In order to introduce the critical temperature and the critical pressure of a working substance (WS), consider the pressure (
) - volume (
) plot diagramed in Figure 1, which highlights the change in volume with the application of pressure for a fixed mass of gas, found at different constant temperatures. The various curves thus obtained (also called isotherms) are represented in Figure 1 in blue.
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Consider for example the
isotherm. When initially found at low pressures (point
), the
WS is gaseous. In addition, over the
interval, the
WS generally exhibits the characteristics of a gas. At point
, the
WS is found at saturation, as any slight increase in pressure will result in a change from the vapour state to the liquid state. Afterwards, over the
interval, the
WS can be found as a mix of both vapour and liquid states. Also over the
interval, the pressure is virtually constant, while the volume is decreasing. At point
, the
WS is found entirely in a liquid state. From point
, the graph becomes almost vertical, indicating that a significant further application of pressure leads to a very little change in volume (as expected, as liquids are virtually incompressible).
Consider now the
isotherm. As can be seen from Figure 1, the
isotherm follows a path which is very similar to that of the
isotherm. However, the
interval is smaller than the corresponding
interval. This indicates that the properties of the liquid and gas states of the
WS are becoming increasingly similar, leading to a point where they will coincide. This point is indeed reached for the
isotherm, which does not show any horizontal discontinuity.
Note: The locus of all the
and
pairs corresponding to all the isotherms constructs the gray curve highlighted in Figure 1.
The
temperature is called the critical temperature, while the corresponding pressure is called the critical pressure (
). As can be seen from Figure 1, the critical temperature of a
WS is the temperature at and above which vapours of that
WS cannot be liquefied, no matter how much pressure is applied. The critical pressure is, in turn, the pressure required to liquefy the
WS at its critical temperature.
Every
WS has its own characteristic critical temperature and pressure. For example, for air:
, and
, while for water:
, and
.
Now consider the same process as in Figure 1, but this time plotted on a temperature (
) - entropy (
) diagram (see Figure 2A). The various curves obtained for the different constant pressures (also called isobars) are represented in blue.
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Consider for example the
isobar. The enthalpy corresponding to this constant pressure,
, can be calculated as the area under the
isobar, limited by
and
(the blue shaded area in Figure 2B).
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However, this area can also be calculated as:
where
is the light blue shaded area in Figure 2B.
Now imagine another isobar,
. Similarly, the corresponding enthalpy
can be calculated as the area under the
isobar, limited by
and
(the blue shaded area in Figure 2C).
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However, this area can also be calculated as:
where
is the light blue shaded area in Figure 2C.
From equations (
2) and (
3), the change in enthalpy becomes:
which can also be written as:
where
is the Gibbs free energy (
is tabulated for most
WS).
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