Question

Other state variables useful for characterizing different classes of processes can be defined from \(E_{\text {int }}, S, P\), and \(V\). These include the enthalpy, \(H=E_{\text {int }}+p V\), the Helmholtz free energy, \(A=E_{\text {int }}-T S,\) and the Gibbs free energy, \(G=E_{\text {int }}+p V-T S\). a) Write the differential equations for \(d H, d A,\) and \(d G\). b) All of these are also exact differentials. What relationships follow from this fact? Use the First Law to simplify.

Short answer

Answer: The relationships between the changes in the state functions and other thermodynamic variables are as follows: 1. \(dH = TdS + Vdp\) 2. \(dA = -pdV - SdT\) 3. \(dG = Vdp - SdT\)

Step-by-step solution

Find the differential equations

To find the differential equations for H, A, and G, we will differentiate their definitions with respect to the respective variables. Enthalpy (H): \(H=E_{\text {int }}+p V\) Differentiate with respect to its variables: \(dH = dE_{\text{int}} + pdV + Vdp\) Helmholtz free energy (A): \(A=E_{\text{int}}-T S\) Differentiate with respect to its variables: \(dA = dE_{\text{int}} - TdS - SdT\) Gibbs free energy (G): \(G=E_{\text{int }}+p V-T S\) Differentiate with respect to its variables: \(dG = dE_{\text{int}} + pdV + Vdp - TdS - SdT\)

Use exact differentials

Exact differentials preserve the work done in a thermodynamic process. Being exact differentials, they don't depend on path but solely on the initial and final states. The given state functions are exact differentials, and we use it to find relationships between them. According to the First Law of Thermodynamics, \(dE_{\text{int}} = TdS - pdV\) Now, substitute the expression of \(dE_{\text {int }}\) into the expressions for \(dH, dA,\) and \(dG\): \(dH = (TdS - pdV) + pdV + Vdp = TdS + Vdp\) \(dA = (TdS - pdV) - TdS - SdT = -pdV - SdT\) \(dG = (TdS - pdV) + pdV + Vdp - TdS - SdT = Vdp - SdT\) These are the relationships we were looking for: 1. \(dH = TdS + Vdp\) 2. \(dA = -pdV - SdT\) 3. \(dG = Vdp - SdT\) These differential forms give us relationships between the changes in the state functions (enthalpy, Helmholtz free energy, and Gibbs free energy) and other thermodynamic variables (entropy, pressure, volume, and temperature).

Key concepts

Enthalpy

Enthalpy, denoted as \( H \), is a fundamental thermodynamic potential. It provides insights into the energetics of processes that occur at constant pressure. Defined as \( H = E_{\text{int}} + pV \), it combines the internal energy \( E_{\text{int}} \) with the product of pressure \( p \) and volume \( V \).

When we differentiate the enthalpy with respect to its natural variables, we find: \( dH = dE_{\text{int}} + pdV + Vdp \). However, applying the First Law of Thermodynamics, which states \( dE_{\text{int}} = TdS - pdV \), simplifies this to \( dH = TdS + Vdp \).

• \( TdS \) represents the heat added to or removed from the system.
• \( Vdp \) encapsulates the work done on or by the system via volume change.

These insights are crucial for understanding processes in open systems, such as chemical reactions at atmospheric pressure.

Helmholtz Free Energy

The Helmholtz free energy, symbolized as \( A \), is vital for systems where temperature and volume are constant. It indicates the ability to perform work and is given by the expression \( A = E_{\text{int}} - TS \). Here, \( T \) is the temperature, and \( S \) is the entropy.

Differentiating \( A \) with respect to its variables, we get \( dA = dE_{\text{int}} - TdS - SdT \). Incorporating the First Law of Thermodynamics leads to a further simplification: \( dA = -pdV - SdT \).

• \(-pdV \) represents the mechanical work done by the system.
• \(-SdT \) accounts for the change in entropy due to temperature fluctuations.

Therefore, Helmholtz free energy is useful to describe processes occurring in closed systems at constant volume, like gas compression.

Gibbs Free Energy

Gibbs free energy, denoted as \( G \), plays an essential role in chemical thermodynamics, especially when reactions occur at constant temperature and pressure. It is expressed as \( G = E_{\text{int}} + pV - TS \).

By differentiating \( G \), we obtain \( dG = dE_{\text{int}} + pdV + Vdp - TdS - SdT \). Applying the First Law gives us the simplified form: \( dG = Vdp - SdT \).

• \( Vdp \) captures the contribution due to pressure changes.
• \(-SdT \) represents the influence of temperature changes on entropy.

This makes Gibbs free energy an excellent potential for predicting reaction spontaneity and equilibrium in open systems, such as biochemical processes in living organisms.

First Law of Thermodynamics

The First Law of Thermodynamics is a cornerstone of physics, embodying the principle of conservation of energy. It is often articulated as \( dE_{\text{int}} = \delta Q - \delta W \), where \( \delta Q \) is the heat exchanged, and \( \delta W \) is the work done by the system.

When considering infinitesimal changes, it is expressed in differential form: \( dE_{\text{int}} = TdS - pdV \).

• \( TdS \) accounts for energy changes caused by variation in entropy.
• \(-pdV \) indicates energy changes due to volume work.

This law ensures energy balance in all thermodynamic processes. It is central to deriving the differential forms of enthalpy, Helmholtz free energy, and Gibbs free energy, assisting in transforming internal energy relationships into forms that are more useful in specific conditions, like constant temperature or pressure.