One mole of a non-ideal gas undergoes a change of state from \((1.0 \mathrm{~atm}, 3.0 \mathrm{~L}, 200 \mathrm{~K})\) to \((4.0\) atm, \(5.0 \mathrm{~L}, 250 \mathrm{~K}\) ) with a change in internal energy \((\Delta U)=40 \mathrm{~L}\) -atm. The change in enthalpy of the process in L-atm : (a) 43 (b) 57 (c) 42 (d) None of these

The first law of thermodynamics is a fundamental principle that provides the foundation for understanding energy interactions within a system. It is sometimes referred to as the law of energy conservation, which states that energy cannot be created or destroyed, only transformed from one form to another. This principle is vital when studying how energy changes during physical and chemical processes, such as the change in enthalpy of a gas.

In the context of our exercise, the first law helps us to determine the relationship between the change in internal energy (\( \text{Δ}U \)), the work done by or on the system, and the change in enthalpy (\( \text{Δ}H \)). When considering a gas under constant pressure, the work done is equivalent to the pressure (\( P \)) multiplied by the change in volume (\( \text{Δ}V \)). This work corresponds to the energy needed to expand or compress the gas, and thus is intricately tied to the concept of enthalpy, a measure that includes both internal energy and the work involved in the volume change of the system.

Internal energy, symbolized as \( U \text{,} \) is the total energy contained within a system, which includes kinetic and potential energies of the particles within the system. This energy can be altered either by transferring heat to or from the system (\( Q \text{,} \)) or by doing work (\( W \text{,} \)). The change in internal energy (\( \text{Δ}U \)) of a system is thus the net result of these heat and work exchanges.

In thermodynamics problems, such as the one in our exercise, calculating the change in internal energy is crucial for understanding the state changes in a system. This change can provide insights into the total energy flow, especially when we are trying to find the enthalpy change. As we have learned from the solution, the exercise asks us to calculate the internal energy change and use it along with the work performed due to volume change to determine the change in enthalpy.

While the ideal gas law is a great approximation for many gas behaviors under a variety of conditions, real gases often deviate from this ideal behavior. Non-ideal or real gas behavior can result due to interactions between gas particles and the finite volume that these particles occupy, which are not accounted for in the ideal gas law. These deviations become substantial at high pressures and low temperatures where molecules are closer together.

Factors such as these are essential in understanding the behavior of gases in practical scenarios and are accounted for with more complex equations of state, like the Van der Waals equation. When working with non-ideal gases, it is crucial to consider these behaviors, especially when calculating the work done by the gas during a change in volume. In exercises like the one provided, assuming ideal gas behavior may lead to incorrect answers. Therefore, when the exercise explicitly states that a gas is non-ideal, it is a signal to consider a more detailed analysis or use the appropriate correction factors to find accurate results.