature at constant pressure means gas expands, $n$ decreases, leading to longer $\lambda$. However, if volume is constant, $n$ is constant, so $\lambda$ is independent of $T$.
Molecular Diameter: — $\lambda \propto 1/d^2$. Larger molecules present a bigger target, leading to more collisions and shorter $\lambda$.

Example: If the pressure of a gas is halved at constant temperature, $\lambda$ will double. If the molecular diameter is halved, $\lambda$ will become four times larger. Always convert temperature to Kelvin for calculations. Understanding these dependencies and the conditions under which they apply is critical for solving both numerical and conceptual NEET problems.

Prelims Revision Notes

Mean Free Path ($\lambda$)

Definition: The average distance a gas molecule travels between successive collisions with other molecules.

Key Formulas:

1
In terms of number density ($n$) and molecular diameter ($d$):

$$\lambda = \frac{1}{\sqrt{2} n \pi d^2}$$

where: * $n = N/V$ (number of molecules per unit volume) * $d$ = molecular diameter * $\pi d^2$ = collision cross-section ($\sigma$)

1
In terms of pressure ($P$) and temperature ($T$):

$$\lambda = \frac{kT}{\sqrt{2} P \pi d^2}$$

where: * $k$ = Boltzmann's constant ($1.38 \times 10^{-23}\,\text{J/K}$) * $T$ = absolute temperature (in Kelvin) * $P$ = pressure (in Pascals)

Proportionality Relationships (Crucial for NEET):

With Number Density ($n$): — $\lambda \propto 1/n$

* Higher $n$ (more crowded) $\implies$ shorter $\lambda$.

With Pressure ($P$): — $\lambda \propto 1/P$ (at constant $T$)

* Higher $P$ $\implies$ higher $n$ $\implies$ shorter $\lambda$.

**With Temperature ($T$):**

* At **constant Pressure ($P$):** $\lambda \propto T$ * Higher $T$ $\implies$ gas expands $\implies$ lower $n$ $\implies$ longer $\lambda$. * At **constant Volume ($V$):** $\lambda$ is independent of $T$. * Constant $V$ means $n$ is constant. Since $\lambda$ depends only on $n$ and $d$, it doesn't change with $T$.

With Molecular Diameter ($d$): — $\lambda \propto 1/d^2$

* Larger $d$ (bigger target) $\implies$ more collisions $\implies$ shorter $\lambda$.

Units:

$\lambda$ in meters (m)
$n$ in $\text{m}^{-3}$
$d$ in meters (m)
$P$ in Pascals (Pa)
$T$ in Kelvin (K)

Common Mistakes to Avoid:

Forgetting to convert temperature to Kelvin.
Confusing the temperature dependence at constant pressure vs. constant volume.
Incorrectly applying the square dependence for molecular diameter (e.g., $\lambda \propto 1/d$ instead of $1/d^2$).
Confusing mean free path with the average distance between molecules.

Vyyuha Quick Recall

To remember the factors affecting mean free path ($\lambda$):

Large Targets Pack Densely, Shortening Lambda.

Large Targets: Larger molecular diameter ($d$) means shorter $\lambda$ ($\lambda \propto 1/d^2$).
Pack Densely: Higher number density ($n$) or pressure ($P$) means shorter $\lambda$ ($\lambda \propto 1/n$, $\lambda \propto 1/P$).
Shortening Lambda: All these factors lead to a shorter mean free path.

For temperature: Temperature Lengthens Lambda (at constant P). Higher T, longer $\lambda$ (if P is constant).