Question

If 12.5 grams of the original sample of cesium-137 remained after 90.6 years, what was the mass of the original sample?

Answer 1

The mass of the original sample of cesium-137 after 90.6 years was 100 grams.

Step-by-step explanation:

To determine the mass of the original sample of cesium-137, we can use the concept of radioactive decay.

Cesium-137 has a half-life of 30.1 years, which means that every 30.1 years, half of the sample decays.

Given that 12.5 grams of the original sample remains after 90.6 years, we can use this information to calculate the original mass.

Since the half-life is 30.1 years, we can divide the total time of 90.6 years by the half-life to determine the number of half-lives that have occurred.

In this case, 90.6 years ÷ 30.1 years ≈ 3 half-lives.

Since the amount remaining after one half-life is one-half of the original amount, we can calculate the original mass by multiplying the remaining mass by 2 (for each half-life).

Original Mass = Remaining Mass × 2³ = 12.5 g × 2³ = 12.5 g × 8 = 100 g

So therefore the mass of the original sample was 100 grams.

Answer 2

Mass of original sample = 100 g

Step-by-step explanation:

Half life of cesium-137 = 30.17 years

`t_(1/2)=\frac {ln\ 2}{k}`

Where, k is rate constant

So,

`k=(\ln2)/(t_(1/2))`

`k=(\ln2)/(30.17)\ year^(-1)`

The rate constant, k = 0.02297 year⁻¹

Time = 90.6 years

Using integrated rate law for first order kinetics as:

`[A_t]=[A_0]e^(-kt)`

Where,

`[A_t]` is the concentration at time t

`[A_0]` is the initial concentration

Initial concentration
`[A_0]` = ?

Final concentration
`[A_t]` = 12.5 grams

Applying in the above equation, we get that:-

`12.5\ g=[A_0]e^(-0.02297* 90.6)`

`[A_0]=(12.5)/(e^(-0.02297* 90.6))\ g=100\ g`

Mass of original sample = 100 g