Question

The molecular weight of a gas which diffuses through a porous plug at \(1 / 6^{\text {th }}\) of the speed of hydrogen under identical conditions is: (a) 27 (b) 72 (c) 36 (d) 48

Short answer

The molecular weight of the gas is 72.

Step-by-step solution

Understanding Graham's Law of Effusion

Graham's Law of Effusion states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass (M). Thus, if a gas effuses at 1/6 the speed of hydrogen (H2), we can express this relationship as: Rate of Gas / Rate of H2 = sqrt(Molar Mass of H2 / Molar Mass of Gas).

Applying the relationship to Hydrogen

Because Hydrogen (H2) has a molar mass of approximately 2 g/mol, we can substitute this value into the relationship: 1/6 = sqrt(2 / Molar Mass of Gas).

Solving for the Molar Mass of the Gas

Square both sides of the equation to remove the square root: (1/6)^2 = 2 / Molar Mass of Gas. Then solve for the Molar Mass of Gas: Molar Mass of Gas = 2 / (1/6)^2.

Calculating the Molecular Weight

Perform the calculation: Molar Mass of Gas = 2 / (1/36), which simplifies to Molar Mass of Gas = 2 * 36 = 72.

Key concepts

Molar Mass

Molar mass is a fundamental concept in chemistry, defined as the mass of one mole of particles - atoms, molecules, or ions - of a substance. The molar mass is generally reported in grams per mole (g/mol) and can be calculated by summing the atomic masses of all atoms in a molecule. For gases, molar mass becomes particularly useful when studying their effusion and diffusion behaviors.

Molar mass serves as a bridge between the microscopic properties of atoms and molecules and their macroscopic physical behaviors. It's essential for understanding Graham's Law of Effusion because this relationship relies on comparing the molar masses of different gases to predict rates of effusion or diffusion.

Rate of Effusion

The rate of effusion of a gas refers to how quickly it moves through a small opening or porous barrier into a vacuum. According to Graham's Law of Effusion, this rate is inversely proportional to the square root of the gas's molar mass. This implies that lighter gases effuse faster than heavier gases.

Understanding the rate of effusion is crucial for a wide range of applications, including the separation of isotopes and manufacturing processes where precise control of gas movement is necessary. Using Graham's Law, we can predict the relative speeds at which different gases will effuse, provided they're under similar physical conditions.

Gas Diffusion

Gas diffusion is the process by which gas molecules spread from an area of higher concentration to one of lower concentration. Diffusion, like effusion, relies on the motion of gas particles, but unlike effusion, it does not involve the passage through a barrier.

While Graham's Law primarily describes effusion, it also applies to diffusion because the underlying principle is the same: lighter gases will spread out more rapidly. The ability of gases to diffuse quickly and mix thoroughly is used in various contexts, from perfuming a room to delivering oxygen through the atmosphere to our lungs.

Molecular Weight Calculation

Calculating molecular weight, also known as molecular mass, involves the sum of the atomic weights of all the atoms within a molecule. The molecular weight can be deduced from the periodic table and is often used interchangeably with 'molar mass' especially when describing the mass of gaseous substances.

The solution provided employs molecular weight calculation when applying Graham's Law. By understanding the relationship between molar mass and rate of effusion, we can calculate the unknown molecular weight of a gas, as demonstrated by comparing it with a known gas such as hydrogen. This technique is beneficial in identifying unknown substances in both educational settings and professional chemical analysis.