Problem 43 - Physics

Question

A monatomic ideal gas \((\gamma=1.67)\) is contained within a box whose volume is \(2.5 \mathrm{~m}^{3}\). The pressure of the gas is \(3.5 \times 10^{5} \mathrm{~Pa}\). The total mass of the gas is \(2.3 \mathrm{~kg}\). Find the speed of sound in the gas.

Step-by-Step Solution

Answer

The speed of sound in the gas is approximately 797.1 m/s.

Step 1: Understand the Formula for Speed of Sound in a Gas

The speed of sound in a gas is given by the formula \( v = \sqrt{\frac{\gamma \cdot P}{\rho}} \), where \( v \) is the speed of sound, \( \gamma \) is the adiabatic index (1.67 for a monatomic ideal gas), \( P \) is the pressure of the gas, and \( \rho \) is the density of the gas.

Step 2: Calculate the Density of the Gas

Density \( \rho \) is given by \( \rho = \frac{m}{V} \), where \( m \) is the mass and \( V \) is the volume. Here, \( m = 2.3 \mathrm{\, kg} \) and \( V = 2.5 \mathrm{\, m}^3 \), so:\[ \rho = \frac{2.3}{2.5} = 0.92 \mathrm{\, kg/m}^3 \]

Step 3: Substitute Values into the Speed of Sound Formula

Using the speed of sound formula \( v = \sqrt{\frac{\gamma \cdot P}{\rho}} \), substitute \( \gamma = 1.67 \), \( P = 3.5 \times 10^5 \mathrm{\, Pa} \), and \( \rho = 0.92 \mathrm{\, kg/m}^3 \):\[ v = \sqrt{\frac{1.67 \cdot 3.5 \times 10^5}{0.92}} \]

Step 4: Calculate the Speed of Sound

Perform the calculation:\[ v = \sqrt{\frac{584,500}{0.92}} \approx \sqrt{635326.09} \approx 797.1 \mathrm{\, m/s} \]

Key Concepts

Ideal Gas Law

The ideal gas law is a cornerstone concept in understanding gases. It describes how pressure, volume, temperature, and the number of particles interact in a gas. The law is usually stated as \( PV = nRT \), where:

• \( P \) is the pressure of the gas.
• \( V \) is the volume of the gas.
• \( n \) is the number of moles of gas.
• \( R \) is the ideal gas constant.
• \( T \) is the absolute temperature of the gas in Kelvin.

This equation assumes that the gas behaves ideally, meaning the particles do not interact except during elastic collisions, and their volume is negligible compared to the volume of the container.
In the context of the speed of sound, the ideal gas law helps determine important properties like pressure and density that contribute to the calculation.

Monatomic Gas

Monatomic gases consist of single atoms per molecule. Examples include the noble gases such as helium, neon, and argon. These gases have specific characteristics:

• They generally exhibit low chemical reactivity.
• They have a simpler motion modeling since there's no internal structure to account for.
• For a monatomic ideal gas, the adiabatic index \( \gamma \) is approximately 1.67, which measures the heat capacity ratio.

Understanding these features is crucial when applying the speed of sound formula for a monatomic gas, since the value of \( \gamma \) affects the speed directly.

Adiabatic Process

An adiabatic process is a key concept where no heat is exchanged with the surroundings. Such processes occur typically when energy transfer happens so quickly that there is no time for heat exchange, or when the system is perfectly insulated.
For a gas, the adiabatic process is described by the relationship \( PV^\gamma = \text{constant} \), indicating that pressure and volume are linked in a way impacted by \( \gamma \), the adiabatic index.
This process is important in calculating the speed of sound, as it incorporates the adiabatic index into the equation, directly influencing the speed through the pressure and density terms.

Pressure and Density Relationship

In gases, pressure and density are intimately connected. Pressure is the force exerted by gas particles colliding with the walls of its container, while density describes how much mass is present in a given volume.

• Density \( \rho \) can be calculated using \( \rho = \frac{m}{V} \), where \( m \) is mass and \( V \) is volume.
• From the ideal gas law, pressure and density relate since \( P = nRT/V \) and \( n = \frac{m}{M} \), where \( M \) is molar mass.
• In the speed of sound calculation, the formula \( v = \sqrt{\frac{\gamma \cdot P}{\rho}} \) shows this direct relationship, where increased pressure or decreased density results in higher speed of sound.

By understanding this relationship, you gain insight into how conditions within a gas influence how quickly sound can travel through it.