Fundamentals

These ten fundamental equations, which represent the foundation of thermodynamics, are listed below.

EquationDescription#Pg.Development of the first main principle

(HeatdQimparted to any body whatsoever acts to increase

the quantity of heatdHand the quantity of workdLdone)I (27) Forces against which work is done

(the quantity of workdLmay be divided into two classes: (a) internal workdJ, those which the molecules of the body exert among themselves, and which depend on the nature of the body itself; (b) external workdW, those which which arise from external influences, to which the body is subjected)II (27) First main principle

(first law of thermodynamics)III (31) Pressure-volume work

(Case in which the only external force is a uniform pressure normal to the surface)IV (38) Convenient expression for the second main principle

(if in a reversible cyclical process every element of heat taken in (positive or negative) be divided by the absolute temperature at which it is taken in, and the difference so formed be integrated for the whole course of the process, the integral so obtained is equal to zero)V (89) Second main principle

(second law of thermodynamics)VI (90)

Discussion on the function τ as being the absolute temperature TVII (105) VIII (107) Non-reversible processes

(uncompensated transformations must always be positive)IX (213) X (214)

Of these, equations III and VI represent the “two main principles” (first main principle and second main principle) of the mechanical theory of heat, as discussed in Clausius' chapter five “Formation of the Two fundamental Equations.”

Gibbs fundamental equation

In 1876, American chemical engineer Willard Gibbs expanded on the two main principles, by adding together equations III, IV, and VI, to form what is known as the combined law of thermodynamics:See main: Gibbs fundamental equation

but then went a step further by assuming that other factors, or rather 'forces', could act on the body, thus having an effect on the change of the bodies energy

which is called the "conjugate", with the product of the two intensive extensive variables

whereby the first main principle can be re-written as:

whereby the condition for a process progressing irreversibly is that the "uncompensated transformations will be positive, expressed by saying that the variation of the entropy of the body will increase in each cycle, dS > 0, as defined by equations nine and ten, whereas the 'condition for equilibrium' is that the variation of the entropy dS of the body will be zero at the equilibrium state. The main nine types of conjugate pairs are listed below:

Intensive Variable | Extensive Variable | Energy | Function | Product | Person | |

Pressure P | Volume dV | pressure-volume work | δW | pdV | Clapeyron (1834) | |

Temperature T | Entropy dS | internal work | δQ | TdS | Clausius (1865) | |

Chemical potential μ | Particle number dn | species transfer work | μdn | Gibbs (1876) | ||

Force F | Length dl | elongation/contraction work | Fdl | |||

Electromotive force ε | Charge de | electrical work | εde | |||

Surface tension γ | Surface area dA | surface work | γdA | |||

Gravitational potential ψ | Mass dm | gravitation work | ψdm | |||

Electric field E | Electric dipole moment dp | electric polarization | Edp | |||

Magnetic field B | Magnetic moment dm | magnetic polarization | Bdm |

What this table says is that when a system does work or has work done on it the system internal energy is effected, and this effect can be quantified by conjugate variable pairs. To go through one example, when the system has a volume d

In short, what Gibbs did was to add on the 'other conditions' to Clausius' first main principle (III) by which the equilibrium of the system could be effected, as described in his now-famous treatise

Condition

Description

Reaction or process is spontaneous in the forward direction. Reaction or process is nonspontaneous (reaction is favored in the opposite direction). System is at equilibrium (there is no net change).

which are simplified rules which quantify and predict the "spontaneity" of the said reaction or process under study.

See also

● Quantity

● Characteristic function

● Characteristic function notation table

References

1. Clausius, Rudolf. (1879).

2. Gibbs, Willard. (1876). "On the Equilibrium of Heterogeneous Substances",