caluclate uranium x1 decay constant from its half life how long would it take to decomposes 30% of original sample
Answers
Answer:
Explanation:expresses the time required for half of a sample to undergo radioactive decay. Exponential decay can be expressed mathematically like this:
A
(
t
)
=
A
0
⋅
(
1
2
)
t
t
1/2
(1), where
A
(
t
)
- the amount left after t years;
A
0
- the initial quantity of the substance that will undergo decay;
t
1/2
- the half-life of the decaying quantity.
So, if a problem asks you to calculate an element's half-life, it must provide information about the initial mass, the quantity left after radioactive decay, and the time it took that sample to reach its post-decay value.
Let's say you have a radioactive isotope that undergoes radioactive decay. It started from a mass of 67.0 g and it took 98 years for it to reach 0.01 g. Here's how you would determine its half-life:
Starting from (1), we know that
0.01
=
67.0
⋅
(
1
2
)
98.0
t
1/2
→
0.01
67.0
=
0.000149
=
(
1
2
)
98.0
t
1/2
98.0
t
1/2
=
log
0.5
(
0.000149
)
=
12.7
Therefore, its half-life is
t
1/2
=
98.0
12.7
=
7.72
years
.
So, the initial mass gets halved every 7.72 years.
Sometimes, if the numbers allow it, you can work backwards to determine an element's half-life. Let's say you started with 100 g and ended up with 25 g after 1,000 years.
In this case, since 25 represents 1/4th of 100, two hal-life cycles must have passed in 1,000 years, since
100.0
2
=
50.0
g
after the first
t
1/2
,
50.0
2
=
25.0
g
after another
t
1/2
.
So,
2
⋅
t
1/2
=
1000
→
t
1/2
=
1000
2
=
500
years
.
Given:
Uranium sample.
Percentage decomposition = 30 %
To Find:
The decay constant and time to decompose 30 % of original sample.
Calculation:
- We know that the half life of Uranium, T1/2 = 4.5 billion years = 4.5 × 10⁹ years
- As radioactive decay is first-order reaction, we have:
k = 0.693/T1/2
⇒ k = 0.693/4.5 × 10⁹
⇒ k = 0.154 × 10⁻⁹ yr⁻¹
- After 30% of decomposition, sample left = 70% of original sample = 0.7[A]₀
- Time taken = (2.303/k) log [A]₀/[A]
⇒ T = (2.303/0.154 × 10⁻⁹) log [A]₀/0.7[A]₀
⇒ T = (14.955 × 10⁹) log 1/0.7
⇒ T = (14.955 × 10⁹) (log 10 - log 7)
⇒ T = (14.955 × 10⁹) (1 - 0.845)
⇒ T = 14.955 × 10⁹ × 0.155