First Law of Thermodynamics - State Functional Formulation For Infinitesimal Processes

State Functional Formulation For Infinitesimal Processes

When the heat and work transfers in the equations above are infinitesimal in magnitude, they are often denoted by δ, rather than exact differentials denoted by "d", as a reminder that heat and work 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 a chemical or physical change is known as a thermodynamic process.

For a homogeneous system, with a well-defined temperature and pressure, the expression for dU can be written in terms of exact differentials, if the work that the system does is equal to its pressure times the infinitesimal increase in its volume. Here one assumes that the changes are quasistatic, so slow that there is at each instant negligible departure from thermodynamic equilibrium within the system. In other words, δW = -PdV where P is pressure and V is volume. As such a quasistatic process in a homogeneous system is reversible, the total amount of heat added to a closed system can be expressed as δQ =TdS where T is the temperature and S the entropy of the system. Therefore, for closed, homogeneous systems:

The above equation is known as the fundamental thermodynamic relation, for which the independent variables are taken as S and V, with respect to which T and P are partial derivatives of U. While this has been derived for quasistatic changes, it is valid in general, as U can be considered as a thermodynamic state function of the independent variables S and V.

E.g., suppose that the system is initially in a state of thermal equilibrium defined by S and V, and then the system is suddenly perturbed so that thermal equilibrium breaks down and no temperature and pressure can be defined. Then the system settles down again to a state of thermal equilibrium, defined by an entropy and a volume which differ infinitesimally from the initial values. The infinitesimal difference in internal energy between the initial and final state will then satisfy the above equation. The work done and heat added to the system will then not satisfy the above expressions, they will instead satisfy the inequalities: δQ < TdS' and δW < PdV'.

In the case of a closed system in which the particles of the system are of different types and, because chemical reactions may occur, their respective numbers are not necessarily constant, the expression for dU becomes:

where dN_i is the (small) increase in amount of type-i particles in the reaction, and μ_i is known as the chemical potential of the type-i particles in the system. If dN_i is expressed in kg then μ_i is expressed in J/kg. The statement of the first law, using exact differentials is now:

If the system has more external mechanical 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. The parameters X_i are independent of the size of the system and are called intensive parameters and the x_i are proportional to the size and called extensive parameters.

For an open system, there can be transfers of particles as well as energy into or out of the system during a process. For this case, the first law of thermodynamics still holds, in the form that the internal energy is a function of state and the change of internal energy in a process is a function only of its initial and final states, as noted in the section below headed First law of thermodynamics for open systems.

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 dVis 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 out of the system as a result of the process. If one were to make this term negative then this would be the work done on the system.

It is useful to view the TdS term in the same light: 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 drives a chemical reaction that changes the numbers of particles, and the corresponding product is the amount of chemical potential energy transformed in 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 rate of 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.