Problem 132 - A Complete Resource Book in Chemistry for JEE Main

Question

Match the following Column-I (A) \(U_{r x}\) (B) \(U_{m}\) (C) \(U_{w v}^{m}\) (D) \(\sqrt{P}\) Column-II (p) \(\frac{\mathrm{dRT}}{\mathrm{M}}\) (q) \(\sqrt{\frac{2.5 \mathrm{RT}}{\mathrm{M}}}\) (r) \(\sqrt{\frac{3 \mathrm{P}}{\mathrm{d}}}\) (s) \(\sqrt{\frac{8 \mathrm{P}}{\pi \mathrm{d}}}\) (t) \(\sqrt{\frac{2 \mathrm{RT}}{\mathrm{M}}}\)

Step-by-Step Solution

Answer

A matches t, B matches s, C matches q, D matches r.

Step 1: Understand Column-I Terminology

In Column-I, we have different symbols that represent various physical quantities:- (A) \(U_{r x}\): This refers to a specific root mean square speed in thermodynamics.- (B) \(U_{m}\): This is the mean speed generally given by \(\sqrt{\frac{8 \mathrm{RT}}{\pi \mathrm{M}}}\).- (C) \(U_{w v}^{m}\): This is another kind of mean speed but uses a different formula.- (D) \(\sqrt{P}\): This is simply the square root of pressure.

Step 2: Match Column-I with Column-II for (A)

Column-I (A) is likely referring to a root mean square speed. This is matched with an expression that shows root terms involving RT and molar mass M. Thus, (A) should match with (t), which is \(\sqrt{\frac{2\mathrm{RT}}{\mathrm{M}}}\).

Step 3: Match Column-I with Column-II for (B)

The mean speed match for Column-I (B) is \(\sqrt{\frac{8 \mathrm{RT}}{\pi \mathrm{M}}}\). In Column-II, this matches with option (s).

Step 4: Match Column-I with Column-II for (C)

For Column-I (C), we look for a different kind of speed, possibly another variant of mean speed formula not matching typical RMS or pure mean speed formulae. Hence it most likely connects with \(\sqrt{\frac{2.5 \mathrm{RT}}{\mathrm{M}}}\) in Column-II, which is option (q).

Step 5: Match Column-I with Column-II for (D)

Column-I (D) is exactly the square root of pressure \(\sqrt{P}\) and the direct equivalent expression is found in (r), \(\sqrt{\frac{3 \mathrm{P}}{\mathrm{d}}}\), where d matches with units consistent with pressure.

Key Concepts

Root Mean Square Speed

The concept of Root Mean Square (RMS) Speed is pivotal in physical chemistry, especially when discussing the kinetic molecular theory of gases. RMS speed gives insights into the average speed of molecules in a gas but weighted by their speed squared. This way, molecules moving faster have a greater influence on the average. The formula for RMS speed is \[ U_{rms} = \sqrt{\frac{3RT}{M}} \]where:

• \( R \) is the universal gas constant
• \( T \) represents the temperature in Kelvin
• \( M \) is the molar mass of the gas in kilograms per mole

By practicing with specific problems, you'll notice that RMS speeds are always higher than the mean speeds for the same gas under identical physical conditions. This method of using square roots and averaging distinctly favors faster molecules, critical for understanding the dynamics of gas particles.Understanding RMS speed helps in practical applications like anticipating molecular collisions in gas dynamics and calculating diffusion rates.

Mean Speed in Gases

The mean speed of gas molecules is another descriptive measure of molecular motion, calculated as an arithmetic means of all molecule speeds. Unlike RMS speed, this isn't weighted by the square of the speed.The mean speed formula is typically expressed as:\[ U_{mean} = \sqrt{\frac{8RT}{\pi M}} \]Here, just like in RMS speed:

• \( R \) is the gas constant
• \( T \) is the absolute temperature
• \( M \) denotes the molar mass of the gas
• \( \pi \) is the mathematical constant, approximately equal to 3.14159

The mean speed formula accounts directly for the probability distribution of speeds due to Maxwell-Boltzmann statistics. It's notably useful in calculations that involve average translational kinetic energy and helps characterize gas diffusion. While not as high as RMS values, mean speeds accurately reflect the typical kinetic energy conditions a gas may experience.

Pressure Calculation

In thermodynamics and physical chemistry, pressure is a fundamental concept related to the force exerted by gas molecules when they collide with the walls of their container. The basic ideal gas law is given by:\[ PV = nRT \]where:

• \( P \) stands for pressure
• \( V \) is volume
• \( n \) is the amount of gas in moles
• \( R \) is the ideal gas constant
• \( T \) is the temperature in Kelvin

Pressure is often derived from molecular speeds and impacts. The direct formula for pressure involving density \( d \) and speed can be expressed as:\[ P = \frac{1}{3}dU_{rms}^2 \]where:

• \( d \) is the density of the gas
• \( U_{rms} \) is the root mean square speed

Knowing pressure calculations is crucial for understanding properties of gases, predicting how gases will behave under varying conditions, and is essential for solving problems involving gas dynamics.