Editing Entropy (section)

=== 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>