Editing Isothermal process (section)

== Calculation of work ==
[[File:Isothermal process.svg|thumb|'''Figure 2.''' The purple area represents the [[Work (thermodynamics)|work]] for this isothermal change.]]

In thermodynamics, the reversible work involved when a gas changes from state ''A'' to state ''B'' is<ref name=":Atkins">{{cite book|last=Atkins |first=Peter |date=1997 |title=Physical Chemistry |url=https://archive.org/details/physicalchemistr00atki |url-access=registration |edition=6th |chapter=Chapter 2: The first law: the concepts |location=New York, NY |publisher=W. H. Freeman and Co |isbn=0-7167-2871-0}}</ref>

:<math>W_{A\to B} = -\int_{V_A}^{V_B}p\,dV</math>

where ''p'' for gas pressure and ''V'' for gas volume. For an isothermal (constant temperature ''T''), [[Reversible process (thermodynamics)|reversible process]], this integral equals the area under the relevant PV (pressure-volume) isotherm, and is indicated in purple in Figure 2 for an ideal gas. Again, ''p''&nbsp;=&nbsp;{{sfrac|''nRT''|''V''}} applies and with ''T'' being constant (as this is an isothermal process), the expression for work becomes:

:<math>W_{A\to B} = -\int_{V_A}^{V_B}p\,dV = -\int_{V_A}^{V_B}\frac{nRT}{V}dV = -nRT\int_{V_A}^{V_B}\frac{1}{V}dV = -nRT\ln{\frac{V_B}{V_A}}</math>

In [[IUPAC]] convention, work is defined as work on a system by its surroundings. If, for example, the system is compressed, then the work is done on the system by the surrounding so the work is positive and the internal energy of the system increases. Conversely, if the system expands (i.e., system surrounding expansion, so [[Free Expansion|free expansions]] not the case), then the work is negative as the system does work on the surroundings.

It is also worth noting that for ideal gases, if the temperature is held constant, the internal energy of the system ''U'' also is constant, and so Δ''U''&nbsp;=&nbsp;0. Since the [[first law of thermodynamics]] states that Δ''U''&nbsp;=&nbsp;''Q''&nbsp;+&nbsp;''W'' in [[IUPAC]] convention, it follows that ''Q''&nbsp;=&nbsp;−''W'' for the isothermal compression or expansion of ideal gases.