Thermodynamics and Chemistry

CHAPTER 7 PURE SUBSTANCES IN SINGLE PHASES

7.3 THERMALPROPERTIES 173

At constantp, the energy equivalent is equal toCpDÅH=.T 2 T 1 /, and the final expres-
sion foris the same as that given by Eq.7.3.26.
To obtain values ofk=andT 1 for use in Eq.7.3.26, we need the slopes of the heating
curve in time intervals (rating periods) just beforet 1 and just aftert 2. Consider the case of
constantvolume. In these intervals,∂wel=dtis zero and dU=dtequalsk.TT 1 /(from
Eq.7.3.23). The heat capacity at constant volume isCV DdU=dT. The sloperin general
is then given by

rD

dT

dt

D

dT

dU

dU

dt

D.k=/.TT 1 / (7.3.29)

Applying this relation to the points at timest 1 andt 2 , we have the following simultaneous
equations in the unknownsk=andT 1 :

r 1 D.k=/.T 1 T 1 / r 2 D.k=/.T 2 T 1 / (7.3.30)

The solutions are

.k=/D

r 1 r 2

T 2 T 1

T 1 D

r 1 T 2 r 2 T 1

r 1 r 2

(7.3.31)

Finally,kis given by

kD.k=/D



r 1 r 2

T 2 T 1



 (7.3.32)

When thepressureis constant, this procedure yields the same relations fork=,T 1 , andk.

Continuous-flow calorimeters

A flow calorimeter is a third type of calorimeter used to measure the heat capacity of a fluid
phase. The gas or liquid flows through a tube at a known constant rate past an electrical
heater of known constant power input. After a steady state has been achieved in the tube,
the temperature increaseÅTat the heater is measured.
If∂wel=dtis the rate at which electrical work is performed (the electric power) and
dm=dt is the mass flow rate, then in time intervalÅta quantityw D.∂wel=dt/Åtof
work is performed on an amountnD.dm=dt/Åt=Mof the fluid (whereMis the molar
mass). If heat flow is negligible, the molar heat capacity of the substance is given by

Cp;mD

w

nÅT

D

M.∂wel=dt/

ÅT .dm=dt/

(7.3.33)

To correct for the effects of heat flow,ÅTis usually measured over a range of flow rates
and the results extrapolated to infinite flow rate.

7.3.3 Typical values

Figure7.5on the next page shows the temperature dependence ofCp;mfor several sub-
stances. The discontinuities seen at certain temperatures occur at equilibrium phase transi-
tions. At these temperatures the heat capacity is in effect infinite, since the phase transition
of a pure substance involves finite heat with zero temperature change.