A container of volume \(1 \mathrm{~m}^{3}\) is divided into two equal parts by a partition. One part has an ideal diatomic gas at \(300 \mathrm{~K}\) and the other part has vacuum. The whole system is isolated from the surrounding. When the partition is removed, the gas expands to occupy the whole volume. Its temperature will be (a) \(300 \mathrm{~K}\) (b) \(227.5^{\circ} \mathrm{C}\) (c) \(455 \mathrm{~K}\) (d) \(455^{\circ} \mathrm{C}\)

Imagine a gas in a container that suddenly has more space to spread out because a partition is removed. This kind of spreading out of the gas without any exchange of heat or work being done is what we call free expansion. In physics, this process is also known as Joule expansion. It's a fascinating case where a gas expands into an evacuated space entirely on its own, without pushing against anything.

Since there is no external pressure working against the expansion, the gas doesn't do any work during this process. This means no energy is transferred as work. In a perfectly isolated system, also no heat comes in or goes out. When you have an ideal gas—a gas that perfectly follows a set of theoretical behaviors—this entire process doesn't change its temperature. This is counterintuitive for many students, who might expect that expanding a gas would have a cooling effect.

Real-life Applications

The principles of free expansion can be seen in phenomena such as when a punctured tire deflates. Although real gases don't exhibit perfectly ideal behavior, the concept still offers a good approximation for understanding the key principles involved.

What does it mean when we say that a system undergoes an isothermal process? 'Iso' means equal or same, and 'thermal' relates to temperature, so it's a process in which the temperature of the system stays exactly the same throughout. We typically see this in situations where there's incredibly slow expansion or compression of a gas, allowing heat to transfer in or out of the system so that temperature remains constant.

In the case of our exercise, the free expansion of the gas is an isothermal process, where the gas's temperature before and after expansion did not change despite the change in volume. This seemingly simple concept can perplex students. It is critical to differentiate it from adiabatic processes, which also have no heat exchange but occur rapidly enough that temperature can change. It's important to highlight that an isothermal process only holds the temperature constant, not the internal energy or the pressure of the gas.

The first law of thermodynamics, often dubbed the law of energy conservation, states that energy can neither be created nor destroyed—it can only change forms. In the context of our gas in the container, it means that the total energy in our isolated system remains constant throughout the free expansion process.

In the traditional equation form, the first law of thermodynamics is represented as \(\Delta U = Q - W\), where \(\Delta U\) is the change in internal energy of the system, \(Q\) is the heat added to the system, and \(W\) is the work done by the system. During free expansion, since no external work is done (\(W = 0\)), and no heat is exchanged (\(Q = 0\)), the change in internal energy (\(\Delta U\)) is also zero. This conservation principle is a cornerstone of understanding physical processes and allows students to predict the behavior of systems under various conditions.

When we talk about an ideal gas, we're referring to a hypothetical gas that perfectly adheres to a set of given laws—Boyle's Law, Charles's Law, Avogadro's Law, and more collectively known as the ideal gas law. These principles state that the gas molecules don't interact with each other and don't occupy any volume on their own, among other characteristics.

The behavior of an ideal gas can be summarized in the ideal gas law equation, given as \(PV = nRT\), where \(P\) stands for the pressure of the gas, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the ideal gas constant, and \(T\) is the temperature. Our exercise assumes ideal gas behavior to hold that the internal energy of the gas depends solely on its temperature. As such, even when the volume of our gas doubles, its temperature should stay the same if it behaves as an ideal gas. This simplification allows us to understand and predict how the gas will act during expansion.