Physical Chemistry Third Edition

10.1 The Macroscopic Description of Nonequilibrium States 443

one term in equations like Eq. (10.1-5) is nonzero. We assume that the time derivatives

of the temperature, pressure, and concentrations are dependent variables, so that it is

not necessary to specify the time derivatives to specify the state of the system.

EXAMPLE10.1

Assume that the concentration of substance number 2 is represented by the function

c 2 c 2 (z,t)c 0 +acos(bz)e−t/τ

wherec 0 ,a,b, andτare constants and wheretrepresents the time. Write the expressions for

∇c 2 and for∂c 2 /∂t.

Solution

∇c 2 −kabsin(bz)e−t/τ

∂c 2

∂t

atcos(bz)e−t/τ

wherekis the unit vector in thezdirection.

If a fluid system is flowing the flow velocity is also needed to specify the state of

the system. The flow velocityuis a vector quantity that depends on position and time:

uu(r,t)u(x,y,z,t)iux(r,t)+juy(r,t)+kuz(r,t) (10.1-6)

Just as with the temperature, pressure, and concentration, we need a specification of

how strongly the flow velocity depends on position. Each term in Eq. (10.1-6) can be

differentiated with respect tox,y, andz, so the gradient ofuhas nine components. We

will avoid a full discussion of this gradient and focus on single terms. For example, the

derivative∂uy/∂zgives the rate of change of theycomponent of the velocity in thez

direction. This quantity is called therate of shear, or the rate at which one layer of the

fluid is sliding (shearing) past an adjacent layer.

PROBLEMS

Section 10.1: The Macroscopic Description of
Nonequilibrium States

10.1 Write a formula for the gradient of each of the following
functions, wherea,b, andcrepresent constants.
a.fe−x

(^2) +y (^2) +z 2
b. frcos(θ) (transform to Cartesian coordinates or use
Eq. (B-46) in Appendix B).
c.fsin(bx) cos(cy)
d.fe−az
2
sin(bx)
10.2Assume that in a two-component solution the temperature
is given by
TT 0 +Bcos(a 1 z)e−t/b^1
and the concentration of component 2 is given by
c 2 c 0 +Csin(a 2 z)e−t/b^2
whereB,C,a 1 ,a 2 ,b 1 , andb 2 are constants and wheret
represents the time.
a.Write the expressions for the gradient of the
temperature and the gradient of the concentration of
component 2.
b.Write the expressions for the time derivatives of the
temperature and the concentration of component 2.