Two spherical nuclei have mass numbers 64 and 8 with their radii R1 and R2 respectively. Then calculate the ratio of R1 to R2.Correct answer is '2'. Can you explain this answer?

Question

Answer

We know radius of nuclei is

given for one nuclei
A₁ =64
and other A₂= 8
So,

Answer

Given:
- Mass number of nucleus 1 (A1) = 64
- Mass number of nucleus 2 (A2) = 8

To find:
- Ratio of radii of the two nuclei (R1/R2)

Solution:
The radius of a nucleus can be estimated using the empirical formula:
R = R0 * A^(1/3)

Where:
- R0 is a constant with a value of approximately 1.2 x 10^(-15) m
- A is the mass number of the nucleus

Calculating the radii of the two nuclei:
- Radius of nucleus 1 (R1) = R0 * A1^(1/3)
- Radius of nucleus 2 (R2) = R0 * A2^(1/3)

Substituting the given values:
- R1 = R0 * 64^(1/3)
- R2 = R0 * 8^(1/3)

Calculating the ratio of radii:
- Ratio of radii (R1/R2) = (R0 * 64^(1/3)) / (R0 * 8^(1/3))

Canceling out R0:
- Ratio of radii (R1/R2) = (64^(1/3)) / (8^(1/3))

Simplifying the expression:
- Ratio of radii (R1/R2) = (4^(1/3) * 4^(1/3) * 4^(1/3)) / (2^(1/3) * 2^(1/3) * 2^(1/3))

Using the exponent property (a^(m/n) = (a^m)^(1/n)):
- Ratio of radii (R1/R2) = (4 * 4 * 4) / (2 * 2 * 2)

Simplifying the expression:
- Ratio of radii (R1/R2) = 64 / 8

Final result:
- Ratio of radii (R1/R2) = 8

Therefore, the ratio of the radii of the two nuclei is 8.

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