Alexander Wood
The Regional College of Technology
Temperature-Entropy Diagrams
Liverpool, England
The satisfactory grasp of the concept of free energy is a matter of some difficulty to some students, in view of the abstract way in which the quantity is usually defined. The author believes that the introduction of the use of temperature-entropy diagrams greatly facilitates the development of the concept for many beginners. Temperature-entropy diagrams (sometimes called "tephigrams," since temperature (T) is plotted against entropy, sometimes represented by the Greek letter +) find some usage in certain branches of science and technology and can be employed advantageously for teaching purposes in chemistry. The diagram shows typical plots for the liquid-vapor region. (The latter region high is convenient, since it allows both condensed and vapor phases to be represented 2 - - .- - - - - - - together and so compared.) W c The inclusion of both highpressure and low-pressure forms is necessary, since the entropy of any system depends on the pressure under I which this quantity is meaENTROPY sured: in the well-known Typical temperature-entropy manner, entropy is obtained plots for the liquid-vopor region. by the integration of C,/T with respect to T, the absolute temperature, and values of the heat capacity C, depend ifcourse on the pressure under which they are measured. (The latter point may be illustrated readily for teaching purposes by quoting selected heat capacity values from the literature, such as the compilation of data by Din'.) The horizontal linear section indicates a phase change, such as a melting-point or boiling-point, where the temperature remains constant while the entropy increases. Most other sections are concave-up, as shown, but the graphs become tangential to the tem~ e r a t u r eaxis a t absolute zero. The relationship between entropy and pressure is brought out clearly by the diagram. At any given temperature, a low-pressure system has more entropy than the same system a t a high pressure. In the case of condensed phases, it is clear that the effect of pressure is ~ u . . . m s l l
It may be helpful also to point out to students that the area under any section of a curve (such as part AB) represents the increase in enthalpy, m,in transferring the system from A to B. 1 DIN, I?., (Edilo~),"Thermodynamio Functions of Gases," Butterworth and Co. Ltd., London, 1962.
Area under c u m = Js" T.dS =
J
C,.dT = AH T*
As an additional advantage, the diagram emphasizes the connection between heat content (enthalpy) and pressure. The actual enthalpy of any system under any given conditions of temperature and pressure is the sum of the enthalpy represented by the area under the curve (drawn for that pressure) from absolute zero to temperature T together with any heat content a t absolute zero. Such a quantity, in general, cannot be evaluated, of course, and so it is the normal practice to select a convenient origin from which the enthalpy can be measured. In chemistry, the reference conditions for elements in their standard states are 25°C and a pressure of one atmosphere. Clearly, also, the area under any horizontal section, for example a t the boilingpoint of the substance under study, represents the associated latent heat change a t that pressure. Further, since the curves are practically coincident for the condensed state, the diierence in enthalpy between points B and B', say, is given by (Area under dB')
- (Area under AB)
to a good approximation. A is then any suitable point in the condensed-phase region. It may be observed also that the entropy change, AS, a t transitions such as the boiling-point normally decreases with increasing -. pressure. The area subtended between section AB and the tem~eratureaxis re~resentsthe chance in Gibbs' free energy, -AG, observed when the system is transferred from A to B. The negative sign is traditional in chemistry. The usefulness of this approach may be illustrated by the following points. (a) The pressure under which the change occurs must be specified.
(b)
(c)
- AG
=
fTB
S . dT at constant pressure
.--, The effect of pressure on the Gibbs' free energy of con-
densed phases may be neglected. ( d ) Continuing from the above, the difference between the Gibbs, free energy of states and Br may be represented by
-AG = (area between AB' and temperature axis) (area. between AB and temperature axis) (e) The nature of Gibbs' free energy itself is clarified, since changes in this quantity can be followed readily. For example, AG = 0 for constant-temperature changes (such as boiling) at a is heated from room given pressure, Again, when a temperature to 10O0C, say, -G increases, i.e., its free energy decreases, and this change can be seen as an area on a. graph. The actual value of AG, may, of course, be found by direct calculation in the usual way. Volume 47, Number 4, April 1970
/
285
(f) The formula, G = H - T . S may be derived by considering s change such as that represented by moving from'd to B; from the areas
-AG
+ AH
=
-(Gs
- GA) 4-Ha - H A
286 / Journol o f Chemical Education
= T B . SB - T A . SA i.e., GB - GA = HB - H A - ( T B . SB - TA . SA)
The requisite formula. is obtained by defining that G
.
T S.
=
H -
