Question

The molecules of a gas \(\Lambda\) travel four times faster than the molecules of gas \(\mathrm{B}\) at the same temperature . The ratio of their molecular weights \(\left(\frac{M_{\mathrm{A}}}{M_{\mathrm{B}}}\right)\) will be (1) \(1 / 16\) (2) 4 (3) \(1 / 4\) (4) 16

Short answer

\( \frac {M_{\text{A}}}{M_{\text{B}}} = \frac {1}{16} \)

Step-by-step solution

Understanding the Relationship Between Speed and Molecular Weight

The speed of gas molecules is related to their molecular weight through the equation: \[ v \propto \sqrt{ \frac {1}{M}} \]. This implies that the speed of the gas molecules is inversely proportional to the square root of their molecular weight.

Setting up the Given Condition

It's given that the molecules of gas \Lambda travel four times faster than the molecules of gas \mathrm{B}, so: \[ v_{\text{A}} = 4 v_{\text{B}} \].

Applying the Proportional Relationship

Using the proportional relationship and the given condition, we can write: \[ \sqrt{ \frac {1}{M_{\text{A}}}} = 4 \sqrt{ \frac {1}{M_{\text{B}}}} \].

Solving for the Molecular Weight Ratio

Square both sides of the equation to eliminate the square roots: \[ \frac {1}{M_{\text{A}}} = 16 \frac {1}{M_{\text{B}}} \]. Now, rearrange the equation to solve for the ratio of molecular weights: \[ \frac {M_{\text{A}}}{M_{\text{B}}} = \frac {1}{16} \].

Key concepts

Gas Molecules

Gas molecules are tiny particles that are in constant, random motion. These molecules are much smaller compared to the distances between them, meaning they don't interact much except when they collide. In gases, molecules move in every direction, bumping into each other and the walls of their container. The speed at which these molecules move can vary depending on their temperature and molecular weight. Higher temperatures make molecules move faster because they have more energy. Different gases have molecules of different masses, so their speeds vary accordingly. Understanding gas molecules is essential to grasp how changes in conditions affect the behavior of gases.

Molecular Weight Ratio

The molecular weight ratio is a comparison of the masses of two different molecules. It is a key factor in determining the behavior of gas molecules. For gases \( A \) and \( B \), if one molecule is twice as heavy as another, their speeds will differ. This relationship is given by the equation: \( v \propto \sqrt{ \frac{1}{M} } \). Based on this, if gas \(\Lambda\) travels four times faster than gas \( \mathrm{B} \), we find that the speed of \( \Lambda \) is inversely proportional to the square root of its molecular weight. This means the molecular weight ratio between gas \( \Lambda \) and gas \( B \) is \( \frac{1}{16} \). To put it simply: heavier molecules move slower, and lighter molecules move faster.

Kinetic Theory of Gases

The kinetic theory of gases explains how gas molecules move and how this movement relates to the gas’s properties. Key points include:

• Gas molecules are in constant, random motion.
• The temperature of a gas is directly proportional to the average kinetic energy of its molecules.
• Pressure is caused by collisions of gas molecules with the walls of their container.
• The average kinetic energy of gas molecules is the same for all gases at a given temperature.

From this theory, we understand why molecular weight affects the speed of gas molecules: lighter molecules move faster because they have the same kinetic energy as heavier molecules but less mass. This foundational theory helps us predict and explain many gas behaviors, such as diffusion and effusion, using basic principles of physics and mathematics.