91Ó°ÊÓ

The Ideal Gas Law is a fundamental equation that describes the behavior of an ideal gas. It combines several physical laws into one simple equation: \[ PV = nRT \] Where:

• \( P \) is the pressure of the gas
• \( V \) is the volume it occupies
• \( n \) is the number of moles of the gas
• \( R \) is the ideal gas constant (8.314 J/mol·K)
• \( T \) is the temperature in Kelvin
In this particular problem, we're interested in elements of the ideal gas that contribute to the understanding of sound speed, namely pressure and volume. The Ideal Gas Law helps establish the relationship between these variables when considering constant quantities of gas in a closed system. As pressure or volume changes, the other components will adapt according to this relationship. This equation can be rearranged in various ways depending on the given values and what you are solving for, making it very versatile.

The adiabatic index, denoted as \( \gamma \), is a crucial factor in understanding how gases behave in various thermodynamic processes. It is defined as the ratio of specific heats at constant pressure \( C_p \) and volume \( C_v \):\[ \gamma = \frac{C_p}{C_v} \] In a monatomic gas, \( \gamma \) is often equal to approximately 1.67. This index influences how a gas reacts during adiabatic processes, which involve changes in pressure, volume, and temperature without any heat exchange with the surroundings. For calculating the speed of sound in an ideal gas, the adiabatic index is critical because it helps determine how compressible the gas is when sound waves travel through it. The speed of sound is faster in gases with higher \( \gamma \) because they are generally more resistant to compression. Thus, in our problem, using \( \gamma = 1.67 \) ensures that we accurately reflect the properties of the monatomic ideal gas in our calculations.

Pressure and volume are core concepts in understanding how gases function within a container. Pressure \( P \) is the force exerted by gas molecules against the walls of their container per unit area.Volume \( V \), on the other hand, is the amount of space that a gas occupies. In Boyle's Law, which is a part of the broader Ideal Gas Law, pressure and volume are inversely related given constant temperature and quantity of gas. This relation can be expressed as:\[ PV = k \]For the calculation of the speed of sound in gases, pressure is a vital component. An increase in pressure, with volume and other variables remain constant, typically increases the speed at which sound travels through the gas. This is because higher pressure implies more forceful molecular collisions, and thus greater energy transferability, facilitating faster propagation of sound waves. In this exercise, this relationship is utilized to find the speed of sound given a known pressure and calculated density.

Density \( \rho \) is the measure of mass per unit volume: \[ \rho = \frac{m}{V} \] Where \( m \) is mass and \( V \) is volume. Density is a crucial factor in determining the speed of sound in a gas. A gas with lower density will generally allow sound to travel faster, as the molecules are spaced further apart, reducing resistance to motion and allowing quicker wave propagation. In this problem, the mass of the gas is provided and, by knowing the volume of the container, we calculate the density using this simple equation. Once the density is known, it can be plugged into the formula for speed of sound in gases: \[ c = \sqrt{\frac{\gamma P}{\rho}} \] Here, it helps us to understand how the physical characteristics of the gas (mass, volume) contribute directly to how fast sound waves can travel through it. This understanding helps predict acoustics, sound speed, and behavior in various substances.