Suppose that \(S\) is regarded as a function of \(p\) and \(\mathrm{T}\). Show that \(T \mathrm{dS}=\) \(C_{\mathrm{p}} \mathrm{d} T-\alpha T V \mathrm{~d} p .\) Hence, show that the energy transferred as heat when the pressure on an incompressible liquid or solid is increased by \(\Delta p\) is equal to \(-\alpha T V \Delta p .\) Evaluate \(q\) when the pressure acting on \(100 \mathrm{~cm}^{3}\) of mercury at \(0^{\circ} \mathrm{C}\) is increased by \(1.0\) kbar. \(\left(\alpha=1.82 \times 10^{-4} \mathrm{~K}^{-1} .\right)\)

Entropy is fundamentally understood as a measure of disorder or randomness in a system. Entropy change, often denoted as \( \Delta S \), refers to the difference in the level of disorder between two states of a system.

In thermodynamics, the change in entropy is closely related to the heat transfer within a system. At a particle level, when a system receives or releases heat, the movement and energy distribution among the particles are altered, which in turn affects its entropy.

The relationship between entropy change and heat transfer is especially important when examining reversible processes, where \( \Delta S = \int \frac{dq_{\text{rev}}}{T} \). For processes under constant pressure, the change in entropy can be represented using the heat capacity at constant pressure \( C_p \), as in the provided thermodynamic identity \( TdS = C_p dT - \alpha T V dp \). This means that when temperature or pressure changes, the entropy of the system changes in proportion to these variations.

The heat capacity at constant pressure, denoted as \( C_p \), is a quantity that describes how much heat energy is required to raise the temperature of a substance by a certain amount under constant pressure conditions.

It's an essential aspect of energy transfer within materials, as it reflects how substances respond to temperature changes. For instance, a high \( C_p \) indicates that a substance can absorb a lot of heat without a significant change in temperature, while a low \( C_p \) means the opposite. In the context of the aforementioned thermodynamic identity, \( C_p \) is the coefficient that scales the rise in temperature to the corresponding entropy change when pressure remains constant.

Engineers and scientists consider \( C_p \) crucial when designing systems where temperature regulation and heat exchange are involved, such as in cooling technologies or thermal insulation materials.

The coefficient of thermal expansion, typically represented by the symbol \( \alpha \), quantifies how the size of a material changes with temperature. Specifically, it describes the fractional change in volume per degree of temperature change.

It's a critical parameter for materials that undergo thermal stresses, as structural components expand and contract with temperature fluctuations – for instance, in bridges or railroad tracks.

In our thermodynamic identity, the term \( -\alpha T V dp \) is associated with the work done against external pressure due to thermal expansion. It implies that even when a substance is incompressible, there can still be work interaction (and hence heat transfer) as a result of pressure changes. Materials with higher values of \( \alpha \) expand more for given temperature increases, influencing not only structural designs, but also the calculation of heat transfers and work in various thermodynamic processes.

When discussing the thermodynamics of incompressible substances, it's crucial to recognize materials such as liquids and solids where volume remains relatively constant under pressure changes. These materials are treated as having negligible compressibility.

In practical terms, for these substances, the association between temperature, pressure, and volume can be simplified since the volume change associated with pressure \( (\Delta V) \) is essentially zero. This simplification is used in the thermodynamic identity \( TdS = C_p dT - \alpha T V dp \), where the term involving \( dp \) (pressure change) becomes negligible in incompressible substances.

Consequently, when analyzing the effects of pressure on an incompressible liquid or solid, such as mercury in the given exercise, the heat transfer becomes primarily dependent on the material's coefficient of thermal expansion (\( \alpha \)) and the amount of pressure applied, rather than volume changes. The result is a direct and predictable interaction between pressure changes and heat transfer which is central to the design of processes involving such materials.