Problem 921 - JEE Physics

Question

What is the ratio of the adiabatic to isothermal elasticities of a triatomic gas ? (A) \((3 / 4)\) (B) \((4 / 3)\) (C) 1 (D) \((5 / 3)\)

Step-by-Step Solution

Answer

The ratio of the adiabatic to isothermal elasticities for a triatomic gas is \( \frac{7}{5} \). However, this option is not available in the given choices. There may be an error in the question or the options.

Step 1: Find the Degrees of Freedom for Triatomic Gas

A triatomic gas has 3 translational and 2 rotational degrees of freedom. Therefore, it has a total of 3 + 2 = 5 degrees of freedom.

Step 2: Calculate Heat Capacity at Constant Volume (Cv)

We calculate Cv using the formula: Cv = (f/2)R, where f is the degrees of freedom, and R is the gas constant. Here, f = 5. Cv = \(\frac{5}{2}\)R

Step 3: Calculate Heat Capacity at Constant Pressure (Cp)

We calculate Cp using the formula: Cp = Cv + R. Cp = \(\frac{5}{2}\)R + R = \(\frac{7}{2}\)R

Step 4: Calculate Adiabatic Elasticity (Cp / Cv)

The adiabatic elasticity is the ratio of Cp to Cv: Adiabatic Elasticity = \( \frac{Cp}{Cv} = \frac{\frac{7}{2}R}{\frac{5}{2}R} \)

Step 5: Simplify the Adiabatic Elasticity

Cancel out the common factor R and simplify: Adiabatic Elasticity = \( \frac{7}{5} \)

Step 6: Calculate Isothermal Elasticity

Isothermal elasticity is always equal to 1 for an ideal gas.

Step 7: Find the Ratio of Adiabatic to Isothermal Elasticity

The ratio of the adiabatic to isothermal elasticity is \( \frac{7}{5} \) divided by 1: Ratio = \( \frac{7}{5} \) Comparing the given options, the answer is not available in the list. There may be an error in the question or the options.

Key Concepts

Triatomic Gas

A triatomic gas is composed of molecules with three atoms. This structure gives it unique properties compared to diatomic or monatomic gases. These molecules have more complex motion and energy distribution, which affects their thermal and mechanical behavior. Triatomic gases show both translational and rotational movements, which contributes to their degrees of freedom. A simple example of a triatomic gas is carbon dioxide (CO extsubscript{2}). The presence of three atoms allows for a variety of energy modes, which need to be considered when analyzing their thermodynamic processes. Understanding triatomic gases helps in calculating other important variables like heat capacities at constant volume and pressure.

Degrees of Freedom

The degrees of freedom for a gas are the number of independent ways the molecules can move. For a triatomic gas, these are the ways the molecules store energy internally. Typically, triatomic gases have:

• 3 translational degrees of freedom: movement along the x, y, and z axes.
• 2 rotational degrees of freedom: rotation around two perpendicular axes.

Adding these gives a total of 5 degrees of freedom for a triatomic gas. The degrees of freedom play a crucial role in determining the gas's capacity to heat, which is essential for understanding how it will behave under different thermal conditions.

Heat Capacity

Heat capacity is the amount of heat energy required to change a gas's temperature. For an ideal gas, it's important to differentiate between heat capacity at constant volume (The heat capacity at constant volume (\( C_v \)) and at constant pressure (The heat capacity at constant pressure (\( C_p \)).Heat capacity depends directly on the degrees of freedom. For a triatomic gas with 5 degrees of freedom, the heat capacity at constant volume is calculated by the formula:\( C_v = \frac{f}{2}R \), where \( f \) is the degrees of freedom and \( R \) is the gas constant. For our example, \( f = 5 \), thus \( C_v = \frac{5}{2}R \).Adding \( R \) to \( C_v \), we get the heat capacity at constant pressure:\( C_p = C_v + R = \frac{7}{2}R \).Understanding these calculations is key in determining how the gas reacts in processes like adiabatic and isothermal changes.

Ideal Gas

An ideal gas is a theoretical model that simplifies the behavior of gases, allowing us to predict gas reactions under varying temperature and pressure.Key assumptions for the ideal gas model include:

• The gas consists of a large number of small particles in random motion.
• The volume of individual gas particles is negligible compared to the volume the gas occupies.
• No intermolecular forces are acting between the gas particles.
• Collisions between gas particles and with the walls are perfectly elastic.

By applying these assumptions, we can derive equations like the Ideal Gas Law:\[ PV = nRT \]where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the gas constant, and \( T \) is the temperature.This model is instrumental in describing how gases behave in various thermodynamic processes, including adiabatic and isothermal processes mentioned in the exercise.