Question

If a gas contains only three molecules that move with velocities of \(100,200,500 \mathrm{~ms}^{-1}\), what is the \(\mathrm{rms}\) velocity of the gas is \(\mathrm{ms}^{-1}\) ? (a) \(100 \sqrt{8 / 3}\) (b) \(100 \sqrt{30}\) (c) \(100 \sqrt{10}\) (d) \(800 / 3\)

Short answer

The rms velocity is (c) \(100 \sqrt{10}\).

Step-by-step solution

Understand the RMS Velocity Formula

The root mean square (rms) velocity of a set of particles is given by the formula: \[ v_{rms} = \sqrt{\frac{v_1^2 + v_2^2 + v_3^2 + \ldots}{N}} \] where \( v_1, v_2, v_3, \ldots \) are the velocities of the particles, and \( N \) is the total number of particles.

Substitute the Given Values

Given the velocities of the three gas molecules are 100, 200, and 500 m/s, and there are three molecules, substitute these values into the formula: \[ v_{rms} = \sqrt{\frac{100^2 + 200^2 + 500^2}{3}} \]

Calculate the Squares of the Velocities

Calculate each velocity squared: \( 100^2 = 10000 \), \( 200^2 = 40000 \), and \( 500^2 = 250000 \).

Sum the Squared Values

Add the squared velocities together: \[ 10000 + 40000 + 250000 = 300000 \]

Divide by the Number of Molecules

Divide the sum by the number of molecules (3): \[ \frac{300000}{3} = 100000 \]

Take the Square Root

Take the square root of the result from Step 5: \[ v_{rms} = \sqrt{100000} = 100\sqrt{10} \]

Match the Answer to the Options

The calculated \( v_{rms} = 100\sqrt{10} \) matches option (c).

Key concepts

Kinetic Theory of Gases

The kinetic theory of gases is a fundamental theory that describes the behavior of gases from a microscopic perspective. It considers gas molecules in perpetual, random motion, all while following Newton's laws of motion. This theory serves as a vital framework for understanding how properties of gases, such as pressure and temperature, arise from their molecular activity. According to this theory, several key assumptions are made about gas molecules:

• The gas consists of a large number of molecules moving in random directions.
• Molecules experience elastic collisions with each other and the walls of their container.
• The intermolecular forces are negligible, except during collisions.
• The actual volume occupied by the molecules themselves is much smaller compared to the total volume of the gas.

The kinetic theory helps explain macroscopic properties like pressure and temperature. When gas molecules collide with the walls of a container, they exert force, which we observe as pressure. The temperature relates directly to the average kinetic energy of these molecules. Hence, understanding the kinetic theory gives insight into the macroscale behavior of gases via their microscale characteristics.

Molecular Velocities

The concept of molecular velocities provides insight into the distribution and movement pace of individual gas molecules. Different molecules in a gas sample do not move at the same speed. Instead, they exhibit a range of velocities. This variability leads to several statistical measures being useful, such as the average velocity, most probable velocity, and the root mean square velocity.

Molecular velocity distribution is often visualized using a Maxwell-Boltzmann distribution curve, which depicts how the velocities of molecules spread out. Three important velocities to consider in this context are:

• **Average Velocity (\(v_{avg}\))**: The mean speed of all molecules.
• **Most Probable Velocity (\(v_{mp}\))**: The speed that a maximum number of molecules will most likely have.
• **Root Mean Square Velocity (\(v_{rms}\))**: The square root of the average of the squares of the velocities. It is a significant measure since it accounts for both the velocity and the energy content of the molecules.

Understanding these velocities provides more than just numbers; they show how kinetic energy, molecular mass, and temperature interrelate within a gas system. Each type of velocity gives essential insights into how molecules move and behave in gas form.

Root Mean Square Speed

Root mean square speed, often abbreviated as \(v_{rms}\), is a critical measurement in the study of gases. It quantifies the speed of gas particles and is useful in connecting molecular movement to the properties of gases. The \(v_{rms}\) is calculated by first squaring the velocities of the particles, taking the average of these squared values, and finally taking the square root of the result.

This measurement is especially useful because it takes into account the kinetic energy associated with molecular motion, providing a deeper understanding of a gas's dynamics. It offers a way to quantify the spread of molecular velocities in a given sample, matching with the observed thermodynamic properties of the gas. The formula for calculating the root mean square speed is:

\[ v_{rms} = \sqrt{\frac{v_1^2 + v_2^2 + v_3^2 + \ldots}{N}} \]

Where \(v_1, v_2, v_3, \ldots\) are the individual velocities, and \(N\) is the number of molecules. This approach of averaging squares rather than raw speeds provides a figure that reflects how energy distributes among molecules.

In practical terms, \(v_{rms}\) provides the most relevant measure for theoretical and experimental work, linking directly to the energy and pressure exerted by gas molecules within a container. Understanding \(v_{rms}\) aids in comprehending how temperature increases lead to faster molecular movement, reflecting directly on real-world gas behavior metrics.