Editing Conservation of energy (section)

==First law of thermodynamics==
{{Main|First law of thermodynamics}}
{{See also|First law of thermodynamics (fluid mechanics)}}
For a [[Closed system#In thermodynamics|closed thermodynamic system]], the first law of thermodynamics may be stated as:

:<math>\delta Q = \mathrm{d}U + \delta W</math>, or equivalently, <math>\mathrm{d}U = \delta Q - \delta W,</math>

where <math>\delta Q</math> is the quantity of [[energy]] added to the system by a [[heat]]ing process, <math>\delta W</math> is the quantity of energy lost by the system due to [[Work (thermodynamics)|work]] done by the system on its surroundings, and <math>\mathrm{d}U</math> is the change in the [[internal energy]] of the system.

The δ's before the heat and work terms are used to indicate that they describe an increment of energy which is to be interpreted somewhat differently than the <math>\mathrm{d}U</math> increment of internal energy (see [[Inexact differential]]). Work and heat refer to kinds of process which add or subtract energy to or from a system, while the internal energy <math>U</math> is a property of a particular state of the system when it is in unchanging thermodynamic equilibrium. Thus the term "heat energy" for <math>\delta Q</math> means "that amount of energy added as a result of heating" rather than referring to a particular form of energy. Likewise, the term "work energy" for <math>\delta W</math> means "that amount of energy lost as a result of work". Thus one can state the amount of internal energy possessed by a thermodynamic system that one knows is presently in a given state, but one cannot tell, just from knowledge of the given present state, how much energy has in the past flowed into or out of the system as a result of its being heated or cooled, nor as a result of work being performed on or by the system.

[[Entropy (classical thermodynamics)|Entropy]] is a function of the state of a system which tells of limitations of the possibility of conversion of heat into work.

For a simple compressible system, the work performed by the system may be written:

:<math>\delta W = P\,\mathrm{d}V,</math>

where <math>P</math> is the [[pressure]] and <math>dV</math> is a small change in the [[volume]] of the system, each of which are system variables. In the fictive case in which the process is idealized and infinitely slow, so as to be called ''quasi-static'', and regarded as reversible, the heat being transferred from a source with temperature infinitesimally above the system temperature, the heat energy may be written

:<math>\delta Q = T\,\mathrm{d}S,</math>

where <math>T</math> is the [[temperature]] and <math>\mathrm{d}S</math> is a small change in the entropy of the system. Temperature and entropy are variables of the state of a system.

If an open system (in which mass may be exchanged with the environment) has several walls such that the mass transfer is through rigid walls separate from the heat and work transfers, then the first law may be written as<ref>{{cite journal | url=https://pubs.acs.org/doi/full/10.1021/ed200405k | doi=10.1021/ed200405k | title=On the Relation between the Fundamental Equation of Thermodynamics and the Energy Balance Equation in the Context of Closed and Open Systems | year=2012 | last1=Knuiman | first1=Jan T. | last2=Barneveld | first2=Peter A. | last3=Besseling | first3=Nicolaas A. M. | journal=Journal of Chemical Education | volume=89 | issue=8 | pages=968–972 | bibcode=2012JChEd..89..968K }}</ref>

:<math>\mathrm{d}U = \delta Q - \delta W + \sum_i h_i\,dM_i,</math>

where <math>dM_i</math> is the added mass of species <math>i</math> and <math>h_i</math> is the corresponding enthalpy per unit mass. Note that generally <math>dS\neq\delta Q/T</math> in this case, as matter carries its own entropy. Instead, <math>dS=\delta Q/T+\textstyle{\sum_{i}}s_i\,dM_i</math>, where <math>s_i</math> is the entropy per unit mass of type <math>i</math>, from which we recover the [[fundamental thermodynamic relation]]

:<math>\mathrm{d}U = T\,dS - P\,dV + \sum_i\mu_i\,dN_i</math>

because the chemical potential <math>\mu_i</math> is the partial molar Gibbs free energy of species <math>i</math> and the Gibbs free energy <math>G\equiv H-TS</math>.