When a system undergoes a change, the change is accompanied by a change in the energy content of the system. It is the purpose of thermodynamics to be able to describe the change in energy content, and to relate it to the performance of work and the transfer of heat between the system and the surroundings. For any change in the system (for a defined sytem), we can define an initial state and a final state, and we must look for parameters which define the energy content of the system in terms of values which are a unique function of the state of the system. Such parameters are called variables of state. For example, the operation of an "ideal gas" engine can be described in terms of a cycle of states, each of which has a unique set of values for P, V, T, and hence energy content, E.

Because the values for variables of state are a unique function of the state, they are independent of the pathway by which a change in state occured. Consequently, variables of state have the property that the sum of changes in value, on going through a series of changes of state which form a cycle, is zero. This is obviously the case, since in a cycle, the initial and final states are the same.


The First Law is a conservation law,- Energy can neither be created nor destroyed. It requires that we balance the energy budget when we describe a change in state. A change in energy content (dE) is accompanied by the performance of work (w), and/or the transfer of heat (q) between the thermodynamic system under consideration, and its surroundings.

System. A set of entities which undergoes a change, and can be formally separated out from the surroundings. A thermodynamic system might be a simple weight, a chemical reaction, an organism, an engine, a solar system, etc. In order to describe a change, we must be able to define the system unambiguously.

Thermodynamic state. In order to describe a change in the system, we have to be able to define and measure the properties of the system which might change, and which characterize it in terms of energy, work and heat content. Variables of state are parameters which have specific values which define the state of the system.

The First Law requires that for any change of state of a system:

dE = dEsystem = - dEsurroundings

dE = w + q

(w and q are positive for work done on, or heat transfered to the system from the surroundings)

Experience tells us that w and q do not have fixed values for a change of a system between known states;- they will vary depending in the pathway by which the change of state occurs, and are therefore not variables of state. For example, we can allow 2 mol H2 to react with 1 mol O2 to form 2 mol water, at normal temperature and pressure, either by explosion (w is zero, q = dE), or in a fuel cell (w is non-zero, q < dE). The same general principle applies to any change of state.

The development of thermodynamics owes much to the Industrial Revolution, and the need to understand how to get the most work from steam engines. A major achievement of classical thermodynamics was to show that a change in state could, in principle, be characterized in terms of changes in energy content, maximal capacity for work (work potential), and heat exchange under reversible conditions, which were independent of the path. This came about through a consideration of the maximal work which could be obtained from an engine for a given input of fuel. From the equation above, it is clear that maximal work can be obtained when minimal heat is exchanged. By introducing the concept of an "ideal" engine, operating through expansion and contraction of ideal gases as heat was put into or removed from the system, the operation of the engine could be considered as a series of partial processes, each of which could be precisely defined in terms of the Ideal Gas Laws, so that the work could be calculated. A pathway between states which minimizes heat exchange and maximizes work is called a "reversible" pathway.

When a change of state occurs through a "reversible" pathway, the work performed (wmax) is maximal and the heat transfered (qrev) is minimal.

dE = wmax + qrev

(for a "reversible" process)

The parameters, wmax and qrev, which define the reversible process, are of special importance in thermodynamics, because, unlike the work and heat exchanged in an irreversible (or "spontaneous") process, they have unique values for a defined change in state; they are variables of state. Further analysis of the heat exchange led to the concept of entropy.

It is important to realize that in these equations, unless otherwise indicated, the terms refer to parameters of the system.

At constant T,

dE = dA + TdS (dS = qrev/T , Entropy change; TdS is the reversible heat)

dA = dE - TdS (Helmholtz free energy change, or maximal work)

dG = dE - (-PdV) - TdS (Gibbs free energy change, or useful work)

dG = dH - TdS (dH = dE - (-PdV) , Enthalpy change)

dG = dA - (-PdV) (Useful work is maximal work minus the work done against the atmosphere)

The Gibbs free energy differs from the Helmholtz free energy by the work (-PdV) performed against the atmosphere, which is not regarded as "useful". The Gibbs free energy is sometimes called the "useful" work.

Like P, V and T for an ideal gas system, the Gibbs free energy, Helmholtz free energy, enthalpy and entropy are variables of state; they have unique values for particular states, and the sum of changes round a cyclic process (the cyclic integral) is zero.

All spontaneous processes occur at the sacrifice of work potential; work can only be obtained from a spontaneous process; all spontaneous processes have a negative dG; a process with a positive dG can occur only if coupled to a process with a negative dG of greater absolute value. The actual work obtained when a system changes state at finite rate is always less than the maximal work, since some extra heat is generated by "friction", which is transferred to the surroundings as an entropy increase(TdSsurroundings, or qirrev).

In general, the exchange of energy, either through performance of work or heat flow, involves a change in either one or both of two factors, an intensive (or intensity) factor (the driving force, gradient, etc.), and an extensive (or capacity) factor (dependent on the size of the system, or amount of substance, etc.). For each type of work, there is a particular pair of intensity and capacity factors, as detailed in the table below:
Energy Exchange Intensity Factor  Capacity Factor  Work or Heat Terms 
Type of Work
Mechanical(1)  force (f) distance (l) fdl ldf
Mechanical(2)  pressure (P=f/A)  volume (V=A x l) -PdV -VdP 
Chemical  chemical potential (µ)  amount of substance (n) µdn ndµ 
Electrical  electrical potential (E or y quantity of charge (Q or nzF) EdQ or ydnzF QdE or nzFdy 
Electrochemical  electrochemical potential (µ amount of substance (n) µdn ndµ 
Heat Temperature (T)  entropy (S) TdS SdT

The maximal work will be the sum of all work terms involved in the change of state by the "reversible" pathway.

The intensity factor is the rate of change of free energy with change in capacity factor, when all other terms are constant.

©Copyright 1996, Antony Crofts, University of Illinois at Urbana-Champaign,