Temperature of one mole of an ideal gas is increased by one degree at constant pressure. Work done by the gas is (a) \(R\) (b) \(2 R\) (c) \(R / 2\) (d) \(3 R\)

The Ideal Gas Law is a cornerstone of understanding how gases behave under various conditions. Simply stated, it relates the pressure (P), volume (V), and temperature (T) of an ideal gas to the number of moles (n) and the ideal gas constant (R) through the equation PV = nRT. This equation assumes that the gas particles are in constant, random motion, and interact with each other and the container walls only through elastic collisions.

For a given amount of gas, if we change one of the variables, at least one other variable must also change to satisfy the equation. This relationship is crucial in calculating the work done by an ideal gas when it undergoes a change in state, such as a temperature increase, while keeping the pressure constant.

In the realm of thermodynamics, work is a transfer of energy that results from a process, such as the expansion or compression of a gas. For an ideal gas, the work done during expansion or contraction can be calculated using the formula W = P * \(V2 - V1\), where W is the work done, P is the constant pressure, V1 is the initial volume, and V2 is the final volume.

It's important to remember that work is a path function, meaning its value depends on the specific process—how a system goes from one state to another. The idea of work connects thermodynamics with mechanical processes, providing a bridge between the microscopic interactions of particles and macroscopic observable phenomena.

A constant pressure process, often called an isobaric process, is a transformation of a system in which the pressure remains unchanged. When dealing with gases, a constant pressure process means that the gas may change volume and temperature while the external pressure remains steady.

This situation simplifies calculations since the work done by the gas is directly proportional to the change in volume, as expressed in the formula W = P \((V2 - V1)\). In isobaric processes, the area under the pressure-volume (P-V) curve graph represents the work done by or on the gas. Constant pressure processes are very common in everyday applications, such as inflating a balloon or operating a piston.

For an ideal gas undergoing a process with constant pressure, there's a direct relationship between volume and temperature known as Charles's Law. This law states that if the pressure is held constant, the volume of a gas is directly proportional to its absolute temperature (measured in Kelvin).

Mathematically, it can be represented as V1/T1 = V2/T2, where V1 and V2 are the initial and final volumes, respectively, and T1 and T2 are the initial and final temperatures. This means if the temperature of the gas rises, so does its volume, and conversely, if the temperature decreases, the volume diminishes, provided the pressure remains unchanged.