# 15.1: Differential Forms of Fundamental Equations

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The fundamental thermodynamic equations follow from five primary thermodynamic definitions and describe internal energy, enthalpy, Helmholtz energy, and Gibbs energy in terms of their natural variables. Here they will be presented in their differential forms.

## Introduction

The fundamental thermodynamic equations describe the thermodynamic quantities U, H, G, and A in terms of their natural variables. The term "natural variable" simply denotes a variable that is one of the convenient variables to describe U, H, G, or A. When considered as a whole, the four fundamental equations demonstrate how four important thermodynamic quantities depend on variables that can be controlled and measured experimentally. Thus, they are essentially equations of state, and using the fundamental equations, experimental data can be used to determine sought-after quantities like $$G$$ or $$H$$.

## First Law of Thermodynamics

The first law of thermodynamics is represented below in its differential form

$dU = dq+dw$

where

• $$U$$ is the internal energy of the system,
• $$q$$ is heat flow of the system, and
• $$w$$ is the work of the system.

Recall that $$U$$ is a state function, while $$q$$ and $$w$$ are path functions. The first law states that internal energy changes occur only as a result of heat flow and work done.

It is assumed that w refers only to PV work, where

$w = -\int{PdV}$

## The Principle of Clausius

The Principle of Clausius states that the entropy change of a system is equal to the ratio of heat flow in a reversible process to the temperature at which the process occurs. Mathematically this is written as

$dS = \dfrac{dq_{rev}}{T}$

where

• $$S$$ is the entropy of the system,
• $$q_{rev}$$ is the heat flow of a reversible process, and
• $$T$$ is the temperature in Kelvin.

## Internal Energy

The fundamental thermodynamic equation for internal energy follows directly from the first law and the principle of Clausius:

$dU = dq + dw$

$dS = \dfrac{dq_{rev}}{T}$

we have

$dU = TdS + dw$

Since only $$PV$$ work is performed,

$dU = TdS - PdV \label{DefU}$

The above equation is the fundamental equation for $$U$$ with natural variables of entropy $$S$$ and volume$$V$$.

## Enthalpy

Mathematically, enthalpy is defined as

$H = U + PV \label{DefEnth}$

where $$H$$ is enthalpy of the system, p is pressure, and V is volume. The fundamental thermodynamic equation for enthalpy follows directly from it definition (Equation $$\ref{DefEnth}$$) and the fundamental equation for internal energy (Equation $$\ref{DefU}$$) :

$dH = dU + d(PV)$

$dH = dU + PdV + VdP$

Because $$dU = TdS - PdV$$, the enthalpy equation becomes:

$dH = TdS - PdV + PdV + VdP$

$dH = TdS + VdP$

The above equation is the fundamental equation for H. The natural variables of enthalpy are S and P, entropy and pressure.

## Gibbs Energy

The mathematical description of Gibbs energy is as follows

$G = U + PV - TS = H - TS \label{Defgibbs}$

where $$G$$ is the Gibbs energy of the system. The fundamental thermodynamic equation for Gibbs Energy follows directly from its definition $$\ref{Defgibbs}$$ and the fundamental equation for enthalpy $$\ref{DefEnth}$$:

$dG = dH - d(TS)$

$dG = dH - TdS - SdT$

Since $$dH = TdS + VdP$$,

$dG = TdS + VdP - TdS - SdT$

$dG = VdP - SdT$

The above equation is the fundamental equation for G. The natural variables of Gibbs energy are P and T, pressure and temperature.

## Helmholtz Energy

Mathematically, Helmholtz energy is defined as

$A = U - TS \label{DefHelm}$

where $$A$$ is the Helmholtz energy of the system, which sometimes also written as the symbol $$F$$. The fundamental thermodynamic equation for Helmholtz energy follows directly from its definition (Equation $$\ref{DefHelm}$$) and the fundamental equation for internal energy (Equation $$\ref{DefU}$$):

$dA = dU - d(TS)$

$dA = dU - TdS - SdT$

Since $$dU = TdS - PdV$$,

$dA = TdS - PdV -TdS - SdT$

$dA = -PdV - SdT$

The above equation is the fundamental equation for A with natural variables of $$V$$ and $$T$$.

## Importance/Relevance of Fundamental Equations

The differential fundamental equations describe U, H, G, and A in terms of their natural variables. The natural variables become useful in understanding not only how thermodynamic quantities are related to each other, but also in analyzing relationships between measurable quantities (i.e. P, V, T) in order to learn about the thermodynamics of a system. Below is a table summarizing the natural variables for U, H, G, and A:

Thermodynamic Quantity Natural Variables
U (internal energy) S, V
H (enthalpy) S, P
G (Gibbs energy) T, P
A (Helmholtz energy) T, V

For these definitions to hold, it is assumed that only PV work is done and that only reversible processes are used. These assumptions are required for the first law and the principle of Clausius to remain valid. Also, these equations do not account include $$n$$, the number of moles, as a variable. When $$n$$ is included, the equations appear different, but the essence of their meaning is captured without including the $$n$$-dependence.

## References

1. DOI: 10.1063/1.1749582
2. DOI: 10.1063/1.1749549
3. DOI:10.1103/PhysRev.3.273
4. A Treatise on Physical Chemistry, 3rd ed.; Taylor, H. S. and Glasstone, S., Eds.; D. Van Nostrand Company: New York, 1942; Vol. 1; p 454-485.