Thermodynamics and Chemistry

CHAPTER 7 PURE SUBSTANCES IN SINGLE PHASES

7.3 THERMALPROPERTIES 172

interval before timet 1 or after timet 2 , the system behaves as if it is approaching a steady
state of constant temperatureT 1 (called the convergence temperature), which it would
eventually reach if the experiment were continued without closing the heater circuit.T 1 is
greater thanTextbecause of the energy transferred to the system by stirring and electrical
temperature measurement. By setting dU=dtand∂wel=dtequal to zero andT equal to
T 1 in Eq.7.3.22, we obtain∂wcont=dtDk.T 1 Text/. We assume∂wcont=dtis constant.
Substituting this expression into Eq.7.3.22gives us a general expression for the rate at
whichUchanges in terms of the unknown quantitieskandT 1 :

dU

dt

Dk.TT 1 /C

∂wel

dt

(7.3.23)

(constantV)

This relation is valid throughout the experiment, not only while the heater circuit is closed.
If we multiply by dtand integrate fromt 1 tot 2 , we obtain the internal energy change in the
time interval fromt 1 tot 2 :

ÅUDk

Zt 2

t 1

.TT 1 /dtCwel (7.3.24)

(constantV)

All the intermittent workwelis performed in this time interval.

The derivation of Eq.7.3.24is a general one. The equation can be applied also

to a isothermal-jacket calorimeter in which a reaction is occurring. Section11.5.2

will mention the use of this equation for an internal energy correction of a reaction

calorimeter with an isothermal jacket.

The average value of the energy equivalent in the temperature rangeT 1 toT 2 is

D

ÅU

T 2 T 1

D

.k=/

Zt 2

t 1

.TT 1 /dtCwel

T 2 T 1

(7.3.25)

Solving for, we obtain

D

wel

.T 2 T 1 /C.k=/

Zt 2

t 1

.TT 1 /dt

(7.3.26)

The value ofwelis known fromwelDI^2 RelÅt, whereÅtis the time interval during which
the heater circuit is closed. The integral can be evaluated numerically onceT 1 is known.
For heating at constantpressure, dHis equal to dUCpdV, and we can write
dH
dt

D

dU

dt

Cp

dV

dt

Dk.TText/C

∂wel

dt

C

∂wcont

dt

(7.3.27)

(constantp)

which is analogous to Eq.7.3.22. By the procedure described above for the case of constant
V, we obtain

ÅHDk

Zt 2

t 1

.TT 1 /dtCwel (7.3.28)

(constantp)