Problem 929 - JEE Physics

Question

The ratio of two specific heats of gas \(\mathrm{C}_{\mathrm{P}} / \mathrm{C}_{\mathrm{V}}\) for Argon is \(1.6\) and for hydrogen is \(1.4 .\) Adiabatic elasticity of Argon at pressure P is E. Adiabatic elasticity of hydrogen will also be equal to \(\mathrm{E}\) at the pressure. (A) \(\mathrm{P}\) (B) \((7 / 8) \mathrm{P}\) (C) \((8 / 7) \mathrm{P}\) (D) \(1.4 \mathrm{P}\)

Step-by-Step Solution

Answer

The correct answer is (C) \((8/7)P\).

Step 1: Write down the formula for adiabatic elasticity

We know that \(E = P(\gamma)\), where \(E\) is the adiabatic elasticity, \(P\) is the pressure, and \(\gamma\) is the specific heat ratio (\(C_p/C_v\)).

Step 2: Write down adiabatic elasticity for Argon and Hydrogen

The adiabatic elasticity for Argon can be written as \(E_A = P_A(\gamma_A)\) and for Hydrogen, it can be written as \(E_H = P_H(\gamma_H)\).

Step 3: Substitute the given specific heat ratios

Given, \(\gamma_A = 1.6\) and \(\gamma_H = 1.4\). Substituting these values, we get: \(E_A = P_A(1.6)\) and \(E_H = P_H(1.4)\).

Step 4: Set adiabatic elasticities equal

Since adiabatic elasticity of Argon and Hydrogen is equal at a specific pressure, we can equate the two expressions: \(E_A = E_H\) \(P_A(1.6) = P_H(1.4)\)

Step 5: Solve for the pressure ratio

To find the pressure at which adiabatic elasticity is equal, we will solve for the pressure ratio of Hydrogen to Argon: \(\frac{P_H}{P_A} = \frac{1.6}{1.4}\) \(\frac{P_H}{P_A} = \frac{8}{7}\)

Step 6: Identify the correct answer choice

According to the problem, Hydrogen has the same adiabatic elasticity as Argon at a pressure of \(\frac{8}{7}P_A\). Thus, the correct answer choice is: (C) \((8/7)P\)

Key Concepts

Specific Heat Ratio

The concept of specific heat ratio, often symbolized by \( \gamma \), is paramount in thermodynamics. It is the ratio of the specific heat at constant pressure \( (C_p) \) to the specific heat at constant volume \( (C_v) \). This ratio is crucial because it indicates how a gas behaves under different thermodynamic processes.

• For monoatomic gases like Argon, \( \gamma \) is generally higher because these gases have fewer degrees of freedom for energy distribution.
• For diatomic gases like Hydrogen, \( \gamma \) tends to be lower due to more complex molecular structure and more energy absorption per temperature rise.

In the problem given, Argon has a \( \gamma \) value of \( 1.6 \), while hydrogen has a \( \gamma \) of \( 1.4 \). These values are used to determine how the gases expand and contract under varying pressures.

Argon

Argon is a noble gas, which means it is usually inert and does not easily participate in chemical reactions. It is a monoatomic gas, which means each molecule is a single atom. This implies that its internal energy is primarily tied to translational motion rather than rotational or vibrational modes.

• Because of its simple atomic structure, Argon is often used in lamps and in other environments where chemical reactivity is undesirable.
• In terms of thermodynamic behavior, Argon's specific heat ratio \( \gamma \) of \( 1.6 \) reflects its lower degrees of freedom compared to more complex gases like Hydrogen.

Argon's adiabatic elasticity depends on both the pressure and the specific heat ratio, making it predictable under adiabatic conditions.

Hydrogen

Hydrogen is a diatomic molecule, consisting of two atoms bonded together. This adds complexity to its thermodynamic behavior, as the molecule can store energy in more ways than a monoatomic gas like Argon.

• Hydrogen's specific heat ratio \( \gamma \) is \( 1.4 \), which is lower than Argon’s, due to its additional degrees of freedom (rotational and vibrational).
• This leads to more energy absorption when subject to the same temperature increase.

In the context of the given problem, Hydrogen's adiabatic behavior is assessed at different pressures to determine where its adiabatic elasticity equals that of Argon.

Pressure Ratio

Understanding the concept of pressure ratio is vital when comparing the adiabatic elasticity of different gases. In this context, the pressure ratio describes the relationship between pressures of two gases when their adiabatic elasticities are equal.

• The formula for adiabatic elasticity is \( E = P(\gamma) \), where \( E \) is elasticity, \( P \) is pressure, and \( \gamma \) is the specific heat ratio.
• The problem establishes a context where Argon and Hydrogen have equal adiabatic elasticities: \( P_A(1.6) = P_H(1.4) \).

To solve for the pressure ratio \( \frac{P_H}{P_A} \), we set the elasticities equal and find that \( \frac{P_H}{P_A} = \frac{1.6}{1.4} = \frac{8}{7} \). This allows us to conclude that Hydrogen has the same adiabatic elasticity as Argon when its pressure is \( \frac{8}{7} \) times that of Argon.