The equation of state for one mole of a gas is \(P V=R T+B P\), where \(B\) is a constant, independent of temperature. The internal energy of fixed amount of gas is function of temperature only. If one mole of the above gas is isothermally expanded from \(12 \mathrm{~L}\) to \(22 \mathrm{~L}\) at a constant external pressure of 1 bar at \(400 \mathrm{~K}\), then the change in enthalpy of the gas is approximately \((\mathrm{B}=2 \mathrm{~L} / \mathrm{mol})\) (a) 0 (b) \(-3.32 \mathrm{~J}\) (c) \(-332 \mathrm{~J}\) (d) \(-166 \mathrm{~J}\)

Understanding the First Law of Thermodynamics is key to solving problems involving heat and work in thermodynamic processes. This fundamental law states that energy cannot be created or destroyed in an isolated system. More formally, it is expressed 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.
In the context of our textbook problem, applying the First Law during an isothermal (constant temperature) expansion, where the internal energy does not change (\(\Delta U = 0\)), leads to the conclusion that the heat added to the gas is equal to the work done by it (\(Q = W\)). This provides the foundation to calculate heat and work for processes, which are further used to deduce other thermodynamic quantities such as enthalpy.

Isothermal expansion is a thermodynamic process that occurs at a constant temperature. This aspect is crucial to our gas expansion problem, where the temperature is maintained at 400 K. During isothermal processes, the internal energy of an ideal gas remains unchanged because it only depends on temperature. Hence, the change in internal energy (\(\Delta U\)) is zero.
Isothermal processes are characterized by a distinct relationship between pressure and volume, which follows from the ideal gas law. The work done in an isothermal expansion can generally be calculated using the formula \(W = nRT \ln(\frac{V_2}{V_1})\), where \(n\), \(R\), \(T\), \(V_1\), and \(V_2\) are the number of moles, the universal gas constant, the absolute temperature, the initial volume, and the final volume, respectively. However, due to our exercise involving an additional \(BP\) term in the equation of state, we must account for this additional pressure-volume work while calculating the total work done during expansion.

Pressure-volume work, also known as \(PdV\) work, is the work done by or against a gas when it expands or contracts against a pressure. In thermodynamics, work can be thought of as the energy transfer that expands or compresses a gas.
In the problem, we are specifically looking at the work done by the gas during an isothermal expansion at constant external pressure. It is given by \(W = P_{\text{ext}} \cdot \Delta V\), where \(P_{\text{ext}}\) is the external pressure applied to the gas and \(\Delta V\) is the change in volume. Additionally, since our exercise includes the \(BP\) term in the gas's equation of state, the total work done on the system is not just given by \(P_{\text{ext}} \cdot \Delta V\), but also includes the work due to this extra term. Mathematically, this additional work can be represented as \(W_{BP} = \int_{V_1}^{V_2} BP dV = B \(P_{\text{ext}}\) \cdot \Delta V\), which signifies the additional energy considerations required for gases that deviate from ideal behavior.

An equation of state is a mathematical model that describes the state of matter under a given set of physical conditions. It provides a relationship between thermodynamic quantities such as pressure \(P\), volume \(V\), and temperature \(T\). The most renowned example is the Ideal Gas Law \(PV = nRT\), which describes the behavior of ideal gases.
In the given exercise, the gas is described by a modified equation of state \(PV = RT + BP\). This tells us that the gas does not behave ideally since the \(BP\) term modifies the ideal behavior captured by the simple \(PV\) relationship. The \(B\) factor is an indicator of the gas's deviation from ideality, independent of temperature. Variations in the equation of state from the Ideal Gas Law are vital for understanding real gas behavior, especially when predicting the work done during processes like expansion or compression, and they are crucial when calculating changes in enthalpy or other thermodynamic properties.