Question

Which relation holds good for Graham's law of diffusion? (1) $\frac{r_1}{r_2}=\sqrt{\frac{M_2}{M_1}}$ (at same $P$ and $T$ ) (2) $\frac{r_1}{r_2}=\sqrt{\frac{d_2}{d_1}}$ (at same $P$ and $T$ ) (3) $\frac{r_i}{r_2}=\sqrt{\frac{M_2}{M_1}}\times \frac{P_1}{P_2}$ (at same $T$ ) (4) All

Short answer

The correct relation is Option (1).

Step-by-step solution

- Understand Graham's Law

Graham's Law of diffusion states that the rate of diffusion of a gas is inversely proportional to the square root of its molar mass at constant temperature and pressure.

- Check the Given Options

Review the given options to see which one matches the statement of Graham’s Law.

- Analyze Option (1)

Option (1) states: \ \ $\frac{r_1}{r_2}=\sqrt{\frac{M_2}{M_1}}$ \ \ This matches the formula derived from Graham's Law where r is the rate of diffusion and M is the molar mass.

- Analyze Option (2)

Option (2) states: \ \ $\frac{r_1}{r_2}=\sqrt{\frac{d_2}{d_1}}$ \ \ This can be correct as well if the densities of two gases are known, because the density (d) is directly related to the molar mass (M).

- Analyze Option (3)

Option (3) states: \ \ $\frac{r_i}{r_2}=\sqrt{\frac{M_2}{M_1}}\times \frac{P_1}{P_2}$ \ \ This does not match, since Graham’s Law traditionally only considers constant pressure.

- Decide if All Options (4) Are Correct

Since only Options (1) and (2) are valid under the conditions of constant temperature and pressure, Option (4) cannot be correct.

Key concepts

Rate of Diffusion

The rate of diffusion refers to how quickly a gas spreads out and mixes with another gas in a given environment. According to Graham's Law of Diffusion, the rate at which a gas diffuses or effuses is inversely proportional to the square root of its molar mass, given that temperature and pressure remain constant. Mathematically, this is expressed as $\frac{r_1}{r_2}=\sqrt{\frac{M_2}{M_1}}$, where $r_1$ and $r_2$ are the diffusion rates of gas 1 and gas 2, respectively, and $M_1$ and $M_2$ are their respective molar masses.

This relationship means that lighter gases will diffuse more quickly than heavier gases. For example, hydrogen gas (\

Molar Mass

Molar mass is a fundamental property of chemicals, representing the mass of one mole of a given substance, expressed in grams per mole (g/mol). It is defined based on the atomic or molecular composition of the substance. For example, the molar mass of water (H₂O) is approximately 18 g/mol, derived from the combined molar masses of its constituent atoms (hydrogen and oxygen).

In the context of Graham's Law of Diffusion, molar mass plays a crucial role in determining the rate of diffusion of gases. The formula $\frac{r_1}{r_2}=\sqrt{\frac{M_2}{M_1}}$ indicates that gases with lower molar masses will diffuse faster than those with higher molar masses, provided that the temperature and pressure conditions are constant. This is a direct consequence of the inverse proportionality between the square root of molar mass and the rate of diffusion.

Understanding molar mass helps us predict and explain the behavior of gases in various chemical processes, including reactions and diffusion phenomena.

Density

Density is another important concept when discussing the diffusion of gases. It is defined as the mass per unit volume of a substance, typically expressed in grams per liter (g/L) for gases. For gases at a given temperature and pressure, density can be directly related to molar mass. The formula for density $d$ in terms of molar mass $M$ and the ideal gas law constants is $$d=\frac{MP}{RT}$$, where $P$ is the pressure, $R$ is the gas constant, and $T$ is the temperature in Kelvin.

In the context of Graham's Law, the relationship between density and molar mass becomes evident. Since the rate of diffusion $r$ is inversely proportional to the square root of the molar mass, and considering that density is also influenced by molar mass, we can modify Graham's Law to account for density: $$\frac{r_1}{r_2}=\sqrt{\frac{d_2}{d_1}}$$. Thus, gases with lower densities (which typically correspond to lower molar masses) will diffuse more rapidly than denser gases at constant temperature and pressure.

Understanding the interplay between density and diffusion rates is essential for comprehending various practical applications, such as air purification, chemical reactions, and respiratory processes.