Show that, if the Helmholtz function of a phase is expressed as a function of \(T, V, n_{1}, n_{2}, \ldots, n_{c}\), then: (a) \(d A=-S d T-P d V+\sum \mu_{j} d n_{j}, \quad\) where \(\mu_{j}=\left(\partial A / \partial n_{j}\right)_{T, V, \text { other } n^{\prime} s^{\circ}}\) (b) \(A=-P V+\sum \mu_{j} n_{j}\) (c) \(-S d T+V d P=\sum n_{j} d \mu_{j}\)

Thermodynamics is the study of energy, work, and heat, and how they interrelate. It involves understanding how energy is transferred and transformed in physical systems.
There are four laws of thermodynamics, but here we will focus on the first law. This law states that energy cannot be created or destroyed, only transferred or transformed, often known as the law of energy conservation.
In the context of Helmholtz free energy, thermodynamics helps us understand how the internal energy, temperature, and entropy are related to each other.
Thermodynamics provides key principles that allow us to derive important equations, such as those involving the Helmholtz free energy and its differential form.

Chemical potential, \(\( \mu \)\), describes the change in the Helmholtz energy when an infinitesimal amount of substance is added to the system, at constant temperature and volume. More formally, it is the partial derivative of the Helmholtz free energy with respect to the number of moles, \(\( n_{j} \)\), of a specific component.
The relationship is given as:
\[ \mu_{j} = \left( \frac{\partial A}{\partial n_{j}} \right)_{T, V, \text { other } \ n's} \]
This definition is crucial for deriving many thermodynamic equations and understanding interactions within a multi-component system.
Chemical potential ties into both the differential forms and the equations involving Helmholtz free energy, helping us quantify how substances within a system contribute to its overall energy.

The Gibbs-Duhem equation provides a valuable relationship between changes in temperature, pressure, and composition in a thermodynamic system. It is derived from the differential form of the Helmholtz free energy.
In the context of this exercise, we derive:
\[ -S dT + V dP = \sum_{j} \ n_{j} d \ \mu_{j} \]
This equation tells us how changes in temperature and pressure affect the chemical potentials in a system with multiple components.
Understanding the Gibbs-Duhem equation allows us to explore deeper thermodynamic properties and how different parts of a system interrelate.

The first law of thermodynamics, or the law of energy conservation, states that energy cannot be created or destroyed—only transferred or changed in form. Mathematically, it's given by:
\[ dU = TdS - PdV + \ \sum_{j} \ \mu_{j} dn_{j} \]
Here, \(\( dU \)\) represents the infinitesimal change in internal energy, \(\( TdS \)\) is the heat added to the system, and \(\( -PdV \)\) is the work done by the system. The term \(\( \sum_{j} \mu_{j} dn_{j} \)\) accounts for changes in energy due to variations in the number of particles of each type.
This principle is fundamental in deriving how the Helmholtz free energy changes with temperature, volume, and number of particles in the system.

Differential forms in thermodynamics allow us to understand how small changes in one quantity affect another. For the Helmholtz free energy, taking the differential form helps in deriving key thermodynamic relations.
Starting with the definition of Helmholtz free energy:
\[ A = U - TS \],
Taking its differential we obtain:
\[ dA = dU - TdS - SdT \].
By substituting \(\( dU \)\) from the first law of thermodynamics, we get:
\[ dA = -S dT - P dV + \sum_{j} \ \mu_{j} \ d \ n_{j} \]
This form is crucial for understanding how a system's free energy changes with its parameters. It ties together temperature, volume, and chemical potential into a single framework.