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 \((\mathrm{s})\) among the following is \((\) are \()\) [A] The work done on the gas is maximum when it is compressed irreversibly from \(\left(\mathrm{p}_{2}, \mathrm{~V}_{2}\right)\) to \(\left(\mathrm{p}_{1}, \mathrm{~V}_{1}\right)\) against constant pressure \(\mathrm{p}_{1}\) [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 \(\mathrm{V}_{1}\) to \(\mathrm{V}_{2}\) under isothermal conditions [D] The change in internal energy of the gas is (i) zero, if it is expanded reversibly with \(\mathrm{T}_{1}=\mathrm{T}_{2}\), and (ii) positive, if it is expanded reversibly under adiabatic conditions with \(\mathrm{T}_{1} \neq \mathrm{T}_{2}\)

Short answer

The correct statement is [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.

Step-by-step solution

Analyze Option [A]

According to the First Law of Thermodynamics, the work done on a gas during compression is negative from the gas's perspective. Maximum work is done when the process is reversible because, in an irreversible process, some work is dissipated as heat due to friction and other factors. Comparing reversible and irreversible processes, the work done under a reversible process is greater in magnitude (more negative from the gas’s perspective) because it is done more slowly, allowing for more time under high pressure. Therefore, option [A] incorrectly describes that the work done is maximum during an irreversible compression.

Evaluate Option [B]

A free expansion is a process in which a gas expands into an evacuated space without doing work on the surroundings. This process is indeed adiabatic because no heat is exchanged with the surroundings. However, an isothermal process requires the temperature to remain constant, which involves exchange of heat with the surroundings. Since no heat is exchanged in a free expansion, it cannot be both adiabatic and isothermal simultaneously. Therefore, option [B] is incorrect.

Assess Option [C]

In a reversible adiabatic expansion, no heat is exchanged with the surroundings (Q=0), and all the work done by the gas comes from its internal energy. In an isothermal expansion, the temperature remains constant, so the internal energy does not change, and the heat absorbed from the surroundings is equal to the work done by the gas. Since temperature remains constant in an isothermal process, a larger amount of heat is absorbed, and thus more work is done under isothermal conditions than adiabatic conditions. Hence, option [C] is correct.

Examine Option [D]

The internal energy change \(\Delta U\) for an ideal gas is dependent on its temperature change. In (i), if a process is reversible and isothermal \((T_1 = T_2)\), the internal energy change is zero, since \(\Delta U\) for an ideal gas is a function of temperature change only. In (ii), for a reversible adiabatic expansion, temperature decreases \((T_1 > T_2)\), so internal energy decreases, making \(\Delta U\) negative, not positive. Therefore, part (i) of option [D] is correct but part (ii) is incorrect.

Key concepts

First Law of Thermodynamics

The First Law of Thermodynamics, also known as the Law of Energy Conservation, states that energy can neither be created nor destroyed, only transformed from one form to another. In the context of thermodynamics, this law is often expressed mathematically as \( \Delta U = Q - W \) where \( \Delta U \) represents the change in internal energy of the system, \( Q \) represents the heat added to the system, and \( W \) is the work done by the system on its surroundings.

When applying this fundamental principle to gas expansions or compressions, the focus is on how energy is exchanged and conserved within the system. For an ideal gas, internal energy changes are directly related to temperature changes, as \( \Delta U \) is a function of temperature. Understanding how work and heat relate during different processes can reveal insights about the system's behavior under various conditions.

In the provided exercise, analyzing the different options requires a solid comprehension of the First Law's application. For instance, when a gas is compressed (opposite of expansion), the work done on the gas, from an external perspective, is positive as the system's potential to do work increases. While option [A] misrepresents that the work done is maximum during an irreversible compression, recalling the First Law helps clarify that, in reality, work is maximum when the compression is reversible, despite being expressed as a negative value from the gas's perspective.

Reversible and Irreversible Processes

Reversible Processes: A reversible process is an idealized concept where a system undergoes changes in such a way that the system and its surroundings can be returned to their original states without leaving any net changes in both. In a reversible process, the system is in thermodynamic equilibrium throughout the entire process. This implies that the process can proceed infinitesimally slowly, and the slightest reversal of external conditions can reverse the direction of the process.

Irreversible Processes: In contrast, an irreversible process is one that cannot simply be reversed to restore both the system and its environment to their original states. This could be due to spontaneous changes, friction, unrestrained expansion like in option [B], or a sudden temperature or pressure change. Most real-life thermodynamic processes are irreversible.

Understanding the distinction between these two types of processes is crucial in thermodynamics. For example, when comparing the work done by an ideal gas in different scenarios, as emphasized in option [C], it is critical to note that more work is done during a reversible isothermal expansion than during a reversible adiabatic expansion. This is because, in an isothermal process, the gas can absorb an infinite amount of heat as it expands, due to the infinitely slow nature of the expansion, while in the adiabatic process, the heat exchange is zero, thus limiting the work done by the internal energy of the gas.

Adiabatic and Isothermal Expansions

Adiabatic Expansions: An adiabatic process is one in which there is no heat exchange between the system and its surroundings. During an adiabatic expansion of an ideal gas, the gas performs work on the surroundings, which results in a decrease of the internal energy and, consequently, a decrease in temperature of the gas, provided there is no external heat input, as per the First Law.

Isothermal Expansions: Conversely, an isothermal expansion maintains a constant temperature throughout the process. For an ideal gas, maintaining temperature constant during expansion implies that heat must be added to the system to balance the work done by the gas, thereby ensuring that the internal energy remains unchanged.

Both types of processes are essential in understanding the behavior of gases under different thermodynamic conditions. Option [B] in the exercise mistakenly combines these two distinct processes, suggesting that a free expansion could be both adiabatic and isothermal, which is not possible. On the other hand, option [D] correctly indicates that during an isothermal, reversible expansion, the internal energy remains constant, illustrating the unique characteristics of such expansions.