Answer:

Introduction:
Isotopes are atoms of the same element that have different numbers of neutrons in the nucleus. Each isotope of an element has a different atomic mass. The mass number is the sum of the number of protons and neutrons in an atom. The mean mass number is the weighted average of the mass numbers of all the isotopes of an element.

Given:
- Three isotopes of an element have mass numbers m, (m 1), (m 2).
- Mean mass number is (m 0.5).

To find:
Which of the following ratios may be accepted for m1, (m 1), (m 2) in that order?

Solution:

The mean mass number is calculated using the formula:
mean mass number = (relative abundance of isotope 1 x mass number of isotope 1) + (relative abundance of isotope 2 x mass number of isotope 2) + (relative abundance of isotope 3 x mass number of isotope 3)

Given that the mean mass number is (m 0.5), we can write the equation as:
(m 0.5) = (x m) + ((1-x) (m 1)) + (m 2)
where x is the relative abundance of isotope 1.

Simplifying this equation, we get:
(m 0.5) = (m 1) + x(m-m1-m2)

Thus, the ratio (m-m1-m2)/(m 1) is equal to (m 0.5 - m 1) / (m - m1 - m2). We can use this ratio to determine which of the given options is correct.

Option 1: m 1/m = 1/4, (m 2)/m = 3/8
Using the ratio formula, we get:
(m-m1-m2)/(m 1) = 7/8
(m 0.5 - m 1) / (m - m1 - m2) = 7/8

Therefore, this option is correct.

Option 2: m 1/m = 3/8, (m 2)/m = 1/4
Using the ratio formula, we get:
(m-m1-m2)/(m 1) = 7/3
(m 0.5 - m 1) / (m - m1 - m2) = -1/7

Therefore, this option is incorrect.

Option 3: m 1/m = 1/3, (m 2)/m = 1/6
Using the ratio formula, we get:
(m-m1-m2)/(m 1) = 5/3
(m 0.5 - m 1) / (m - m1 - m2) = -5/3

Therefore, this option is incorrect.

Option 4: m 1/m = 3/4, (m 2)/m = 1/8
Using the ratio formula, we get:
(m-m1-m2)/(m 1) = 7/3
(m 0.5 - m 1) / (

0:1:1