Problem 130 - A Complete Resource Book in Chemistry for JEE Main

Question

In the given sample of a gas all molecules do not possess same speed. Due to frequent molecular collisions, the molecules move with ever changing speeds and also with changing direction. There are three types of velocities (i) root mean square velocity (ii) average velocity and (iii) most probable velocity. Oxygen has a density of \(1.429 \mathrm{gm} / \mathrm{L}\) at STP. The RMS velocity of \(\mathrm{O}_{2}\) molecules in \(\mathrm{cms}_{-1}\) (a) \(4.61 \times 10_{3}\) (b) \(4.16 \times 10_{3}\) (c) \(46.1 \times 10_{3}\) (d) \(6.41 \times 10_{3}^{3}\)

Step-by-Step Solution

Answer

The RMS velocity of \(O_2\) molecules is \(4.61 \times 10^3\) cm/s, matching option (a).

Step 1: Identify the Formula for RMS Velocity

The root mean square velocity ( \(v_{rms}\)) of gas molecules is calculated using the equation \(v_{rms} = \sqrt{\frac{3RT}{M}}\), where \(R\) is the universal gas constant, \(T\) is the temperature in Kelvin, and \(M\) is the molar mass of the gas in kg/mol.

Step 2: Gather Required Values

For calculations at Standard Temperature and Pressure (STP), the temperature is \(T = 273.15\) K. The universal gas constant \(R\) is \(8.314\) J/mol K. The molar mass \(M\) for \(O_2\) is \(32\) g/mol, which is \(0.032\) kg/mol.

Step 3: Plug Values into the RMS Formula

Using the formula: \[ v_{rms} = \sqrt{\frac{3 \times 8.314 \times 273.15}{0.032}} \]. Substitute the values: \(R = 8.314\), \(T = 273.15\), and \(M = 0.032\).

Step 4: Calculate RMS Velocity

First calculate the numerator: \(3 \times 8.314 \times 273.15 = 6802.1739\). Divide by the molar mass: \(\frac{6802.1739}{0.032} = 212568.59\). Take the square root: \(v_{rms} = \sqrt{212568.59} \approx 4610.5\) cm/s.

Step 5: Determine the Closest Answer Choice

The calculated root mean square velocity of \(O_2\) is approximately \(4610.5\) cm/s, which matches closest to the given option (a) \(4.61 \times 10^3\) cm/s.

Key Concepts

Molecular Speed

Molecular speed in a gas sample varies greatly among individual molecules. This is due to constant collisions, causing changes in speed and direction. In any gas, at any given moment, not all molecules are moving at the same speed. Instead, there exists a distribution of molecular speeds, where some molecules move fast, some slow, and many at moderate speeds.

This speed distribution can be described using three key types of velocities:

• Average Velocity: This is the simple mean of all the molecular speeds present within the gas. It gives an overall sense of how fast the molecules are moving, but doesn't capture the spread or distribution of speeds.
• Most Probable Velocity: The speed at which the highest number of molecules is moving. It represents the peak of the speed distribution curve, often known as the Maxwell-Boltzmann distribution.
• Root Mean Square Velocity (RMS): Considered a more useful measure for comparing speeds. It accounts for the energies of the molecules, as it's influenced by mass and speed squared.

Each of these velocities provides insight into molecular behavior in gases, playing crucial roles in predicting gas properties and behaviors under different conditions.

Root Mean Square Velocity

Root mean square velocity (RMS velocity) is a crucial concept in understanding the kinetic theory of gases. Defined by the formula:\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]where:

• \(v_{rms}\) represents the root mean square velocity,
• \(R\) is the universal gas constant with a value of approximately 8.314 J/mol K,
• \(T\) denotes the absolute temperature measured in Kelvin,
• \(M\) is the molar mass of the gas in kilograms per mole.

RMS velocity provides an average speed of gas molecules, weighted more heavily by the faster-moving ones, because speed is squared in this measure. Thus, it aligns with concepts in kinetic energy, as directly influenced by both mass and speed squared.

To determine the RMS velocity, the temperature and molar mass are crucial, as higher temperatures typically increase molecular speed, while greater molar mass typically decreases it. For instance, in the provided exercise, oxygen's molar mass and standard temperature condition are used. The calculated RMS velocity gave us clarity about how fast the molecules move on average at standard conditions.

Ideal Gas Law

The ideal gas law is a fundamental equation connecting the macroscopic properties of gases. It states:\[ PV = nRT \]where:

• \(P\) represents the pressure of the gas,
• \(V\) is the volume occupied by the gas,
• \(n\) signifies the number of moles of the gas,
• \(R\) is the universal gas constant,
• \(T\) is the absolute temperature measured in Kelvin.

This law assumes the gas behaves ideally, meaning attractions between molecules are negligible and collisions are perfectly elastic. While real gases can deviate under high pressure or low temperatures, the ideal gas law is incredibly useful for approximations.

In the context of molecular speeds and kinetic theory, the ideal gas law helps determine kinetic properties like pressure and temperature, influencing calculations of RMS velocity. Importantly, it allows for easy interrelationships between different properties of gas, explaining why temperature conditions and gas mass are key in RMS velocity calculations. Understanding these connections affords a deeper grasp of gas behavior and molecular dynamics.