Problem 86 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

Two gases \(\mathrm{X}\) and \(\mathrm{Y}\) have their molecular speed in ratio of \(3: 1\) at certain temperature. The ratio of their molecular masses \(M_{x}: M_{y}\) is (a) \(1: 3\) (b) \(3: 1\) (c) \(1: 9\) (d) \(9: 1\)

Step-by-Step Solution

Answer

The ratio of their molecular masses is \(1:9\).

Step 1: Understand the relationship between molecular speed and mass

The relationship between the molecular speeds and masses of gases can be described by the equation derived from the kinetic theory of gases: \(v_{x}/v_{y} = \sqrt{M_{y}/M_{x}}\), where \(v_{x}\) and \(v_{y}\) are the speeds of gases X and Y, respectively, and \(M_{x}\) and \(M_{y}\) are their molecular masses.

Step 2: Substitute the given speed ratio

The given speed ratio of gas X to gas Y is 3:1, which means \(v_{x}/v_{y} = 3/1\). Substituting this into the equation from Step 1, we get \(3/1 = \sqrt{M_{y}/M_{x}}\).

Step 3: Square both sides of the equation

To get rid of the square root, square both sides of the equation: \((3/1)^2 = (\sqrt{M_{y}/M_{x}})^2\) which simplifies to \(9/1 = M_{y}/M_{x}\).

Step 4: Express the ratio of molecular masses

The ratio of molecular masses can be written as \(M_{x} : M_{y}\), which from the previous step is found to be \(1:9\). Therefore, the correct answer is (c) \(1:9\).

Key Concepts

Kinetic Theory of Gases

The kinetic theory of gases is a fundamental principle that describes the behavior of gases in terms of the movement of their particles. According to this theory, gases consist of tiny particles in constant, random motion. The energy of these particles is directly related to the temperature of the gas; higher temperatures indicate faster moving particles.

A crucial aspect of the kinetic theory is the understanding that the pressure exerted by a gas in a container results from collisions of the gas particles with the container's walls. This pressure is also affected by the speed of the molecules and their mass. A formula that ties these variables together is derived from the kinetic theory: \( v_x / v_y = \sqrt{M_y / M_x} \), where \(v_x \) and \(v_y \) are the speeds of two different gases, and \(M_x \) and \(M_y \) are their respective molecular masses.

Molecular Mass

Molecular mass, also known as molecular weight, is the sum of the atomic masses of all the atoms in a molecule. It's an essential factor when comparing different gases because it directly influences a molecule's speed at a given temperature. According to the kinetic theory of gases, lighter molecules move faster than heavier ones at the same temperature.

In the case of comparing gases, it's often useful to consider the relative molecular masses. For instance, in our original exercise, the ratio between two gases \(M_x : M_y\) is sought. This ratio can provide insight into the relative speeds of the gases, as seen in the step-by-step solution. Understanding how molecular mass impacts the behavior and properties of gases is crucial in fields like chemistry and physics.

Speed of Gases

The speed of gas molecules is an important concept within the kinetic theory of gases. It explains how molecular speed can be related to the kinetic energy and temperature of the gas. At a given temperature, lighter gas molecules (with lower molecular mass) tend to move faster than heavier molecules.

When we analyze two different gases at the same temperature and apply the kinetic theory, we can compare the speed of one gas to another using the relationship \(v_x / v_y = \sqrt{M_y / M_x}\), as in the textbook solution. The ratio of the speeds given in the problem (\(3:1\)) can be used to find the ratio of their molecular masses. This concept explains why the molecular speed ratio provided in the example leads us to the molecular mass ratio between the two gases being \(1:9\), as seen in step 4 of the given solution.