Editing Isothermal process (section)

== Example of an isothermal process ==
[[File:Isothermal expansion of an ideal gas.png|thumb|upright=1.80|'''Figure 3.''' Isothermal expansion of an [[ideal gas]].  Black line indicates continuously reversible expansion, while the red line indicates stepwise and nearly reversible expansion at each incremental drop in pressure of 0.1 atm of the working gas.]]

The reversible expansion of an [[ideal gas]] can be used as an example of work produced by an isothermal process. Of particular interest is the extent to which heat is converted to usable work, and the relationship between the confining [[force]] and the extent of expansion.

During isothermal expansion of an ideal gas, both {{math|''p''}} and {{math|''V''}} change along an isotherm with a constant {{math|''pV''}} product (i.e., constant ''T''). Consider a working gas in a cylindrical chamber 1 m high and 1 m<sup>2</sup> area (so 1m<sup>3</sup> volume) at 400 K in [[static equilibrium]]. The [[surroundings]] consist of air at 300 K and 1 atm pressure (designated as  {{math|''p<sub>surr</sub>''}}). The working gas is confined by a piston connected to a mechanical device that exerts a force sufficient to create a working gas pressure of 2 atm (state {{mvar|A}}). For any change in state {{mvar|A}} that causes a force decrease, the gas will expand and perform work on the surroundings. Isothermal expansion continues as long as the applied force decreases and appropriate heat is added to keep {{math|''pV''}} = 2 [atm·m<sup>3</sup>] (= 2 atm × 1 m<sup>3</sup>). The expansion is said to be internally reversible if the piston motion is sufficiently slow such that at each instant during the expansion the gas temperature and pressure is uniform and conform to the [[ideal gas law]]. Figure 3 shows the {{math|''p''–''V''}} relationship for {{math|''pV''}} = 2 [atm·m<sup>3</sup>] for isothermal expansion from 2 atm (state {{mvar|A}}) to 1 atm (state {{mvar|B}}).

The work done (designated <math>W_{A\to B}</math>) has two components. First, ''expansion'' work against the surrounding atmosphere pressure (designated as {{math|''W''<sub>''p''Δ''V''</sub>}}), and second, usable ''mechanical'' work (designated as {{math|''W''<sub>mech</sub>}}). The output {{math|''W''<sub>mech</sub>}} here could be movement of the piston used to turn a crank-arm, which would then turn a pulley capable of lifting water out of [[Salt mining|flooded salt mines]].

:<math>W_{A\to B} = -p\,V\left(\ln\frac{V_B}{V_A}\right) = -W_{p \Delta V} -W_{\rm mech}</math>

The system attains state {{mvar|B}} ({{math|''pV''}} = 2 [atm·m<sup>3</sup>] with {{math|''p''}} = 1 atm and {{math|''V''}} = 2 m<sup>3</sup>) when the applied force reaches zero. At that point, <math>W_{A\to B}</math> equals –140.5 kJ, and {{math|''W''<sub>''p''Δ''V''</sub>}} is –101.3 kJ. By difference, {{math|''W''<sub>mech</sub>}} = –39.1 kJ, which is 27.9% of the heat supplied to the process (- 39.1 kJ / - 140.5 kJ). This is the maximum amount of usable mechanical work obtainable from the process at the stated conditions. The percentage of {{math|''W''<sub>mech</sub>}} is a function of {{math|''pV''}} and {{math|''p''<sub>surr</sub>}}, and approaches 100% as {{math|''p''<sub>surr</sub>}} approaches zero.

To pursue the nature of isothermal expansion further, note the red line on Figure 3. The fixed value of {{math|''pV''}} causes an exponential increase in piston rise vs. pressure decrease. For example, a pressure decrease from 2 to 1.9 atm causes a piston rise of 0.0526 m. In comparison, a pressure decrease from 1.1 to 1 atm causes a piston rise of 0.1818 m.