Question

Since the half-life of 235U (7. 13 x 108 years) is less than that of 238U (4.51 x 109 years), the isotopic abundance of 235U has been steadily decreasing since the earth was fonned about 4.5 billion years ago. How long ago was the isotopic abundance of 235U equal to 3.0 a/o, the enrichment of the uranium used in many nuclear power plants

Answer 1

`\mathtt{ t_1-t_2= In((3)/(y)) * (7.13 * 10^8)/(In2) \ years}`

Step-by-step explanation:

Given that:

The Half-life of
`^(235)U` =
`7.13 * 10^8 \ years` is less than that of
`^(238) U = 4.51 * 10^9 \ years`

Although we are not given any value about the present weight of
`^(235)U`.

So, consider the present weight in the percentage of
`^(235)U` to be y%

Then, the time elapsed to get the present weight of
`^(235)U` =
`t_1`

Therefore;

`N_1 = N_o e^(-\lambda \ t_1)`

here;

`N_1` = Number of radioactive atoms relating to the weight of y of
`^(235)U`

Thus:

`In( (N_1)/(N_o)) = - \lambda t_1`

`In( (N_o)/(N_1)) = \lambda t_1` --- (1)

However, Suppose the time elapsed from the initial stage to arrive at the weight of the percentage of
`^(235)U` to be =
`t_2`

Then:

`In( (N_o)/(N_2)) = \lambda t_2` ---- (2)

here;

`N_2` = Number of radioactive atoms of
`^(235)U` relating to 3.0 a/o weight

Now, equating equation (1) and (2) together, we have:

`In( (N_o)/(N_1)) -In( (N_o)/(N_2)) = \lambda( t_1-t_2)`

replacing the half-life of
`^(235)U` =
`7.13 * 10^8 \ years`

`In( (N_2)/(N_1)) = (In 2)/(7.13 * 10^9)( t_1-t_2)` ( since
`\lambda = (In 2)/(t_(1/2))` )

∴

`\mathtt{In((3)/(y)) * (7.13 * 10^8)/(In2)= t_1-t_2}`

The time elapsed signifies how long the isotopic abundance of 235U equal to 3.0 a/o

Thus, The time elapsed is
`\mathtt{ t_1-t_2= In((3)/(y)) * (7.13 * 10^8)/(In2) \ years}`