2 moles of an ideal monoatomic gas undergoes reversible expansion from (4 \(\mathrm{L}\), \(400 \mathrm{~K}\) ) to \(8 \mathrm{~L}\) such that \(T V^{2}=\) constant. The change in enthalpy of the gas is (a) \(-1500 R\) (b) \(-3000 R\) (c) \(+1500 R\) (d) \(+3000 R\)

Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, radiation, and physical properties of matter. The behavior of these quantities is governed by the four laws of thermodynamics which dictate how energy is transferred between objects and how it affects their properties.

In the context of the polytropic process described in the exercise, thermodynamics provides the framework to understand how energy is transformed when an ideal gas expands or contracts. Such a process involves work being done by or on the gas, and heat being exchanged with the surroundings, impacting the internal energy and temperature of the gas.

The ideal gas law is a fundamental equation in thermodynamics that relates the pressure (P), volume (V), temperature (T), and amount of substance (n) of an ideal gas. Mathematically, it is represented as: \( PV = nRT \), where \(R\) is the universal gas constant. The law assumes that gases consist of many small particles moving in random directions with no interactions other than perfectly elastic collisions.

Using the ideal gas law enables the determination of one state variable if the others are known. In our exercise, the ideal gas law is implicitly used in the assumption that the gas behaves ideally, which simplifies the calculations for changes in temperature and volume during the polytropic process.

Enthalpy change, \(\Delta H\), represents the total heat content change of a system at constant pressure. It’s an important concept in thermodynamics because it accounts for heat absorbed or released in chemical reactions and phase changes.\

In the case of the exercise, the enthalpy change in a monoatomic ideal gas, undergoing a polytropic process, can be calculated with the formula \(\Delta H = nC_p\Delta T\). The exercise illustrates how the enthalpy change is directly proportional to the change in temperature, the number of moles of gas, and the molar heat capacity at constant pressure.

Heat capacity is the amount of heat needed to change the temperature of a substance by a certain temperature interval. It's a property that reflects how a substance absorbs and releases heat.

For an ideal gas, the heat capacity can be specified at constant volume (\(C_V\)) or at constant pressure (\(C_p\)). Molar heat capacity is the heat capacity per mole of a substance. In the given problem, we focus on \(C_p\), which for a monoatomic ideal gas, is given by \(\frac{5}{2}R\). The heat capacity plays a crucial role in defining how much the temperature of the gas changes during a thermodynamic process, and, consequently, its enthalpy change.