Physical Chemistry Third Edition

1246 B Some Useful Mathematics

Thedtterm gives no contribution on the second leg of the path sincetis constant and

dtvanishes on this leg.

The contributions of Eqs. (B-56) and (B-57) are combined and set equal to zero:

z(t′, [B]t′)−z(0, 0)e−k^2 t

′

[B]t′−

K 1 [A] 0

k 2 −k 1

(e(k^2 −k^1 )t

′

−1) 0 (B-58)

Our only interest in the functionzis that it furnishes us with this equation, which is

an algebraic equation that can be solved for [B] as a function oft′, giving the desired

solution:

[B]t

k[A] 0

k 2 −k 1

(e−k^1 t−e−k^2 t) (B-59)

where we omit the prime symbol ont′.

B.5 Complex and Imaginary Quantities

Any complex quantityzcan be written in the form

zx+iy (B-60)

where√ xandyare real quantities and whereiis theimaginary unit,defined to equal

−1. Do not confuse the imaginary unitiwith the unit vectori. The real quantityxis

called thereal partofzand the real quantityyis called theimaginary partofz. The

complex conjugateofzis denoted byz∗(or sometimes byz ̄), and is defined to have

the same real part aszand an imaginary part that is the negative of that ofz.

z∗x−iy (definition ofz∗) (B-61)

A real quantity is equal to its complex conjugate.

Any complex expression can be turned into its complex conjugate by changing the

sign in front of everyiin the expression, although we do not prove this fact. For

example,

(

eiα

)∗

e−iα (B-62)

This can be shown by using the identity

eiαcos(α)+isin(α) (B-63)

and the fact that the cosine is an even function and the sine is an odd function.

The product of any complex number and its complex conjugate is equal to the square

of the magnitude of that complex number, denoted by|z|^2 , and is always a real quantity.

Themagnitudeorabsolute valueofzis the positive square root ofz∗z.

r|z|

√

|z|^2 

√

z∗z (B-64)