### Question 2.27

NCERT Class XI Physics

Estimate the average mass density of a sodium atom assuming its size to be about 2.5 Å. (Use the known values of Avogadro's number and the atomic mass of sodium). Compare it with the mass density of sodium in its crystalline phase : 970 kg m-3. Are the two densities of the same order of magnitude ? If so, why ?

### Video Explanation

(detailed solution given after this video)

Please watch this youtube video for a quick explanation of the solution:

### Solution in Detail

Derive a formula first, then put the data.

Let $\displaystyle N_A$ = avogadro's number

Let $\displaystyle \text{d }$ = diameter of sodium atom

Vol. of 1 mol sodium $\displaystyle = N_A \times \pi d^3/6 $

Let $\displaystyle A $ = mass number of sodium 23 gm

$ \begin{aligned} \therefore \text{ density }\rho &= \frac {\text{mass}}{\text{volume}}\\\\ & = \frac{A}{N_A \times \pi d^3/6}\\\\ & = \frac{6A}{N_A \times \pi d^3}\\\\ & = \frac{6 \times (23 \times 10^{-3}\text{ kg})}{(6.0 \times 10^{23}) \times \pi (2.5 \times 10^{-10})^3}\\\\ & = 4.7 \times 10^3 \text{ kg m}^{-3}\:\underline{Ans} \end{aligned} $

(rounded to 2 SF because A = 23 and diameter 2.5 Å both have been given to 2 significant digits)

$ \begin{aligned} &\underline{\text{Part II}}\\ &\text{calculated } \rho \text{ is } 4.7 \times 10^3 \text{ has order } 3\\ &\text{crystal } \rho \text{ is } 0.970 \times 10^3 \text{ has order } \approx 3\\ &\text{(densities are in kg m}^{-3}\text{)}\\\\ &\text{the order is roughly the same }\\ &\text{because the atoms are closely }\\ &\text{packed in a solid crystal}\:\underline{Ans} \end{aligned} $

### Physical Concept

(for Advanced Students)

Mass of 1 mol (avogadro number) of sodium atoms is same as the atomic mass number of 23 grams.

Volume of 1 atom can be obtained by using the formula for the volume of a sphere. From this we can obtain the volume of 1 mol of sodium atoms.

The density can be obtained as mass/volume.

The calculated density will be of the same order as the density of crystalline sodium because the atoms are tightly packed in crystalline state but there is inter-atomic space that causes the volume to increase, and ultimately the observed density is lower than the theoretical value.

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