A \(1.00-\mathrm{L}\) volume of a gas undergoes first an isochoric process in which its pressure doubles, followed by an isothermal process until the original pressure is reached. Determine the final volume of the gas.

The ideal gas law is a fundamental equation in thermodynamics that relates the pressure, volume, temperature, and number of moles of a gas through the equation
\( PV = nRT \)
where P stands for pressure, V is the volume, n represents the number of moles, R is the universal gas constant, and T is the temperature in Kelvins. This equation describes how gases tend to behave under different sets of conditions, assuming the gas particles do not interact and take up no space, which is a good approximation for many gases under normal conditions.

In the context of the exercise, we use the ideal gas law to find relationships between the gas's pressure, volume, and temperature during both isochoric and isothermal processes. It allows us to determine changes in one variable while keeping others constant.

Thermodynamics is the branch of physics that deals with heat, work, and the forms of energy involved in chemical and physical processes. At the core of thermodynamics are laws governing these interactions. The first law of thermodynamics, also known as the law of energy conservation, states that energy cannot be created or destroyed, only transformed from one form to another.

In an isochoric process, the volume of the gas remains constant. If heat is added to the system, it results in an increase in pressure and temperature but no work is done because volume does not change. Conversely, in an isothermal process, the temperature stays constant, which means that any heat transfer into or out of the system must result in work done, causing a change in volume while maintaining pressure-temperature equilibrium. These principles are directly applied when we solve problems related to gas behavior under different thermodynamic processes.

The relationship between gas volume and pressure is described by Boyle's Law, one of the gas laws stating that, for a given amount of gas at a constant temperature, the volume of the gas is inversely proportional to its pressure. This can be mathematically expressed as
\( P \times V = \text{constant} \)
This law is particularly relevant during isothermal processes where temperature is constant. In the given exercise, after the gas undergoes an isochoric process (constant volume), it enters an isothermal process. During the isothermal phase, as the original pressure is restored, the volume must change to compensate and maintain the product of pressure and volume. Understanding this inverse relationship allows students to predict the behavior of a gas when either its volume or pressure is altered while temperature remains stable.