Problem 94 - A Complete Resource Book in Chemistry for JEE Main

Question

One mole of monatomic ideal gas at \(\mathrm{T}(\mathrm{K})\) is expanded from \(1 \mathrm{~L}\) to \(2 \mathrm{~L}\) adiabatically under a constant external pressure of 1 atm the final temperature of the gas in Kelvin is (a) \(\mathrm{T}\) (b) \(\frac{\mathrm{T}}{2^{5 / 3-2}}\) (c) \(\mathrm{T}-\frac{2}{3 \times 0.0821}\) (d) \(\mathrm{T}+\frac{3}{2 \times 0.0821}\)

Step-by-Step Solution

Answer

The final temperature is \(T_2 = \frac{T}{2^{2/3}}\), which is option (b).

Step 1: Understand the Process

The problem describes an adiabatic expansion of a monatomic ideal gas under constant external pressure. For adiabatic processes, there is no heat exchange with the surroundings.

Step 2: Identify the Adiabatic Condition

For adiabatic processes, the equation \( P V^\gamma = \text{constant} \) holds for an ideal gas, where \( \gamma = \frac{C_p}{C_v} \). For a monatomic ideal gas, \( C_v = \frac{3}{2}R \) and \( C_p = \frac{5}{2}R \), so \( \gamma = \frac{5}{3} \).

Step 3: Apply the Adiabatic Equation

Using the adiabatic condition \( TV^{\gamma-1} = \text{constant} \), and given the volume changes from \(1 \text{ L}\) to \(2 \text{ L}\), we set up the relationship \( T_1(1)^{2/3} = T_2(2)^{2/3} \).

Step 4: Solve for Final Temperature \(T_2\)

Rearranging the equation from the previous step, we get \( T_2 = T_1 \times \frac{1^{2/3}}{2^{2/3}} \). Since \( 1^{2/3} = 1\), this simplifies to \( T_2 = T_1 \times 2^{-2/3} \).

Step 5: Simplify the Expression

The expression \( 2^{-2/3} \) can be approximated or evaluated to understand its relation to \( T \). Noting that \( 2^{5/3 - 2} = 2^{-1/3} \), thus \( T_2 = \frac{T}{2^{2/3}} \), matching option (b).

Key Concepts

Monatomic Ideal Gas

A monatomic ideal gas is a theoretical gas composed of single-atom molecules. Understanding this concept helps in grasping various thermodynamic processes.
Monatomic gases like helium, neon, and argon typically behave close to ideal conditions.

• In an ideal gas, the particles do not interact except when they collide elastically.
• There are no forces of attraction or repulsion, and the volume of the gas particles themselves is negligible compared to the space the gas occupies.

These assumptions simplify calculations and help establish equations that describe gas behavior under various conditions.
Due to these properties, the internal energy of a monatomic ideal gas is expressed as a function of its temperature alone. This characteristic is utilized in analyzing thermodynamic processes, such as adiabatic expansion.

Adiabatic Process

In an adiabatic process, a gas expands or contracts without exchanging heat with its surroundings. This means that the process is insulated from external influence.

• The term 'adiabatic' implies no heat transfer, making these processes ideal for studying energy conservation within a closed system.
• For an adiabatic expansion, the gas does the work on its surroundings using its internal energy, resulting in a temperature change.

For an ideal gas undergoing an adiabatic process, the relationship between pressure, volume, and temperature is given by the equation \[ P V^\gamma = \text{constant} \] where \(\gamma\) is the heat capacity ratio \(\frac{C_p}{C_v}\).
In the case of monatomic gases, \(\gamma\) is typically \(\frac{5}{3}\), which is crucial for calculating outcomes like temperature change during expansion or compression.

Ideal Gas Constant

The ideal gas constant \(R\) is a critical factor in the equations governing the behavior of ideal gases. It appears in the ideal gas law \[ PV = nRT \] and also in thermodynamic calculations involving energy and capacity.

• Its value is approximately 8.314 J/(mol·K) in SI units and 0.0821 atm·L/(mol·K) in the units often used for Chemistry problems.

This constant helps link mechanical and thermal processes in thermodynamics, enabling the prediction of how gases respond to changes in pressure, volume, and temperature.
In the context of an adiabatic process, \(R\) is essential for determining specific heat capacities like \(C_p\) and \(C_v\), which are used to find the heat capacity ratio \(\gamma\). Understanding \(R\) simplifies the analysis of an adiabatic process.

Thermodynamics

Thermodynamics is the study of heat, energy, and work in systems. It involves several laws governing how energy is transferred or transformed.

• The first law of thermodynamics, also known as the law of energy conservation, implies that energy cannot be created or destroyed, only transferred or changed in form.
• It's crucial in understanding how internal energy changes in processes like adiabatic expansion, where no heat is transferred but work is done.

In our context of an adiabatic process, this law helps explain how internal energy changes result in temperature variations without heat exchange.
Applying thermodynamic principles allows us to understand phenomena like how expanding gases can cause a cooling effect, an insight leveraged in many practical applications from engines to refrigeration technologies.