Calculate the work done during the process when one mole of gas is allowed to expand freely into vacuum.

When we talk about free expansion in thermodynamics, we're referring to a fairly unique scenario. Imagine a gas that's confined to one side of a container with a moveable barrier. When the barrier is removed, the gas expands into the available space without encountering any resistance. This is because it expands into a vacuum. A vacuum offers no opposition, so despite the gas molecules moving and filling the new volume, they do not perform measurable work on the surroundings, as work in the thermodynamic sense requires a force to act through a distance against an opposing pressure.

Consider this in layman's terms: if you're pushing against nothing, you're not really using any effort, even if you're moving your arms. That's the concept behind free expansion; the gas is allowed to 'push against nothing', and so, from a physics standpoint, no work is being performed during this process.

Thermodynamics is the branch of physical science that deals with the relations between heat and other forms of energy (such as mechanical, electrical, or chemical energy). It plays a crucial role in understanding the behavior of gases and the concept of work within these systems.

According to the first law of thermodynamics, energy cannot be created or destroyed in an isolated system. However, it can change from one form to another. For example, when a gas expands against an external pressure, it converts some of its internal energy into work. But in the case of free expansion, no energy is converted into work since there is no external pressure to do work against. Moreover, learning about thermodynamics helps students, especially those preparing for competitive exams like the IIT-JEE, to grasp fundamental concepts that are widely applicable in physical chemistry and other scientific disciplines.

Pressure-volume work, or PV work, is an important concept in thermodynamics that refers to the work done by or on a system during a volume change under constant pressure. The formula for calculating such work is: \(W = P \times \Delta V\), where \(W\) is work, \(P\) is the external pressure, and \(\Delta V\) is the change in volume.

The idea is straightforward: if the gas expands, it does work on its surroundings, and when it is compressed, the surroundings do work on the gas. However, it's critical to note that if the external pressure is zero, as in free expansion, the result of the work calculation will also be zero since anything multiplied by zero is zero. This precise and accurate assessment of work is vital in understanding energy changes within physical and chemical processes.

The IIT-JEE, or the Indian Institute of Technology Joint Entrance Examination, is an academic challenge that many students undergo in pursuit of admission to engineering courses in India. Physical chemistry is a substantial part of this exam, covering the transformation of physical properties and energy within chemical systems.

To prepare for IIT-JEE physical chemistry, one must be well-versed with concepts like thermodynamics, chemical equilibria, kinetics, and others. Understanding processes such as free expansion, which don't align with everyday intuition, is paramount. It's situations like these where students are tested not just on their memorization of formulas, but also on their deep comprehension of the underlying principles, including why work done is zero in a free expansion despite a gas's volume change.