The first law of thermodynamics, an expression of the principle of conservation of energy, states that energy can be transformed (changed from one form to another), but it can neither be created nor destroyed. Alternatively:

The increase in the internal energy of a system is equal to the amount of energy added by heating the system, minus the amount lost as a result of the work done by the system on its surroundings.

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 δwis the infinitesimal amount of work done by the system. The infinitesimal heat and work are denoted by δ rather than d  because, in mathematical terms, they are not exact differentials. In other words, they do not describe the state of any system. The integral of an inexact differential 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.

An expression of the first law can be written in terms of exact differentials by realizing that the work that a system does is, in case of a reversible process, equal to its pressure times the infinitesimal change in its volume. In other words δw = PdV where P is pressure and V isvolume. Also, for a reversible process, the total amount of heat added to a system can be expressed as δQ = TdS where T is temperature and Sis entropy. Therefore, for a reversible process, :

Since U, S and V are thermodynamic functions of state, the above relation holds also for non-reversible changes. The above equation is known as the fundamental thermodynamic relation.

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 dN_i is the (small) number of type-i particles added to the system, and μ_i 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. μ_i is known as thechemical potential of the type-i particles in the system. The statement of the first law, using exact differentials is now:

If the system has more external variables than just the volume that can change, the fundamental thermodynamic relation generalizes to:

Here the X_i are the generalized forces corresponding to the external variables x_i.

A useful idea from mechanics is that 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 trasfer 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.