Understanding the Ratio of Specific Heats for Air

The question asks about the ratio of the specific heat of air at constant pressure ($C_p$) to the specific heat of air at constant volume ($C_v$). This ratio is a fundamental property of gases in thermodynamics and is denoted by the Greek letter gamma ($\gamma$).

The formula for this ratio is:

\(\gamma = \frac{C_p}{C_v}\)

This ratio, $\gamma$, is also commonly known as the adiabatic index or the Poisson constant. Its value depends on the molecular structure of the gas:

• For monatomic gases (like Helium, Neon): \(\gamma \approx 1.67\)
• For diatomic gases (like Oxygen, Nitrogen, which make up most of air): \(\gamma \approx 1.40\)
• For polyatomic gases (like Carbon Dioxide, water vapor): \(\gamma\) is typically lower than 1.40 and varies more with temperature.

Air is primarily composed of diatomic gases (about 78% Nitrogen and 21% Oxygen). Therefore, air behaves very closely to a diatomic gas in terms of its thermodynamic properties.

The theoretical value of $\gamma$ for an ideal diatomic gas is 1.4 (\(C_p = \frac{7}{2}R\) and \(C_v = \frac{5}{2}R\), where R is the ideal gas constant, so \(\gamma = \frac{7/2}{5/2} = \frac{7}{5} = 1.4\)).

In real-world conditions, the value for air is slightly different from the ideal value due to various factors, including temperature effects and the presence of small amounts of other gases. Experimental values for the specific heat ratio of dry air at room temperature are very close to 1.4.

Let's look at the given options:

• 1.41
• 1.89
• 1.01
• 2.2

Comparing the theoretical and experimental values for air with the options, the value closest to 1.4 and commonly accepted for air under standard conditions is 1.41.

Statement Analysis

The question asks for the ratio \(C_p/C_v\) for air.

• This ratio is represented by \(\gamma\).
• Air is predominantly a diatomic gas.
• For diatomic gases, the theoretical ideal value is 1.4.
• Real values for air are very close to 1.4.
• Among the options, 1.41 is the closest and most appropriate value for the specific heat ratio of air.

Determining the Correct Ratio for Air

Based on the composition of air and thermodynamic principles, the specific heat ratio \(\gamma\) for air is approximately 1.4. Looking at the provided options, 1.41 is the value that aligns best with the properties of air.

Therefore, the ratio of specific heat of air at constant pressure to the specific heat of air at constant volume is approximately 1.41.

Revision Table: Key Thermodynamics Concepts

Additional Information: Ideal Gas Relationships

For an ideal gas, there is a relationship between \(C_p\), \(C_v\), and the ideal gas constant \(R\) (specifically, the specific gas constant, \(R_{specific} = R_{universal} / M\), where M is the molar mass):

Meyer's relation states: \(C_p - C_v = R_{specific}\)

Using this relation, we can also express \(C_p\) and \(C_v\) in terms of \(\gamma\) and \(R_{specific}\):

• From \(\gamma = C_p / C_v\), we have \(C_p = \gamma C_v\).
• Substituting into Meyer's relation: \(\gamma C_v - C_v = R_{specific}\)
• \(C_v (\gamma - 1) = R_{specific}\), so \(C_v = \frac{R_{specific}}{\gamma - 1}\)
• And \(C_p = \gamma C_v = \frac{\gamma R_{specific}}{\gamma - 1}\)

These relationships highlight how the specific heat ratio \(\gamma\) is interconnected with the specific heats themselves and the gas constant, providing further insight into the thermodynamic behavior of gases.