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Solution - Estimate the Average Mass Density of a Sodium Atom Assuming Its Size to Be About 2.5 Compare It with the Density of Sodium in Its Crystalline Phase: 970 Kg M–3. Are the Two Densities of the Same Order of Magnitude - CBSE (Science) Class 11 - Physics

ConceptDimensional Formulae and Dimensional Equations

Question

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 density of sodium in its crystalline phase: 970 kg m–3. Are the two densities of the same order of magnitude? If so, why?

Solution 1

Diameter of sodium atom = Size of sodium atom = 2.5 Å

Radius of sodium atom, r  = 1/2 xx 2.5 Å = 1.25 Å

= 1.25 × 10–10 m

Volume of sodium atom, `V = 4/3 pir^3`

=`4/3 xx 3.14 xx(1.25 xx 10^(-10))^3`

According to the Avogadro hypothesis, one mole of sodium contains 6.023 × 1023 atoms and has a mass of 23 g or 23 × 10–3 kg.

∴ Mass of one atom = `(23xx10^(-3))/(6.023 xx 10^(23)) kg`

Density of sodiuym atom,p = `((23xx10^(-3))/(6.023xx10^(23)))/(4/3xx3.14xx(1.25xx10^(-10))^3)` = `4.67 xx 10^(-3) kg m^(-3)`

It is given that the density of sodium in crystalline phase is 970 kg m–3.

Hence, the density of sodium atom and the density of sodium in its crystalline phase are not in the same order. This is because in solid phase, atoms are closely packed. Thus, the inter-atomic separation is very small in the crystalline phase.

Solution 2

It is given that radius of sodium atom, R = 2.5 A = 2.5 x 10-10 m

Volume of one mole atom of sodium, V = NA .4/3 π R3

V = 6.023 x 1023 x –4/3 x 3.14 x (2.5 x 10-10)3 m3and mass of one mole atom of sodium, M = 23 g = 23 x 10-3 kg

∴ Average mass density of sodium atom, p = M/V

=(23 x 10-3/6.023 x 1023 x 4/3 x 3.14 x (2.5 x 10-10))

= 6.96 x 102 kg m-3 = 0.7 x 10-3 kg m-3

The density of sodium in its crystalline phase = 970 kg m-3

= 0.97 x 103 kg m-3

Obviously the two densities are of the same order of magnitude (= 103 kg m-3). It is on account of the fact that in solid phase atoms are tightly packed and so the atomic mass density is close to the mass density of solid.

Is there an error in this question or solution?

APPEARS IN

 NCERT Physics Textbook for Class 11 Part 1 (with solutions)
Chapter 2: Units and Measurements
Q: 27 | Page no. 37

Reference Material

Solution for question: Estimate the Average Mass Density of a Sodium Atom Assuming Its Size to Be About 2.5 Compare It with the Density of Sodium in Its Crystalline Phase: 970 Kg M–3. Are the Two Densities of the Same Order of Magnitude concept: Dimensional Formulae and Dimensional Equations. For the courses CBSE (Science), CBSE (Arts), CBSE (Commerce)
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