Show that, for a perfect gas, \((\partial U / \partial S)_{v}=T\) and \((\partial U / \partial V)_{\mathrm{S}}=-p\).

Understanding the relationship between internal energy and entropy is fundamental for students tackling thermodynamics, especially within the context of perfect gases. Internal energy, denoted as U, is the total energy contained within a system, which in a perfect gas only depends on temperature. Entropy, symbolized by S, is a measure of a system's disorder - how energy is spread out among particles.

For a perfect gas, an increase in entropy typically means energy is more dispersed within the system. Since the internal energy of a perfect gas is dependent on temperature, which in turn is influenced by how energy is distributed (entropy), there is a direct link between the two. To clarify, when entropy increases at a constant volume, the system must absorb heat - hence, the temperature and internal energy increase correspondingly.

Thermodynamic partial derivatives are immensely valuable in understanding how changing one variable affects another while holding a different variable constant. In the context of perfect gases, they highlight how changes in entropy or volume can influence internal energy.

A partial derivative of the internal energy with respect to entropy or volume tells us how internal energy will change as entropy or volume change, independently of one another. The formula \( (\frac{\text{\textpartial} U}{\text{\textpartial} S})_{V} = T \) reveals that at a constant volume, any change in entropy directly alters the temperature. Conversely, \( (\frac{\text{\textpartial} U}{\text{\textpartial} V})_{S} = -p \) explains that at a constant entropy, any change in volume inversely changes the pressure within the system.

The fundamental thermodynamic relation for a perfect gas, expressed as \( dU = TdS - pdV \) serves as an essential equation in understanding thermodynamics. This equation ties together internal energy (U), temperature (T), entropy (S), pressure (p), and volume (V).

It illustrates how changes in entropy and volume can result in changes to the internal energy of a gas. If the volume remains constant, only the entropy part \( TdS \) contributes to the change in internal energy, whereas if the entropy remains constant, only the volume part \( -pdV \) affects the change. This relation helps in painting a clearer picture of how thermodynamic processes occur at the molecular level.

The relationship between entropy and volume is particularly intriguing in the realm of perfect gases. As volume increases while temperature remains constant, it can be deduced that the system's entropy must also increase. Why? Because the gas molecules have more space to occupy, which increases the number of ways the molecules can be arranged.

This contributes to the 'disorder' of the system, a core aspect of entropy. When we talk about partial derivatives in this context, it's about how tweaking volume—while maintaining constant entropy—can impact other thermodynamic properties. This relationship emphasizes the interconnectivity of thermodynamic variables and is critical for students to grasp for a deeper understanding of how gases behave under different conditions.