For a particular gas with Cp = 840.4 J/kg-K and Cv = 651.5 J/kg-K, what is the speed of sound when it is in an adiabatic state at a temperature of 377 K?
Engineering · College · Mon Jan 18 2021
Answered on
To find the speed of sound in a gas that is in an adiabatic state, you can use the following formula:
\[ a = \sqrt{\gamma \cdot R \cdot T} \]
where: - \( a \) is the speed of sound in the gas, - \( \gamma \) (gamma) is the ratio of specific heats, \( \frac{C_p}{C_v} \), - \( R \) is the specific gas constant for the gas, and - \( T \) is the absolute temperature of the gas in kelvin (K).
For a particular gas, \( C_p \) and \( C_v \) are given, so first we need to calculate the ratio of specific heats, \( \gamma \):
\[ \gamma = \frac{C_p}{C_v} = \frac{840.4 \, \text{J/kg-K}}{651.5 \, \text{J/kg-K}} \]
Calculate \( \gamma \):
\[ \gamma \approx 1.2897 \]
Next, using the ideal gas law, we can relate the gas constant \( R \) to \( C_p \) and \( C_v \):
\[ R = C_p - C_v \]
Calculate \( R \):
\[ R = 840.4 \, \text{J/kg-K} - 651.5 \, \text{J/kg-K} \] \[ R \approx 188.9 \, \text{J/kg-K} \]
Now, plug in the values for \( \gamma \), \( R \), and \( T \) into the speed of sound formula:
\[ a = \sqrt{1.2897 \cdot 188.9 \, \text{J/kg-K} \cdot 377 \, \text{K}} \]
Solve for \( a \):
\[ a = \sqrt{1.2897 \cdot 188.9 \cdot 377} \] \[ a \approx \sqrt{91805.13} \] \[ a \approx 303.0 \, \text{m/s} \]
So, the speed of sound in the gas at an adiabatic state and a temperature of 377 K is approximately 303.0 meters per second.
Extra: The concept of the speed of sound in a gas relates to how fast sound waves travel through the medium. Sound is a mechanical wave that propagates due to particle interactions in the medium. In gases, sound waves travel as longitudinal waves where gas particles vibrate along the direction of wave travel.
The speed of sound is affected by the medium's density and the medium's compressibility. In general, sound travels faster through mediums with higher elasticity and lower density. For gases, the speed of sound increases with the temperature of the gas because the particles move faster at higher temperatures, which facilitates quicker energy transfer between particles.
The ratio of specific heats (\( \gamma \)) is important because it reflects adiabatic processes, where no heat is exchanged with the surroundings. This factor arises in the derivation of the speed of sound formula from the first principles of thermodynamics and fluid mechanics.
Specific heats (\( C_p \) and \( C_v \)) are properties that denote the amount of heat required to raise the temperature of a unit mass of a substance by one degree Celsius (or one kelvin) at constant pressure or constant volume, respectively. The difference between these specific heats is what defines the specific gas constant (\( R \)) for a given gas, and this constant inherently relates to the ideal gas law, which is a key equation in thermodynamics and fluid mechanics.