Two litre of \(\mathrm{N}_{2}\) at \(0^{\circ} \mathrm{C}\) and 5 atm pressure are expanded isothermally against a constant external pressure of 1 atm until the pressure of gas reaches 1 atm. Assuming gas to be ideal, calculate work of expansion.

The ideal gas law is a fundamental equation in thermodynamics and physical chemistry, providing a relationship between the pressure (P), volume (V), temperature (T), and number of moles (n) of an ideal gas. The law is expressed as the equation \( PV = nRT \), where R is the universal gas constant. In this context, 'ideal' refers to a hypothetical gas that perfectly follows this relationship with no intermolecular forces or volume occupied by the gas particles themselves.

For example, when a gas expands isothermally, implying that the temperature remains constant, the ideal gas law can be used to determine changes in volume as a result of changes in pressure. This principle is applied in our exercise to calculate the initial and final volumes of nitrogen gas during expansion.

In thermodynamics, the work done by a gas during expansion or contraction is related to the pressure and volume changes the gas undergoes. For isothermal processes, where the temperature remains constant, the work (W) is calculated by the formula \( W = -P_{\text{ext}} \times \triangle V \), where \( P_{\text{ext}} \) is the external pressure that the gas is doing work against, and \( \triangle V \) is the change in volume.

It's important to note the negative sign in the equation signifies that the work done by the gas on the surroundings is considered when it expands, thus doing positive work from the system's perspective but negative when viewed in terms of the surroundings. The concept of work becomes tangible when you consider scenarios such as a gas expanding to push a piston in an engine.

Pressure-volume changes are central to understanding thermodynamic processes. Boyle's Law, for instance, states that for a given mass of ideal gas at constant temperature (isothermal conditions), the pressure is inversely proportional to the volume. When a gas expands against an external pressure, its volume increases while its internal pressure decreases until it matches the external pressure.

In the given exercise, the nitrogen gas expands against a constant external pressure until it equilibrates with it. This kind of isothermal expansion demonstrates the inverse relationship between pressure and volume, which is captured by the ideal gas law. When solving thermodynamic problems, it is crucial to clearly establish the kind of process (isothermal, adiabatic, isotropic, etc.) and to apply the corresponding laws to find the necessary parameters.