Answer

Verified

413.7k+ views

**Hint:**Volume of the nucleus of radius $r$ is given by $V = \dfrac{4}{3}\pi {r^3}$ . Use the given expression $r = {r_0}{A^{1/3}}$ to find the value of radius of the nucleus.

Density of nucleus is given by, ${\text{Density}} = \dfrac{{{\text{Mass of nucleus}}}}{{{\text{Volume of nucleus}}}}$ . Here the mass of the nucleus will be in kg and not in amu.

**Complete step by step answer:**

We first use the given relation $r = {r_0}{A^{1/3}}$ in finding the volume of the nucleus.

As we know that the volume of the nucleus of radius $r$ is given by $V = \dfrac{4}{3}\pi {r^3}$

So, substituting the expression of radius in the formula of volume we have

$\implies$ $V = \dfrac{4}{3}\pi {\left( {{r_0}{A^{1/3}}} \right)^3} = \dfrac{4}{3}\pi A{r_0}^3$

Now, as we know that the mass of nucleus can be obtained through mass number after converting the unit amu to kg i.e. $M = A{\text{ amu}} = A \times 1.67 \times {10^{ - 27}}{\text{ kg}}$

Now, we know that the density of nucleus is given by $\rho = \dfrac{{{\text{Mass of nucleus}}}}{{{\text{Volume of nucleus}}}}$ . Here the mass of the nucleus will be in kg.

So, substituting the values in the above expression we have

$\implies$ $\rho = \dfrac{{A \times 1.67 \times {{10}^{ - 27}}}}{{\dfrac{4}{3}\pi A{r_0}^3}}$

As given in the question ${r_0} = 1.2f$ . So, substituting this in the expression we have

$\implies$ $\rho = \dfrac{{3 \times 1.67 \times {{10}^{ - 27}}}}{{4 \times 3.14 \times {{\left( {1.2 \times {{10}^{ - 15}}} \right)}^3}}} = \dfrac{{5.01 \times {{10}^{18}}}}{{21.71}} = 2.3 \times {10^{17}}{\text{ kg}}{{\text{m}}^{{\text{ - 3}}}}$

Hence, the mass density of sodium nucleus is $2.3 \times {10^{17}}{\text{ kg}}{{\text{m}}^{ - 3}}$ .

**Additional Information:**

There is a device called Nuclear Density Gauge (NZO) which is used to find the density of several materials that can be further used in the construction of roads, highways and airports runways. This device works with the help of gamma radiations and uses its interaction with materials to find its density. These devices are also used in the petroleum industry.

**Note:**In the expression for radius of nucleus $r = {r_0}{A^{1/3}}$ , ${r_0}$ is known as empirical constant. We can find in the expression of density of the nucleus that it is independent of the mass no. of the element or atom.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Why Are Noble Gases NonReactive class 11 chemistry CBSE

Let X and Y be the sets of all positive divisors of class 11 maths CBSE

Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE

Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE

Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Guru Purnima speech in English in 100 words class 7 english CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Change the following sentences into negative and interrogative class 10 english CBSE

Difference Between Plant Cell and Animal Cell

Three liquids are given to you One is hydrochloric class 11 chemistry CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE