Question

The radius of a nucleus of mass number 64 is 4.8 fermi. Then the mass number of another nucleus having radius of 4 fermi is \(\frac{1000}{x}\) , where x is _____.

Answer 1

Correct Answer: 27

Given:

\[ R = R_0 A^{1/3} \]

We know that:

\[ R^3 \propto A \]

For the first nucleus:

\[ \frac{4.8^3}{A} = \frac{4^3}{64} \]

Rearranging and simplifying:

\[ \frac{64}{A} = \left(\frac{4}{4.8}\right)^3 \]

Calculating:

\[ \frac{64}{A} = (1.2)^3 \]

\[ \frac{64}{A} = 1.44 \times 1.2 \]

Next, equating for the second nucleus:

\[ \frac{1000}{x} = 64 \times 1.44 \times 12 \]

Calculating:

\[ x = 27 \]

Answer 2

The problem provides the radius and mass number of one nucleus and the radius of a second nucleus. We need to find the mass number of the second nucleus and then use it to determine the value of 'x' in the given expression.

Concept Used:

The radius \(R\) of a nucleus is empirically related to its mass number \(A\) by the formula:

\[ R = R_0 A^{1/3} \]

where \(R_0\) is a constant of proportionality, approximately \(1.2 \times 10^{-15}\) m (or 1.2 fermi). For two different nuclei with radii \(R_1, R_2\) and mass numbers \(A_1, A_2\), the ratio of their radii can be expressed as:

\[ \frac{R_1}{R_2} = \frac{R_0 A_1^{1/3}}{R_0 A_2^{1/3}} = \left(\frac{A_1}{A_2}\right)^{1/3} \]

This relationship allows us to find the mass number of one nucleus if the properties of the other are known.

Step-by-Step Solution:

Step 1: Identify the given values for the two nuclei.

For the first nucleus:
Mass number, \(A_1 = 64\)
Radius, \(R_1 = 4.8\) fermi

For the second nucleus:
Radius, \(R_2 = 4\) fermi
Mass number, \(A_2 = ?\)

Step 2: Set up the ratio of the radii using the formula from the concept section.

\[ \frac{R_1}{R_2} = \left(\frac{A_1}{A_2}\right)^{1/3} \]

Step 3: Substitute the known values into the equation.

\[ \frac{4.8}{4} = \left(\frac{64}{A_2}\right)^{1/3} \]

Step 4: Simplify the equation and solve for \(A_2\).

First, simplify the ratio of the radii:

\[ 1.2 = \left(\frac{64}{A_2}\right)^{1/3} \]

To eliminate the cube root, we cube both sides of the equation:

\[ (1.2)^3 = \frac{64}{A_2} \]

Alternatively, we can write 1.2 as a fraction \( \frac{6}{5} \):

\[ \left(\frac{6}{5}\right)^3 = \frac{64}{A_2} \] \[ \frac{6^3}{5^3} = \frac{216}{125} = \frac{64}{A_2} \]

Now, solve for \(A_2\):

\[ A_2 = \frac{64 \times 125}{216} \]

We can simplify the fraction by dividing 64 and 216 by their greatest common divisor, which is 8:

\[ A_2 = \frac{(8 \times 8) \times 125}{(27 \times 8)} = \frac{8 \times 125}{27} \] \[ A_2 = \frac{1000}{27} \]

Final Computation & Result:

We have found the mass number of the second nucleus to be \(A_2 = \frac{1000}{27}\). The problem states that this mass number is equal to \(\frac{1000}{x}\).

\[ \frac{1000}{27} = \frac{1000}{x} \]

By comparing the two expressions, we can directly find the value of x.

\[ x = 27 \]

Thus, the value of x is 27.