Half-lifes and concentration

[Robert Zellmer]

This e-mail originally came about due to a mistake in the solutions
manual for a question (the 11th edition).

The publisher's solution to EOC problem 14.88 in the 11th edition is
wrong.  It erroneously states the half-lifes for both first and second order
reactions are constant over the course of the reaction.

This is EOC exercise 14.102 in the 12th ed. and 14.96 in the 13th and
14th eds. However, the solutions manuals for these editions have the
correct answer.

This question deals with half-lifes and how they depend on conc.
(do they vary with conc and what happens to consecutive half-lifes
as the reaction proceeds).  Remember, the half-life for a 1st-order
reaction does NOT depend on conc. so consecutive half-lifes are constant.
Reactions which are other orders (zero, 2nd, etc.) have half-lifes which
depend on conc. and consecutive half-lifes change as the reaction
proceeds.

For a zero-order reaction the half-life is proportional to the initial conc.
([A]o/2k) and consecutive half-lifes decrease (prove each one is cut in
half, e.g. 10 s,  5 s, 2.5 s, etc.).

For a first-order reaction the half-life is constant and doesn't depend
on conc (0.693/k) and consecutive half-lifes are constant (10 s, 10 s,
etc.).

For a second-order reaction the half-life is inversely proportional to the
initial conc. (1/k[A]o) and consecutive half-lifes increase (prove each
one doubles, e.g. 10 s, 20 s, 40 s, etc.).

The only rate law for which the half-life is constant and does NOT depend
on conc. is first order.  The half-life for all other orders depends on 
conc.
and thus changes as the reaction proceeds.

Also, note all the half-life expressions are inversely proportional to
the rate constant.  A larger k means a shorter half-life.  That's because
a larger k means a faster reaction and thus it takes less time to get to
half the original conc.

If you find mistakes please let me know.  Make sure you tell me
which edition of the book if you are using an older edition to work
from.

Dr. Zellmer
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