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 is\(150 \text{ K} \).

Step 1: Convert Initial Temperature to Kelvin

The initial temperature is given as \(27^{\circ} \text{C}\). To convert this to Kelvin, use the formula \(T(K) = T(^{\circ}C) + 273.15\). Thus, the initial temperature \(T_1 = 27 + 273.15 = 300.15 \text{ K}\).

Step 2: Understand Adiabatic Process

For an adiabatic process, \( \Delta Q = 0 \). Thus, the work done by the gas is equal to the negative change in internal energy: \( W = - \Delta U \). For an ideal gas, \( \Delta U = nC_v(T_2-T_1) \).

Step 3: Use the Work Done Formula

The work done, \( W = 3 \text{ kJ} = 3000 \text{ J} \), is given. Thus, \( 3000 = nC_v(T_1 - T_2) \). Since \( n = 1 \text{ mol} \), \( C_v = 20 \text{ J/K/mol} \), we find that \( 3000 = 20(T_1 - T_2) \).

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

Rearrange the equation: \( T_1 - T_2 = \frac{3000}{20} \). Calculating this gives \( T_1 - T_2 = 150 \). Therefore, \( T_2 = T_1 - 150 = 300.15 - 150 = 150.15 \text{ K} \).

Step 5: Round to Nearest Integer

According to the answer choices, we round the calculated \( T_2 = 150.15 \text{ K} \) to the nearest whole number, resulting in \( 150 \text{ K} \).

Key Concepts

Understanding Ideal Gas Laws

Ideal Gas Laws are fundamental principles in physics that describe the behavior of ideal gases. These gases are theoretical substances where particles do not interact except through elastic collisions. We express the relationship among pressure, volume, and temperature using the equation \( PV = nRT \), where:

• \( P \) is the pressure of the gas.
• \( V \) is the volume it occupies.
• \( n \) is the number of moles present.
• \( R \) is the ideal gas constant.
• \( T \) is the absolute temperature.

Adiabatic processes make use of specific conditions of ideal gases where no heat is transferred unto or from the system. This means the ideal gas laws help outline how temperature, pressure, and volume interact with no external heat gain or loss.

The ideal gas laws form the backbone of understanding an adiabatic change, ensuring that any variation in thermodynamic properties arises only due to internal adjustments within the system, not from external influence.

Essentials of Thermodynamics

Thermodynamics is the branch of physics that deals with heat, work, and the forms of energy involved in physical and chemical processes. In an adiabatic process, one of the core concepts is that there is no heat exchange with the surroundings. This implies that the energy change is entirely due to work done on or by the system.

Key Points about Thermodynamics:

• An adiabatic process can be visually understood in a Pressure-Volume graph as a steeper curve compared to isothermal processes.
• The First Law of Thermodynamics assures conservation of energy, positing that energy change within the system is the sum of heat and work involved.
• For adiabatic processes, the energy change is purely the result of work done within the system, since \( \Delta Q = 0 \).

Thermodynamics also teaches about the efficiency of processes and helps us understand phenomena like why expanding gases cool down, as seen in adiabatic expansion.

Diving into Internal Energy

Internal Energy is the total energy stored within a system due to the motion and intermolecular forces of particles. It's a crucial concept when considering thermodynamic processes as it indicates how much energy is available to do work.

In an adiabatic process, the internal energy change \( \Delta U \) takes center stage:

• The equation \( \Delta U = nC_v(T_2 - T_1) \) helps quantify this change.
• This equation shows that in absence of heat exchange, any energy change corresponds to a change in temperature and therefore affects your outcomes.
• For ideal gases, \( C_v \) is the molar specific heat capacity under constant volume. It is crucial in determining how much the temperature of a gas changes with internal energy shifts.

By understanding internal energy, one can deduce information about particle kinetic movements and potential energy, crucial in predicting outcomes for adiabatic processes.

Calculating Work Done in Thermodynamics

In thermodynamics, work done on or by a system during processes involves energy transfer. During adiabatic processes, the work done translates directly to changes in internal energy since no heat is exchanged.

To calculate the work done in such scenarios, we use the relationship \( W = - \Delta U \). The equation modifies to \( W = nC_v(T_1 - T_2) \) for ideal gases:

• Work is positive when the gas expands (as it does work on the surroundings).
• Work done is negative during compression (when surroundings do work on the gas).
• This formula empowers us to solve for unknown variables like final temperature \( T_2 \) given the work done and initial conditions.

Understanding work done in adiabatic processes allows us to delve deeper into how energy conservation influences outcome, helping comprehend thermodynamics with real-world examples.