91Ó°ÊÓ

The adiabatic index, often represented as \( \gamma \), is an important concept in thermodynamics. It is the ratio of specific heats of a gas at constant pressure \( C_p \) and constant volume \( C_v \). Mathematically, it is expressed as:\[ \gamma = \frac{C_p}{C_v}\]This ratio indicates how a gas will behave under adiabatic processes, where no heat is exchanged with the surroundings. When the adiabatic index has a higher value, it means the gas expands more easily when heated. This is crucial for calculations involving sound through gases because it affects the speed at which pressure waves move.
In this problem, we work with the same \( \gamma = 1.40 \) for both air and hydrogen, simplifying our calculations. Using a constant \( \gamma \), along with the known speeds of sound, allows us to relate the properties of different gases under the same conditions, especially when temperature and pressure do not vary.

• The knowledge of \( \gamma \) helps in deriving relationships between temperature, pressure, and the speed of sound among different gases.
• Because \( \gamma \) values differ for various gases, it affects how compressible gases are under pressure.

Specific gravity is a measure that indicates the ratio of the density of a substance to a reference substance. Commonly, the density of water is used as a reference for liquids and solids, while for gases, air is typically the reference. In our case, specific gravity of hydrogen relative to air is given as 0.0690. This means hydrogen is significantly lighter than air. Specific gravity is useful because it allows us to understand differences in behavior between different gases.

• For gases, the specific gravity can directly correlate to molar mass at standard conditions of temperature and pressure, due to the Ideal Gas Law.
• For example, a specific gravity of 0.0690 implies that hydrogen is much less dense, which is crucial for calculating sound speed, as less dense means faster propagation of sound.

By understanding the specific gravity, we were able to substitute it into the formula that relates the speed of sound in hydrogen with air. This provided a straightforward route to compute the desired speed in hydrogen without needing comprehensive other properties like actual gas Density or Molar masses.

The universal gas constant \( R \) is a fundamental constant in physical chemistry that appears in various important equations in thermodynamics, most notably the Ideal Gas Law:\[ PV = nRT\]Where \( P \) is pressure, \( V \) is volume, \( n \) is moles of gas, and \( T \) is temperature in Kelvin. The gas constant \( R \) links the amount of gas and its temperature to pressure and volume changes.In problems involving the speed of sound like in this exercise, though \( R \) does not explicitly appear in the final ratio of speeds, its presence is implicit in understanding how temperature and molar properties affect sound speed.

• The constant allows us to generalize behaviors of gases under different conditions, establishing that certain variables remain constant across different gases (e.g., temperature and pressure).
• While \( R \) remains a constant for all gases, its calculation alongside molar mass and temperature gives a foundation that influences how the adiabatic process calculates sound speed.

An inherent understanding of \( R \) is necessary to comprehend the full scope of gas dynamics, especially when working with gas properties not explicitly defined by the exercise but deeply influential in thermodynamic calculations.