Question

Order of magnitude of density of uranium nucleus is $\left[ {{m_p} = 1.67 \times {{10}^{ - 27}}\,{\text{kg}}} \right]$
A. $4 \times {10^{20\,}}\,{\text{kg}}{{\text{m}}^{{\text{ - 3}}}}$
B. $1.8 \times {10^{17\,}}\,{\text{kg}}{{\text{m}}^{{\text{ - 3}}}}$
C. $4.2 \times {10^{14\,}}\,{\text{kg}}{{\text{m}}^{{\text{ - 3}}}}$
D. $1.4 \times {10^{11}}\,{\text{kg}}{{\text{m}}^{{\text{ - 3}}}}$

Answer 1

Recall the formula for density. Find the volume of the nucleus and mass of the nucleus using the value of mass of proton given in the question. And then use these values to find the density of the uranium nucleus and check the order of which option matches with the calculated value.

Complete step by step answer:
Given, Mass of proton ${m_p} = 1.67 \times {10^{ - 27}}\,{\text{kg}}$. Density is defined as mass by volume. That is,
$D = \dfrac{M}{V}$
Here, $D$ is the density, $M$ is the mass and $V$ is the volume.

We need to find the density of uranium nucleus, and we know the shape of the nucleus is spherical so if $R$ is the radius of the nucleus then volume of the nucleus can be written as,
$V = \dfrac{4}{3}\pi {R^3}$ (i)
We have the formula for radius of a nucleus as,
$R = {R_0}{A^{1/3}}$ where $A$ is the mass number and ${R_0} = 1.2 \times {10^{ - 15}}\,{\text{m}}$.
Putting this value of $R$ in equation (i) we have
$V = \dfrac{4}{3}\pi {\left( {{R_0}{A^{1/3}}} \right)^3}$
$\Rightarrow V = \dfrac{4}{3}\pi {R_0}^3A$ (ii)
Mass of the nucleus can be written as,
$M = {m_p}A$

Therefore, density of uranium nucleus is
$D = \dfrac{M}{V}$
Putting the values of $M$ and $V$ in above equation we get

\Rightarrow D = \dfrac{{{m_p}}}{{\dfrac{4}{3}\pi {R_0}^3}} $$ Now, we put the values of $${R_0}$$ and $${m_p}$$and we get $$D = \dfrac{{1.67 \times {{10}^{ - 27}}}}{{4 \times 3.14 \times {{\left( {1.2 \times {{10}^{ - 15}}} \right)}^3}\,}} \\\ \therefore D = 1.8 \times {10^{17}}\,{\text{kg}}{{\text{m}}^{{\text{ - 3}}}} $$ Therefore, density of uranium nucleus is $$1.8 \times{10^{17}}\,{\text{kg}}{{\text{m}}^{{\text{ - 3}}}}$$. **Hence, the correct answer is option B.** **Note:** The density of a nucleus is independent of the mass number that is density of all nuclei are same and it is of order of $${10^{17}}$$, always remember this value as most of the time questions on density of nucleus are asked. But also remember that the radius of the nucleus is not the same for all nuclei, it depends on the mass number and increases with increase in mass number.