Radioactive disintegration is a first order reaction and its rate depends only upon the nature of nucleus and does not depend upon external factors like temperature and pressure. The rate of radioactive disintegration (Activity) is represented as
`-(dN)/(dt)=lambdaN` Where `lambda=` decay constant, N= number of nuclei at time t, `N_(0)` =intial no. of nuclei. The above equation after integration can be represented as
`lambda=(2.303)/(t)log((N_(0))/(N))`
Half-life period of `U^(2.5xx10^(5)` years. In how much thime will the amount of `U^(237)` remaining be only `25%` of the original amount ?

Options

A. `2.5xx10^(5)` year
B. `1.25xx10^(5)` years
C. `5xx10^(5)` years
D. none of these

Answer

To solve the problem, we need to determine the time it takes for the amount of uranium-237 to decrease to 25% of its original amount, given its half-life.

### Step-by-Step Solution:

1. **Understand the Half-Life Concept**:
The half-life (t_half) is the time required for half of the radioactive nuclei in a sample to decay. For uranium-237, the half-life is given as \(2.5 \times 10^5\) years.

2. **Determine the Fraction Remaining**:
We are looking for the time when the remaining amount of uranium-237 is 25% of the original amount. Mathematically, this can be expressed as:
\[ N = \frac{N_0}{4} \]
This indicates that the original amount \(N_0\) has been reduced to a quarter of its value.

3. **Relate the Fraction Remaining to Half-Lives**:
The relationship between the fraction remaining and the number of half-lives (n) can be expressed as:
\[ \frac{N_0}{N} = 2^n \]
Since we want \(N = \frac{N_0}{4}\), we can set up the equation:
\[ 2^n = 4 \]
This simplifies to:
\[ n = 2 \]
This means that it takes 2 half-lives for the amount of uranium-237 to decrease to 25% of its original amount.

4. **Calculate the Total Time**:
Now, we can calculate the total time (t) it takes for 2 half-lives:
\[ t = n \times t_{half} \]
Substituting the values:
\[ t = 2 \times (2.5 \times 10^5 \text{ years}) = 5 \times 10^5 \text{ years} \]

5. **Final Answer**:
The time required for the amount of uranium-237 to remain at 25% of the original amount is:
\[ t = 5 \times 10^5 \text{ years} \]