Which of the following exprésstons' is 'true for an ideal gas ? (a) \(\left(\frac{\partial V}{\partial T}\right)_{P}=0\) (b) \(\left(\frac{\partial P}{\partial T}\right)_{V: n}=0^{6}\) (c) \(\left(\frac{\partial U}{\partial V}\right)_{T}=0\) (d) \(\left(\frac{\partial U}{\partial T}\right)_{V}=0\)

Understanding the behavior of an ideal gas is fundamental in thermodynamics, particularly when working with gas-related problems. An ideal gas is a hypothetical gas whose molecules occupy negligible space and have no interactions, and it exactly follows the ideal gas law, given by the equation PV = nRT, where P stands for pressure, V for volume, T for temperature, n for the number of moles, and R for the ideal gas constant.

This simple yet powerful equation encapsulates how changes in one state variable, such as temperature, will affect others, if certain conditions remain constant. For example, if we hold pressure constant (isobaric process), the volume will increase as the temperature increases. Conversely, if volume is kept constant (isochoric process), an increase in temperature will lead to an increase in pressure. These relationships are rooted in the kinetic theory of gases, which states that temperature reflects the average kinetic energy of gas molecules. More movement (higher temperature) generally means more pressure or more volume.

It's crucial to note that ideal gas behavior is an approximation. Real gases exhibit ideal behavior only under low pressure and high temperature conditions. At high pressures and low temperatures, the interactions between gas molecules become significant, and the volume occupied by the molecules themselves can no longer be neglected.

In thermodynamics, partial derivatives provide a powerful tool for understanding how a system's properties change with respect to one another while holding other variables constant. They are essential when we analyze the relationships between different state variables of a system, such as pressure, volume, temperature, and internal energy.

For instance, the partial derivative \( \left(\frac{\partial V}{\partial T}\right)_P \) represents how volume V changes with temperature T when pressure P is kept constant. Intuitively, if this derivative is positive, it means that as the temperature increases, the volume increases, which is consistent with Charles's Law. On the other hand, a partial derivative like \( \left(\frac{\partial P}{\partial T}\right)_{V, n} \) indicates how pressure P varies with temperature T when both volume V and the number of moles n are fixed, directly illustrating Gay-Lussac's Law.

The use of subscripts, such as V or P, signifies that these variables remain constant during the differentiation process. Mastery of partial derivatives is essential for students to fully comprehend not just the mathematical formulations, but also the physical implications of various thermodynamic processes.

The concept of internal energy in thermodynamics is pivotal in understanding how energy is stored within a gas. The internal energy, denoted as U, encompasses all the energy contained within a system, which for gases, predominantly comprises the kinetic energy of the moving gas particles. According to the kinetic theory, this energy increases as the temperature rises, implying that the internal energy of an ideal gas is directly related to its temperature.

In mathematical terms, for an ideal gas, the internal energy depends only on temperature and can be represented by \( U = \frac{3}{2}nRT \) for a monatomic ideal gas. This means that when we look at a partial derivative such as \( \left(\frac{\partial U}{\partial V}\right)_T \), we expect it to be zero because changing the volume while keeping temperature constant should not affect the internal energy of an ideal gas. This property simplifies many thermodynamic calculations and is a cornerstone in the understanding of ideal gas processes.

Furthermore, recognizing that an ideal gas does not have potential energy due to particle interactions, we can see why the internal energy of an ideal gas would not be affected by changes in volume at a constant temperature. This understanding is essential for solving problems related to the work done by or on the gas, as well as predicting how energy will be transferred in the form of heat or work during a process.