Problem 38 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

An ideal gas is expanded from \(\left(P_{1}, V_{1}, T_{1}\right)\) to \(\left(P_{2}, V_{2}, T_{2}\right)\) under different conditions. The correct statement(s) among the following is (are) (a) The work done on the gas is maximum when it is compressed irreversibly from \(\left(P_{2}, V_{2}\right)\) to \(\left(P_{1}, V_{1}\right)\) against constant pressure \(P_{1}\) (b) If the expansion is carried out freely, it is simultaneously both isothermal as well as adiabatic (c) The work done by the gas is less when it is expanded reversibly from \(V_{1}\) to \(V_{2}\) under adiabatic conditions as compared to that when expanded reversibly from \(V_{1}\) to \(V_{2}\) under isothermal conditions (d) The change in internal energy of the gas is (i) zero, if it is expanded reversibly with \(T_{1}=T_{2}\), and (ii) positive, if it is expanded reversibly under adiabatic conditions with \(T_{1} \neq T_{2}\)

Step-by-Step Solution

Answer

(a), (b), and (c) are correct; (d-i) is correct, (d-ii) is incorrect.

Step 1: Understanding the Context

We are dealing with an ideal gas being expanded under various conditions, and we want to determine which statements about the process are correct. This involves understanding concepts of thermodynamics like work, internal energy, adiabatic and isothermal processes.

Step 2: Analyzing Statement (a)

Statement (a) talks about compressing the gas irreversibly against a constant pressure. In such irreversible processes, the work done on the gas is indeed maximum when the external pressure is a constant value greater than the final pressure, hence this statement is correct.

Step 3: Evaluating Statement (b)

Statement (b) suggests free expansion which is a process where the gas expands without any obstruction (no external pressure). This process is adiabatic since no heat is exchanged, and due to no work being done, it’s also considered isothermal in terms of no change in temperature, making the system's internal energy change zero. Hence, the statement is correct.

Step 4: Checking Statement (c)

Statement (c) involves comparing work done in adiabatic versus isothermal processes. In an adiabatic reversible expansion, the work done is less than in an isothermal reversible expansion because, in adiabatic processes, energy is used to change the temperature, not just to do work. Thus, this statement is indeed correct.

Step 5: Assessing Statement (d)

Part (i) of statement (d) says the internal energy change is zero if the expansion is isothermal with equal initial and final temperatures, which is correct based on the ideal gas law. Part (ii) states the internal energy change is positive when expanded adiabatically with differing initial and final temperatures, which is incorrect; it should generally decrease if the temperature decreases. Thus, part (i) is true, and part (ii) is false.

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental principle in thermodynamics that relates the pressure, volume, and temperature of an ideal gas. It is expressed in the formula:\[ PV = nRT \]where:

• P is the pressure of the gas
• V is the volume of the gas
• n is the number of moles
• R is the ideal gas constant
• T is the temperature in Kelvin

This law helps us understand how an ideal gas behaves under changes in pressure, volume, or temperature.
It assumes that the gas particles do not interact with each other and occupy no volume, which is a useful approximation for many gases under normal conditions.

Adiabatic Process

An adiabatic process is one where no heat is exchanged with the surroundings. The system is thermally isolated, leading to changes in internal energy only due to work done. For an adiabatic process, the ideal gas law can be modified to:\[PV^\gamma = ext{constant}\], where \(\gamma\) (gamma) is the heat capacity ratio (\(C_p/C_v\)).
In contexts like reversible adiabatic expansion:

• The system's temperature changes as a result of energy being used or work being done internally.
• The work done by or on the gas results in a change in temperature, as seen in the statement about energy use for adiabatic conditions.

Isothermal Process

An isothermal process occurs at a constant temperature. During such a process, any work done on or by the gas results in heat flow to or from the surroundings to maintain that constant temperature:\[W = -\Delta U = nRT \ln\left(\frac{V_f}{V_i}\right)\]

• In an isothermal expansion, the gas absorbs heat from the surroundings.
• In an isothermal compression, heat is expelled to the surroundings.

Since the temperature remains constant, the internal energy change is zero. This condition was reflected in the statement about work done and change in internal energy for isothermal processes, showing zero change if temperature does not vary.

Internal Energy

The internal energy of an ideal gas is directly proportional to its temperature. For an ideal gas, the internal energy is given by:\[U = \frac{3}{2}nRT\]

• In an isothermal process, the temperature, and hence the internal energy, remains constant. That means no change occurs in internal energy unless the temperature changes.
• In adiabatic processes, if temperature changes, the internal energy changes accordingly, often decreasing due to work done by the system without heat exchange.

The statement about internal energy change (zero for isothermal processes with no temperature change, and incorrectly predicted as positive for adiabatic processes when temperature shifts) highlights this concept.

Work Done

Work done during a thermodynamic process varies depending on whether the process is isothermal or adiabatic.
For an isothermal process, work done is calculated using:\[W = nRT \ln\left(\frac{V_f}{V_i}\right)\]This formula assumes constant temperature (isothermal), considering the reversible nature of the expansion.
In adiabatic processes, since there is no heat exchange, the work done results in a change in internal energy rather than a simple transfer of heat:\[W = \frac{P_iV_i - P_fV_f}{\gamma - 1}\]

• Isothermal expansion involves more work as energy is continuously transferred to maintain temperature.
• Adiabatic conditions, however, have less work done due to the energy being partitioned into changing the system's temperature, which is noticed in process comparisons mentioned in statement evaluations.