Explanation:

Given:
- 3/4 part of radioactive sample disintegrates in the first 10 years

To Find:
- Fraction that will disintegrate in the first 20 years

Step 1: Understanding Radioactive Decay
- Radioactive decay is a first-order reaction, meaning a constant fraction of the remaining material disintegrates in a given time period.

Step 2: Fraction Remaining After 10 Years
- If 3/4 of the sample disintegrates in the first 10 years, then 1/4 remains after 10 years.

Step 3: Fraction Disintegrated in the First 10 Years
- Since 3/4 of the sample disintegrates in the first 10 years, the fraction disintegrated is 1 - 1/4 = 3/4.

Step 4: Fraction Disintegrated in the Next 10 Years
- As 3/4 of the sample has already disintegrated, the remaining 1/4 will undergo further decay in the next 10 years.
- Since it is a first-order reaction, a constant fraction of the remaining 1/4 will disintegrate in the next 10 years.

Step 5: Calculation
- Using the formula for radioactive decay: N(t) = N0 * e^(-kt), where N(t) is the amount remaining at time t, N0 is the initial amount, k is the rate constant, and t is time.
- We know that N(10) = 1/4 and N(20) represents the fraction remaining after 20 years.

Step 6: Fraction Disintegrated in the First 20 Years
- By calculating N(20), we can find the fraction that disintegrates in the first 20 years.

Conclusion:
- By following these steps and understanding radioactive decay principles, we can determine the fraction of a radioactive sample that will disintegrate in a given time period.