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Half--Life
Half
HALF-LIFE is the
HALFtime it takes for 1/2 a
sample is disappear.
For 1st order reactions,
the concept of
HALF--LIFE is
HALF
especially useful.
Half--Life
Half
• Reaction is 1st
order
decomposition of
H2O2.
Half--Life
Half
• Reaction after 654
min, 1 halfhalf-life.
• 1/2 of the reactant
remains.
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Half--Life
Half
• Reaction after
1308 min, or 2
half
half--lives.
• 1/4 of the reactant
remains.
Half--Life
Half
• Reaction after 3
half
half--lives, or 1962
min.
• 1/8 of the reactant
remains.
Half--Life
Half
• Reaction after 4
half
half--lives, or 2616
min.
• 1/16 of the reactant
remains.
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Half--Life
Half
Sugar is fermented in a 1st order process
(using an enzyme as a catalyst).
sugar + enzyme -->
--> products
Rate of disappear of sugar = -k[sugar]
k = 3.3 x 10-4 sec-1
What is the half
half--life of this reaction?
Half--Life
Half
Rate = -k[sugar] and k = 3.3 x 10-4 sec-1. What is the
half
half--life of this reaction?
Solution
[A] / [A]0 = fraction remaining = 1/2 when t = t1/2
Therefore, Ln (1/2) = - k • t1/2
- 0.693 = - k • t1/2
t1/2 = 0.693 / k
So, for sugar,
t1/2 = 0.693 / k = 2100 sec
Half--Life
Half
Rate = k[sugar] and k = 3.3 x 10-4 sec-1. HalfHalf-life
is 2100 sec or 35 min. Start with 5.00 g sugar.
How much is left after 2 hr and 20 min (8400 sec
or 140 min)?
Solution
2 hr and 20 min = 4 half
half--lives
Half
Half--life Time Elapsed Mass Left
1st
35 min
2.50 g
2nd
70
1.25 g
3rd
105
0.625 g
4th
140
0.313 g
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OR…
Ln(A/A0) = -kt
Ln(A/5.00 g) = -(3.3 x 10-4 1/sec)(8400 sec)
Ln (A) – Ln 5.00 = -2.77
Ln (A) = -1.16
e-1.16 =0.313 g
Half--Life
Half
Radioactive decay is a first order process.
Tritium ----->
> electron + helium
3H
0 e
3He
-1
t1/2 = 12.3 years
If you have 1.50 mg of tritium, what fraction is
left after 49.2 years?
Half--Life
Half
Start with 1.50 mg of tritium, what fraction is left after
49.2 years? t1/2 = 12.3 years
Solution
ln ([A] / [A]0) = -kt
[A] = ?
[A]0 = 1.50 mg
t = 49.2 y
Need k, so we calculate k from: k = 0.693 / t1/2
Obtain k = 0.0564 y-1
Now ln ([A] / [A]0) = -kt =
- (0.0564 y-1) • (49.2 y) = - 2.77 = ln ([A] / [A]0)
Take antilog: [A] / [A]0 = e-2.77 = 0.0627
0.0627 = fraction remaining
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Start with 1.50 mg of tritium, what fraction is left after
49.2 years? t1/2 = 12.3 years
How many milligrams are left?
Solution
[A] / [A]0 = 0.0627
0.0627 is the fraction remaining!
Because [A]0 = 1.50 mg therefore…
[A] = 0.0940 mg
But notice that 49.2 y = 4.00 half-lives
1.50 mg ---> 0.750 mg after 1 half-life
---> 0.375 mg after 2
---> 0.188 mg after 3
---> 0.0940 mg after 4
TRY ONE MORE
• A substance has a half-life of 4.55 msec and
follows first order kinetics, how much time has
passed when the reaction is 35.0 % complete?
Start by finding “k”
k = 0.693 / t1/2
k = 0.152 1/msec
Since you really have no starting or ending amount,
consider this…35.0 % complete, means that 65.0 % is
still unreacted.
Ln(.65) = - kt
Ln(.65) = - (.152 1/msec)t
t = 2.83 msec
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