First order reaction graph Kinetics

Graphs and equations for a first order chemical reaction, showing the rate constant and slope of the natural log graph

1. Rate Law: First-Order Reaction
A → B
Rate Law:
Rate = k x [A]
B appears at same rate that A
disappears

2. Time (s)[A] [B] K 0.05M/s
01.500.00
Reaction
Begins:
Initial
Concentration
Of A is 1.50 M
0 5 10 15 20 25 30 35 40
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Concentration v Time
A rearranging to B
[A]
[B]
time
C
o
n
c
e
n
t
r
a
t
io
n

(
M
)

3. In first second,
0.08 molar
Reduction in [A]
And 0.08 molar
Increase in [B]
Time [A] [B]
01.500.00
11.430.08
0 5 10 15 20 25 30 35 40
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Concentration v time
A rearranging to B
time (s)
C
o
n
c
e
n
t
r
a
t
io
n
s
(
M
)

4. In second
second, rate is
Somewhat
Lower because
[A] is lower
Time [A] [B]
01.500.00
11.430.08
21.350.15
0 5 10 15 20 25 30 35 40
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Concentration v time
A rearranging to B
time (s)
C
o
n
c
e
n
t
r
a
t
io
n
s
[M
]

5. After 40 seconds it is easy to
see the logarithmic decay
and increase of the reactant
and product.
0 5 10 15 20 25 30 35 40 45
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Concentration v time
A rearranging to B
time (s)
C
o
n
c
e
n
t
r
a
t
io
n
s
(
M
)
Time [A] [B]
01.500.00
51.160.34
100.900.60
150.690.81
200.540.96
250.421.08
300.321.18
350.251.25
400.191.31

6. Over the same time period,
plotting the natural log of the
concentration v time gives a
straight line with a negative
slope.
0 5 10 15 20 25 30 35 40 45
-2
-1.5
-1
-0.5
0
0.5
1
Ln(Concentration) vs time
A rearranging to B
time (s)
L
n

[A
]

Time Ln [A]
0 0.405
5 0.148
10 -0.105
15 -0.371
20 -0.616
25 -0.868
30 -1.139
35 -1.386
40 -1.661

7. You may recall that we set K
for the rate law as 0.050 M/s.
Let's check to see if the
negative of the slope is that.
0 5 10 15 20 25 30 35 40 45
-2
-1.5
-1
-0.5
0
0.5
1
Ln(Concentration) vs time
A rearranging to B
time (s)
L
n

[A
]

Time Ln [A]
0 0.405
5 0.148
10 -0.105
15 -0.371
20 -0.616
25 -0.868
30 -1.139
35 -1.386
40 -1.661

8. Setting our intervals to 5 s,
we see that the slope of the
Ln[A] v time line is, indeed,
-0.05. So the slope is the
negative of the rate constant
Time Ln [A]
0 0.405
5 0.148
10 -0.105
15 -0.371
20 -0.616
25 -0.868
30 -1.139
35 -1.386
40 -1.661
0 5 101520253035 4045
-2
-1.5
-1
-0.5
0
0.5
1
f(x) = -0.05x + 0.41
R² = 1
Natural Log Concentration v time
First Order Reaction
Ln [A]
Linear (Ln [A])
time (s)
L
n

[A
]

9. This is the effect of changing
the rate constant to 0.01 M/s.
The reaction is much
slower.
0 5 10 15 20 25 30 35 40 45
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Concentration v time
A rearranging to B
time (s)
C
o
n
c
e
n
t
r
a
t
io
n
s
(
M
)
Lower Rate Constant

10. This is the effect of changing
the rate constant to 0.10 M/s.
The reaction is much faster.
0 5 10 15 20 25 30 35 40 45
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Concentration v time
A rearranging to B
time (s)
C
o
n
c
e
n
t
r
a
t
io
n
s
(
M
)
Higher Rate Constant