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Sound travels through different mediums by creating waves, and how fast these waves move depends on the medium they travel in. When considering gases, the speed of sound is not constant but varies with factors like temperature and the type of gas. The speed of sound in air, for example, is different from the speed of sound in helium because of the properties of the gases. For gases, the speed of sound can be determined by the formula:

• \( v = \sqrt{\frac{y R T}{M}} \)

where:

• \( v \) is the speed of sound,
• \( y \) is the adiabatic index, which we'll discuss shortly,
• \( R \) is the universal gas constant,
• \( T \) is the temperature in Kelvin,
• \( M \) is the molar mass of the gas.

Understanding and using this formula allows us to calculate important properties of gases from the speed of sound. It's crucial to note this formula because sound speed varies significantly depending on these parameters.

The adiabatic index, represented by \( y \), is a measure of how a gas expands and compresses without losing heat. It is also known as the heat capacity ratio.
Its value varies depending on the specific gas and is defined as the ratio of specific heats:

• \( y = \frac{C_p}{C_v} \)

where:

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

Gases with higher adiabatic indexes tend to have more rigid molecular structures, which affects their thermodynamic properties like sound speed. In the problem we solved, the gas has an adiabatic index of 1.40, a common value for diatomic gases like nitrogen and oxygen, which are the primary components of air. This helps predict how the gas responds to pressure changes.

Temperature plays a crucial role in gas behavior and calculations involving gases, such as determining the speed of sound. However, temperatures in scientific formulas are often expressed in Kelvin, the SI unit, rather than degrees Celsius. The conversion from Celsius to Kelvin is straightforward:

• \( T (\text{Kelvin}) = T (\degree C) + 273.15 \)

This conversion is essential because the Kelvin scale begins at absolute zero, the point where all molecular motion ceases, unlike the Celsius scale, which begins from the freezing point of water. In the problem given, at \( 0^{\circ}C \), the temperature in Kelvin is \( 273.15 \).
By using Kelvin, we achieve more consistent and accurate calculations, especially in thermodynamics and physics, where absolute temperatures are required.