SME
General Chemistry II · Problem type
The activity of a mass of a nuclide
Use this for “Co-60 has a half-life of 5.27 years. What is the activity of 1.00 mg of it?”
Diagram
After 5.27 yr, 1 half-life has passed: 4.19 × 10¹⁰ × (1/2)¹ = 2.09 × 10¹⁰ Bq left.
In the grid 50 of 100 Parent atoms are left; each half-life, half of those still there decay.
This is an example. Type your own values to start a new problem.
m
Mass of the nuclide
example · mg
M
Molar mass
example · g/mol
N
Nuclei in the sample
calculated
t½
Half-life
example · yr
k
Decay constant
calculated · s⁻¹
A
Activity
calculated · Bq
Formulas
N
=
m
÷ 1,000
M
×
6.022 ×
10
23
1.005 ×
10
19
=
1 ÷ 1,000
59.93
×
6.022 ×
10
23
k
=
ln(2)
t
1
2
× 3.156 ×
10
7
4.168 ×
10
-9
=
ln(2)
5.27 × 3.156 ×
10
7
A
=
k
×
N
4.188 ×
10
10
=
4.168 ×
10
-9
×
(1.005
×
10
19
)
Assumptions
•
The sample is the pure nuclide: its moles are its mass over its molar mass, and each mole holds 6.022 × 10²³ atoms.
•
One year is 365.25 days, or 3.156 × 10⁷ s, so k comes out per second and the activity in becquerels.
•
Activity is first order, A = kN: each nucleus has the same chance k each second to decay.
Step-by-step
WE KNOW
m = 1 mg, M = 59.93 g/mol, t½ = 5.27 yr
FIND
nuclei in the sample (N), decay constant (k), activity (A)
Step 1 · Find nuclei in the sample (N)
Use N = (m ÷ 1,000) ÷ M × 6.022 × 10²³
Turn mg into g, divide by the molar mass for moles, then multiply by NA for atoms.
N
=
1 ÷ 1,000
59.93
×
6.022 ×
10
23
N
=
0.001
59.93
×
(6.022
×
10
23
)
N
=
(1.6686
×
10
-5
)
×
(6.022
×
10
23
)
N
=
1.0048 ×
10
19
Step 2 · Find decay constant (k)
Use k = ln(2) ÷ (t½ × 3.156 × 10⁷)
kt
1
2
=
ln
2
with
t
1
2
in
seconds:
a
year
is
3.156 ×
10
7
s.
k
=
ln(2)
5.27 × 3.156 ×
10
7
k
=
ln(2)
1.663 ×
10
8
k
=
0.6931
1.663 ×
10
8
k
=
4.1675 ×
10
-9
s
-1
Step 3 · Find activity (A)
Use A = k × N
Each
nucleus
decays
with
chance
k
each
second:
multiply
by
the
number
of
nuclei.
A
=
4.1675 ×
10
-9
×
(1.0048
×
10
19
)
A
=
4.188 ×
10
10
Bq
Check
Put the numbers back in. Both sides should match.
1.0048 ×
10
19
=
1 ÷ 1,000
59.93
×
6.022 ×
10
23
✓
4.1675 ×
10
-9
=
ln(2)
5.27 × 3.156 ×
10
7
✓
4.188 ×
10
10
=
4.1675 ×
10
-9
×
(1.0048
×
10
19
)
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.