Question 317: Atoms & Nuclei

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 _______.

Accepted Answer

According to the empirical formula relating the radius of a nucleus ($ R $) with its mass number ($ A $), we know that the radius of a nucleus is proportional to the cube root of its mass number. This relationship is given as:
$$ R = R_0 A^{1/3} $$
where $ R_0 $ is a constant with an approximate value of 1.2 fermis.
Given that for a nucleus with a mass number 64 has a radius of 4.8 fermis, we can write:
$$ 4.8 	ext{ fermi} = R_0 	imes 64^{1/3} $$
Now another nucleus has a radius of 4 fermis:
$$ 4 	ext{ fermi} = R_0 	imes A'^{1/3} $$
Where $ A' $ is the mass number of the other nucleus.
Let's solve for $ R_0 $ from the first equation:
$$ R_0 = \frac{4.8 	ext{ fermi}}{64^{1/3}} $$
Now we're going to find the mass number $ A' $ using the second equation and substituting $ R_0 $ from the above:
$$ 4 	ext{ fermi} = \left( \frac{4.8 	ext{ fermi}}{64^{1/3}} ight) 	imes A'^{1/3} $$
Now, we want to find $ A' $ in terms of $ x $ as given by the equation in the question:
$$ A' = \frac{1000}{x} $$
Substitute $ A' $ in the equation above, we get:
$$ 4 = \left( \frac{4.8}{64^{1/3}} ight) 	imes \left( \frac{1000}{x} ight)^{1/3} $$
Let's solve for $ x $:
$$ (4)^3 = \left( \frac{4.8}{64^{1/3}} ight)^3 	imes \frac{1000}{x} $$
$$ 4^3 	imes x = \left( \frac{4.8}{64^{1/3}} ight)^3 	imes 1000 $$
$$ x = \left( \frac{\left( \frac{4.8}{64^{1/3}} ight)^3 	imes 1000}{4^3} ight) $$
Now calculate the values:
$$ x = \left( \frac{\left( \frac{4.8}{4} ight)^3 	imes 1000}{64} ight) $$
$$ x = \left( \frac{1.2^3 	imes 1000}{64} ight) $$
$$ x = \left( \frac{1.728 	imes 1000}{64} ight) $$
$$ x = \left( \frac{1728}{64} ight) $$
$$ x = 27 $$
Therefore, the value of $ x $ is 27.

Atoms & Nuclei

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 _______.

Integer / Numeric Answer Type