Editing Entropy (section)

== Entropy change formulas for simple processes ==
For certain simple transformations in systems of constant composition, the entropy changes are given by simple formulas.<ref>{{cite web |url=http://www.grc.nasa.gov/WWW/k-12/Numbers/Math/Mathematical_Thinking/ideal_gases_under_constant.htm |title=GRC.nasa.gov |publisher=GRC.nasa.gov |date=27 March 2000 |access-date=17 August 2012 |archive-url=https://web.archive.org/web/20110821135844/http://www.grc.nasa.gov/WWW/K-12/Numbers/Math/Mathematical_Thinking/ideal_gases_under_constant.htm |archive-date=21 August 2011 |url-status=dead }}</ref>

=== Isothermal expansion or compression of an ideal gas ===
For the expansion (or compression) of an [[ideal gas]] from an initial volume <math display="inline">V_0</math> and pressure <math display="inline">P_0</math> to a final volume <math display="inline">V</math> and pressure <math display="inline">P</math> at any constant temperature, the change in entropy is given by:<math display="block">\Delta S = n R \ln{\frac{V}{V_0}} = - n R \ln{\frac{P}{P_0}}</math>Here <math display="inline">n</math> is the amount of gas (in [[Mole (unit)|moles]]) and <math display="inline">R</math> is the [[ideal gas constant]]. These equations also apply for expansion into a finite vacuum or a [[throttling process (thermodynamics)|throttling process]], where the temperature, internal energy and enthalpy for an ideal gas remain constant.

=== Cooling and heating ===
For pure heating or cooling of any system (gas, liquid or solid) at constant pressure from an initial temperature <math display="inline">T_0</math> to a final temperature <math display="inline">T</math>, the entropy change is:
:<math display="inline">\Delta S = n C_\mathrm{P} \ln{\frac{T}{T_0}}</math>
provided that the constant-pressure molar [[heat capacity]] (or specific heat) <math display="inline">C_\mathrm{P}</math> is constant and that no [[phase transition]] occurs in this temperature interval.

Similarly at constant volume, the entropy change is:<math display="block">\Delta S = n C_\mathrm{V} \ln{\frac{T}{T_0}}</math>where the constant-volume molar heat capacity <math display="inline">C_\mathrm{V} </math> is constant and there is no phase change.

At low temperatures near absolute zero, [[Debye T3 law|heat capacities of solids quickly drop off to near zero]], so the assumption of constant heat capacity does not apply.<ref>{{cite web|last1=Franzen|first1=Stefan|title=Third Law|url=http://www4.ncsu.edu/~franzen/public_html/CH433/lecture/Third_Law.pdf|publisher=ncsu.edu|archive-url = https://web.archive.org/web/20170709093839/http://www4.ncsu.edu:80/~franzen/public_html/CH433/lecture/Third_Law.pdf |archive-date = 9 July 2017}}</ref>

Since entropy is a [[Functions of state|state function]], the entropy change of any process in which temperature and volume both vary is the same as for a path divided into two steps – heating at constant volume and expansion at constant temperature. For an ideal gas, the total entropy change is:<ref>{{cite web|url=http://www.grc.nasa.gov/WWW/K-12/airplane/entropy.html |title=GRC.nasa.gov |publisher=GRC.nasa.gov |date=11 July 2008 |access-date=17 August 2012}}</ref><math display="block">\Delta S = n C_\mathrm{V} \ln{\frac{T}{T_0}} + n R \ln{\frac{V}{V_0}}</math>Similarly if the temperature and pressure of an ideal gas both vary:<math display="block">\Delta S = n C_\mathrm{P} \ln{\frac{T}{T_0}} - n R \ln{\frac{P}{P_0}}</math>

=== Phase transitions ===
Reversible [[phase transition]]s occur at constant temperature and pressure. The reversible heat is the enthalpy change for the transition, and the entropy change is the enthalpy change divided by the thermodynamic temperature.<ref>{{cite book|first=Michael E.|last=Starzak|date=2010|chapter=Phase Equilibria & Colligative Properties|chapter-url=https://books.google.com/books?id=cw0QV7l559kC&pg=PA138|title=Energy & Entropy: Equilibrium to Stationary States|pages=138–140|isbn=978-1489983671|publisher=Springer Science+Business Media|access-date=5 September 2019}}</ref> For fusion (i.e., [[melting]]) of a solid to a liquid at the melting point <math display="inline">T_\mathsf{m} </math>, the [[entropy of fusion]] is:<math display="block">\Delta S_\mathsf{fus} = \frac{\Delta H_\mathsf{fus}}{T_\mathsf{m}}.</math>Similarly, for [[vaporisation]] of a liquid to a gas at the boiling point <math>T_\mathsf{b}</math>, the [[entropy of vaporization|entropy of vaporisation]] is:<math display="block">\Delta S_\mathsf{vap} = \frac{\Delta H_\mathsf{vap}}{T_\mathsf{b}}</math>