Problem 159 - Master Resource Book in JEE Main Physics

Question

Assertion When speed of sound in a gas is \(c\), then $$ c_{\mathrm{rms}}=\sqrt{\frac{3}{\gamma}} \times c $$ Reason \(c=\sqrt{\frac{\gamma p}{\rho}}\)

Step-by-Step Solution

Answer

Both the assertion and reason are true.

Step 1: Understand the Definitions

First, recognize what the terms in the equations represent. The speed of sound in a gas is given by \( c \) and is determined using the formula \( c = \sqrt{\frac{\gamma p}{\rho}} \), where \( \gamma \) is the heat capacity ratio, \( p \) is the pressure, and \( \rho \) is the density of the gas.

Step 2: Identify the Purpose

Understand that the assertion is about relating the root mean square speed (\( c_{\text{rms}} \)) of gas particles to the speed of sound in the gas. The assertion states that \( c_{\text{rms}} = \sqrt{\frac{3}{\gamma}} \times c \).

Step 3: Derive the Equation for Root Mean Square Speed

The root mean square speed of gas particles is derived from kinetic theory and is \( c_{\text{rms}} = \sqrt{\frac{3kT}{m}} \). Here, \( k \) is the Boltzmann constant, \( T \) is the temperature, and \( m \) is the molecular mass.

Step 4: Link Temperature to Sound Speed

Express temperature in terms of pressure and density using ideal gas laws, \( p = \rho RT/m \). Therefore, \( T = \frac{pm}{\rho R} \), and substitute this into the \( c_{\text{rms}} \) expression. This gives \( c_{\text{rms}} = \sqrt{\frac{3R}{m}} \sqrt{\frac{pm}{\rho R}} = \sqrt{\frac{3p}{\rho}} \).

Step 5: Relate Root Mean Square Speed to Sound Speed

Combine the equations for sound speed and the simplified form of \( c_{\text{rms}} \). Notice that the approximation \( c_{\text{rms}} = \sqrt{\frac{3}{\gamma}} \cdot c \) arises naturally when \( \gamma \) relates the two different expressions for speeds through their dependences on \( p \) and \( \rho \).

Step 6: Conclusion

The reason provided is correct as the expression for sound speed relates to the expressions for \( c_{\text{rms}} \) and the sound speed, providing the necessary relationship outlined by the assertion. Thus, both the assertion and the reason are true.

Key Concepts

Root Mean Square Speed

Root mean square speed (RMS speed) is an important concept in the kinetic theory of gases. It represents the average speed of gas particles and provides insight into their kinetic energy. To calculate the RMS speed, we use the formula:\[ c_{\text{rms}} = \sqrt{\frac{3kT}{m}} \]Here:

• \(k\) is the Boltzmann constant, linking temperature and energy.
• \(T\) denotes the absolute temperature of the gas.
• \(m\) is the molecular mass of the gas particle.

Using the RMS speed helps us understand the behavior of gas particles under varying temperatures and pressures. The higher the temperature, the greater the RMS speed, reflecting increased molecular motion.

Heat Capacity Ratio

The heat capacity ratio, often symbolized by \(\gamma\), is crucial in thermodynamics and the study of sound in gases. This ratio is defined as:\[ \gamma = \frac{C_p}{C_v} \]Where:

• \(C_p\) is the heat capacity at constant pressure.
• \(C_v\) is the heat capacity at constant volume.

This ratio indicates how energy is stored within gas molecules and plays a significant role in determining the speed of sound in gases. When gases expand or compress adiabatically (without heat exchange), \(\gamma\) directly influences the efficiency and outcome of these processes. In practical terms, a higher \(\gamma\) results in faster sound speed due to greater adiabatic efficiency.

Kinetic Theory of Gases

The kinetic theory of gases offers a microscopic view of gas behavior, based on the idea that gases consist of a large number of small particles in constant motion. This theory helps explain macroscopic phenomena such as pressure, volume, and temperature by considering the motion and collisions of gas particles. Fundamental assumptions of this theory include:

• Gas particles are in perpetual and random motion.
• Collisions between gas particles and with the walls of the container are perfectly elastic.
• Gas molecules are considered to have negligible volume compared to the volume of the container.
• There are no significant interactions between molecules except during collisions.

By understanding these principles, students can analyze how changes in conditions such as temperature or pressure affect gas behavior. This knowledge forms the basis for more advanced topics in thermodynamics and fluid dynamics, including deriving expressions like the speed of sound in gases.