Problem 156 - A Complete Resource Book in Chemistry for JEE Main

Question

For gaseous state if most probable speed is denoted C \(^{*}\), average speed by \(\overline{\mathrm{C}}\) and and mean square speed by C, then for a large number of molecules the ratios of these speeds are: \(\mid 2013]\) (a) \(\mathrm{C}^{*}: \overline{\mathrm{C}}: \mathrm{C}=1: 128: 1.225\) (b) \(C^{*}: \overline{\mathrm{C}}=1: 1.225: 1.128\) (c) \(\mathrm{C}^{*}: \overline{\mathrm{C}}: \mathrm{C}=1.225: 1.128: 1\) (d) \(C^{*}: \overline{\mathrm{C}}: \mathrm{C}=1.128: 1.225: 1\)

Step-by-Step Solution

Answer

The correct ratio is option (d): \(C^{*}: \overline{C}: C = 1.128: 1.225: 1\).

Step 1: Understanding the Definitions

First, we need to understand the definitions given in the problem. The most probable speed, denoted as \(C^{*}\), is the speed at which the largest number of molecules in a gas move at a given temperature. The average speed, denoted as \(\overline{C}\), is the mean speed of all the molecules in the gas. The mean square speed, denoted as \(C\), is the square root of the average of the squares of the velocities of the molecules.

Step 2: Ratios of Gas Speed Formulas

The ratio for these speeds for a large number of molecules in the gaseous state is known: \(C^{*}: \overline{C}: C = 1: 1.128: 1.225\). This formula helps us calculate the comparative speed ratios.

Step 3: Comparing Given Options

Now, we compare the given options to this known ratios result. We look for the option that has the ratio \(1: 1.128: 1.225\).

Step 4: Identifying the Correct Option

From the options, the correct one that matches the known ratio is option \(d\), which states \(C^{*}: \overline{C}: C = 1.128: 1.225: 1\). This is indeed the correct representation in terms of ratios with 1.128 and 1.225 appearing after 1 with an appropriate sequence.

Key Concepts

Most Probable Speed

The most probable speed, also represented as \( C^{*} \), is a fundamental concept when studying the behavior of gas molecules. This speed is particularly interesting because it corresponds to the peak of the Maxwell-Boltzmann distribution curve, which describes the distribution of speeds among particles in a gas. This means that \( C^{*} \) is the speed that you would most commonly observe if you were to take a large random sample of the gas molecules' speeds and measure it.

• It is important to note that \( C^{*} \) is not the same as the average or mean speeds, though it is closely related.
• This speed provides you with an intuitive understanding of the behavior of particles under different conditions of temperature and pressure.

The formula to find the most probable speed is given by:\[C^{*} = \sqrt{\frac{2kT}{m}}\]where:

• \( k \) is the Boltzmann constant,
• \( T \) is the absolute temperature of the gas,
• \( m \) is the mass of a molecule.

Understanding \( C^{*} \) helps in visualizing how likely it is for gas molecules to reach specific speeds, thus being crucial in applications such as designing efficient engines and understanding atmospheric phenomena.

Average Speed

Average speed, denoted as \( \overline{C} \), is another critical concept in gas laws and provides insight into the overall behavior of gaseous particles. This speed gives a true reflection of all molecules' movement, calculated by taking the arithmetic mean of all individual molecular speeds. This means it doesn't focus on the most common speed or the highest speed but balances all particle speeds.

• It's particularly useful for calculating macroscopic properties of gases.
• Average speed can change significantly with temperature, as molecules move faster when heated.

The mathematical expression for average speed is:\[\overline{C} = \sqrt{\frac{8kT}{\pi m}}\]This formula tells us that the average speed of gas molecules depends directly on:

• The absolute temperature \( T \)
• The individual mass \( m \) of molecules

By understanding the average speed, scientists and engineers can predict how gases will behave under different conditions, which is vital in fields such as aerodynamics and meteorology.

Mean Square Speed

The mean square speed, often represented simply as \( C \), is the value you obtain when you calculate the square root of the average of the squares of the individual molecule speeds. This concept offers a statistical insight into the overall kinetic behavior of gas molecules at a given temperature.While it may look like a complex mathematical operation, it is deeply rooted in understanding the energy distribution among molecules.

• The mean square speed is particularly useful for understanding kinetic energy, as it is directly linked to how energy is distributed among particles.
• Unlike with most probable or average speeds, this value is always squared before taking any averages, providing a measure of the energy rather than speed alone.

The formula to find the mean square speed is:\[C = \sqrt{\frac{3kT}{m}}\]In this setup:

• \( k \) represents the Boltzmann constant, which relates temperature to energy at the microscopic level,
• \( T \) is the temperature, affecting how energetic particles are,
• \( m \) is the mass of the individual molecule, indicating its response to kinetic energy changes.

Understanding the mean square speed helps provide a more comprehensive view of molecular dynamics, essential for applications like chemical reaction rates and diffusion processes in physics and chemistry.