Consider a uniform substance of variable mass. Suppose that, at any moment, there are \(n\) moles of substance and that the system undergoes an infinitesimal reversible process in which \(n\) changes by \(d n\). For any 1 mol of substance, $$ \mathrm{d} q=T d s=d u+P d \nu $$ and the total heat transfer \(d Q=n\) ? \(q=n T d s\). (a) Prove that $$ \mathrm{d} Q=d U+P d V-h d n $$ where \(U\) and \(V\) refer to the entire system, but \(h\) is the molar enthalpy. (b) From Prob. \(15.11(a)\) show that $$ T d S=d U+P d V-g d n $$ (c) How does \(T\) dS differ from dQ? (d) Show that, if \(T\) and \(P\) remain constant, then ? \(Q=0\) but \(T d S\) does not.

In thermodynamics, an infinitesimal reversible process is a theoretical concept where the process occurs infinitely slowly, ensuring that the system remains in a state of equilibrium throughout. This means during such a process, the systems parameters change only by an infinitesimally small amount at each stage. Because the process proceeds so slowly, it assumes that the system can instantaneously adjust to a new state of equilibrium, making it reversible.

• The concept is primarily used for the sake of simplifying calculations and understanding the fundamental principles of thermodynamics.
• In these kinds of processes, the change in thermodynamic variables like pressure, volume, and temperature happen so gradually that the system can reverse direction at any point without any loss of energy.

The equation for an infinitesimal reversible process is given by:
\text\tex{d} q = T d s\ (where \( T \) represents temperature, \( ds \) stands for differential entropy change) It implies that the heat added to a system (\( dq \)) is equal to the product of the temperature and the small change in entropy (\( ds \)).

Molar enthalpy, also known as specific enthalpy, is the total heat content of a system per mole of a substance. It is represented by the symbol \( h \) and is defined as:
\- h = u + P v \ (where \( u \) is the internal energy per mole, \( P \) is the pressure, and \( v \) is the volume per mole)
Molar enthalpy is a convenient way to express the energy changes in a system because it combines internal energy (\( u \)) and the product of pressure and volume (\( P v \)). This combination is particularly useful in thermodynamic processes involving constant pressure, such as in many chemical reactions.

• When heat is added to the system at constant pressure, the increase in the system's enthalpy is equal to the heat added.
• The change in molar enthalpy can thus tell us how much energy (in the form of heat) is needed to change a system's state.

The change in molar enthalpy (\( \Delta h \)) can be calculated as:
\( \Delta h = h_2 - h_1 \) (where \( h_1 \) and \( h_2 \) are the initial and final enthalpies, respectively)

Gibbs free energy, denoted by \( G \), is a thermodynamic potential that measures the maximum reversible work obtainable from a thermodynamic system at constant temperature and pressure. It is defined as:
\ G = H - TS \ (where \( H \) is enthalpy, \( T \) is temperature, and \( S \) is entropy) Gibbs free energy is a crucial concept in determining whether a reaction will occur spontaneously. A process or reaction at constant pressure and temperature is spontaneous if the change in Gibbs free energy (\( \Delta G \)) is negative.

• A negative \( \Delta G \) indicates that the process is exergonic (energy-releasing and spontaneous).
• A positive \( \Delta G \) signifies that the process is endergonic (energy-absorbing and non-spontaneous).

The expression for Gibbs free energy changes when additional moles of substances are involved, such as in our original exercise, becomes:
\(T dS = dU + PdV - g dn\)
Here, \( g \) refers to the Gibbs free energy per mole, and \( dn \) is the infinitesimal change in the number of moles.

Entropy change (\( \Delta S \)) represents the degree of disorder or randomness in a system. It is a critical concept in understanding the direction of spontaneous processes. Entropy change is calculated using the following basic relation:
\ \Delta S = \int \frac{dQ_{rev}}{T} \ Here, \( dQ_{rev} \) is the infinitesimal amount of heat added reversibly, and \( T \) is the temperature.

• An increase in entropy (\( \Delta S > 0 \)) usually means the system has become more disordered.
• A decrease in entropy (\( \Delta S < 0 \)) suggests the system has become less disordered.

In our textbook solution, we saw that: \( dQ = n T ds \)
This indicates the relationship between heat transfer (\