The first law of thermodynamics is a generalized axiom of nature in relation to the conservation of energy. The most common enunciation of first law of thermodynamics is:

First law of thermodynamics

The first explicit statement of the first law of thermodynamics was given by Rudolf Clausius in 1850: “There is a state function E, called ‘energy’, whose differential equals the work exchanged with the surroundings during an adiabatic process.”

The mathematical statement of the first law is given by:

where dU is the infinitesimal increase in the internal energy of the system, δQ is the infinitesimal amount of heat added to the system, and δW is the infinitesimal amount of work done by the system on the surroundings. The infinitesimal heat and work are denoted by δ rather than d because, in mathematical terms, they are inexact differentials rather than exact differentials. In other words, they do not describe the state of any system.

The integral of an inexact differential is path dependent, i.e. it depends upon the particular “path” taken through the space of thermodynamic parameters while the integral of an exact differential depends only upon the initial and final states. If the initial and final states are the same, then the integral of an inexact differential may or may not be zero, but the integral of an exact differential will always be zero. The path taken by a thermodynamic system through state space is known as a thermodynamic process.

Revesible Process
An expression of the first law can be written in terms of exact differentials by realizing that the work that a system does is equal to its pressure times the infinitesimal change in its volume. In other words, δW = pdV where p is pressure and V is volume. For a reversible process, the total amount of heat added to a system can be expressed as δQ = TdS where T is temperature and S is entropy. For a reversible process, the first law may now be restated:

In the case where the number of particles in the system is not necessarily constant and may be of different types, the first law is written:

where dNi is the (small) number of type-i particles added to the system, and μi is the amount of energy added to the system when one type-i particle is added, where the energy of that particle is such that the volume and entropy of the system remains unchanged. μi is known as the chemical potential of the type-i particles in the system. The statement of the first law for reversible processes, using exact differentials is now:

Force Function
A useful idea, introduced by Willard Gibbs in 1876, is that quantities such as internal energy U and Helmholtz free energy A may be regarded as a kind of force-function. For example, the energy gained by a particle is equal to the force applied to the particle multiplied by the displacement of the particle while that force is applied. Now consider the first law without the heating term: dU = pdV. The pressure p can be viewed as a force (and in fact has units of force per unit area) while dV is the displacement (with units of distance times area). We may say, with respect to this work term, that a pressure difference forces a transfer of volume, and that the product of the two (work) is the amount of energy transferred as a result of the process.

It is useful to view the TdS term in the same light: With respect to this heat term, a temperature difference forces a transfer of entropy, and the product of the two (heat) is the amount of energy transferred as a result of the process. Here, the temperature is known as a “generalized” force (rather than an actual mechanical force) and the entropy is a generalized displacement.

Similarly, a difference in chemical potential between groups of particles in the system forces a transfer of particles, and the corresponding product is the amount of energy transferred as a result of the process. For example, consider a system consisting of two phases: liquid water and water vapor. There is a generalized “force” of evaporation which drives water molecules out of the liquid. There is a generalized “force” of condensation which drives vapor molecules out of the vapor. Only when these two “forces” (or chemical potentials) are equal will there be equilibrium, and the net transfer will be zero.

The two thermodynamic parameters which form a generalized force-displacement pair are termed “conjugate variables”. The two most familiar pairs are, of course, pressure-volume, and temperature-entropy.

Thermodynamics and Engineering

In thermodynamics and engineering, it is natural to think of the system as a heat engine which does work on the surroundings, and to state that the total energy added by heating is equal to the sum of the increase in internal energy plus the work done by the system. Hence δW is the amount of energy lost by the system due to work done by the system on its surroundings. During the portion of the thermodynamic cycle where the engine is doing work, δW is positive, but there will always be a portion of the cycle where δW is negative, e.g., when the working gas is being compressed. When δW represents the work done by the system, the first law is written:

Very occasionally, the sign on the heat may be inverted, so that δQ is the flow of heat out of the system:

Because of this ambiguity, it is vitally important in any discussion involving the first law to explicitly establish the sign convention in use.