Problem 91 - A Complete Resource Book in Chemistry for JEE Main

Question

One mole of an ideal gas is allowed to expand reversibly and adiabatically from a temperature of \(27^{\circ} \mathrm{C}\). If work done during the process is \(3 \mathrm{~kJ}\), then final temperature of the gas is \(\left(\mathrm{C}_{\mathrm{v}}=20 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}\right)\) (a) \(150 \mathrm{~K}\) (b) \(200 \mathrm{~K}\) (c) \(175 \mathrm{~K}\) (d) \(225 \mathrm{~K}\)

Step-by-Step Solution

Answer

The final temperature of the gas is 150 K.

Step 1: Convert Initial Temperature to Kelvin

Initial temperature in Celsius is given as \(27^{\circ}C\). Convert it to Kelvin using the formula \(T(K) = T(^{\circ}C) + 273.15\). Thus, \(T_1 = 27 + 273.15 = 300.15\, K\).

Step 2: Use Adiabatic Process Equation

For an adiabatic process, the equation \(W = C_v \cdot (T_1 - T_2)\) can be used, where \(W\) is the work done, \(C_v\) is the molar heat capacity at constant volume, and \(T_1, T_2\) are initial and final temperatures respectively.

Step 3: Calculate the Change in Temperature

Given that \(W = 3\, kJ = 3000\, J\) and \(C_v = 20\, J/K/mol\), substitute values into the equation: \(3000 = 20 \cdot (300.15 - T_2)\).

Step 4: Solve for Final Temperature

Rearrange the equation to find \(T_2\): \(3000 = 6003 - 20 \cdot T_2\). \(20 \cdot T_2 = 6003 - 3000\), hence \(20 \cdot T_2 = 3003\). Divide both sides by 20 to find \(T_2\): \(T_2 = \frac{3003}{20} = 150.15\, K\).

Step 5: Choose the Correct Answer

Round \(T_2\) to the nearest integer. \(T_2 \approx 150\, K\). Therefore, the correct choice is (a) \(150\, K\).

Key Concepts

Adiabatic Process

In thermodynamics, an adiabatic process is one in which no heat is transferred into or out of the system. This means that the entire energy balance is due to work done by or on the system. The system's thermal energy changes without any exchange with its surroundings, making it a key concept in understanding energy conservation in isolated systems.
Adiabatic processes are characterized by the following features:

• There is no heat transfer, so the heat exchange, denoted by \(Q\), is zero.
• The energy change of the system is purely due to work done, which follows the first law of thermodynamics: \(\Delta U = W\), where \(\Delta U\) is the change in internal energy and \(W\) is the work done.
• In an adiabatic process with an ideal gas, the relation between pressure and volume (and consequently temperature changes) is more complex than in isothermal or isobaric processes.

Understanding how temperature and pressure change without heat transfer is crucial for predicting the behavior of gases under different processes. This concept applies to real-life scenarios such as the compression of air in a piston engine or natural phenomena like the adiabatic cooling that occurs in rising air, crucial for weather patterns.

Reversible Expansion

Reversible expansion is a theoretical concept that describes how systems undergo changes to their states. It is an ideal process, meaning it assumes the system changes in an infinitely slow manner, allowing it to remain in thermodynamic equilibrium. This concept is essential in understanding maximum potential work extraction.

• In a reversible process, changes are carried out so gradually that the system maintains equilibrium at every step.
• The energy transformations are perfectly efficient, allowing the system to do the maximum possible work on the surroundings.
• Mathematically, the work done by a gas during a reversible expansion can be described by the integral \(W = \int_{V_1}^{V_2} P \, dV\), where \(P\) is pressure and \(V\) is volume.

Though a perfect reversible process is unattainable in real life due to inevitable energy dissipation, this model provides the standard against which real processes are measured. It informs us of the potential maximum efficiency and helps optimize industrial processes and engines' performance.

Molar Heat Capacity at Constant Volume

The molar heat capacity at constant volume, denoted as \(C_v\), is an essential concept in thermodynamics describing the heat required to raise the temperature of one mole of a substance by one degree Celsius (or Kelvin), without changing its volume. This form of heat capacity is relevant for processes fixed at constant volume, where no expansion work is done.

• \(C_v\) is useful for calculations involving gases, particularly ideal gases, where volume conditions are controlled.
• It is a direct measure of the energy needed to increase a mole of a gas's internal energy under volume constraints.
• The relationship between heat capacity, change in temperature, and energy change is given by \(\Delta Q = n C_v \Delta T\), where \(n\) is the number of moles.

Understanding molar heat capacity at constant volume is crucial for calculations in thermodynamics, especially under adiabatic conditions where processes occur without heat exchange, relying on changes in \(C_v\) to determine the outcome. This concept directly influences how scientists and engineers design systems to control and utilize temperature changes efficiently.