Question

The specific gravity of the stainless-steel spherical balls used in ball-bearings are 10.2. How many iron atoms are present in each ball of diameter 1 cm if the balls contain 84 per iron by mass? The atomic mass of iron is 56.
(A) \[4.12 \times {10^{21}}\]
(B) \[4.82 \times {10^{22}}\]
(C) \[3.82 \times {10^{22}}\]
(D) None of these

Answer 1

Formula used: \[volume{\text{ }}of{\text{ }}sphere = \dfrac{4}{3}\pi {r^3}\], \[density = \dfrac{{mass}}{{volume}}\], \[specific{\text{ }}gravity{\text{ }}of{\text{ }}iron = \dfrac{{density{\text{ }}of{\text{ }}iron}}{{density{\text{ }}of{\text{ }}water}}\]and \[moles = \dfrac{{given{\text{ }}mass}}{{{\text{molar }}mass}}\]

Hint: Specific Gravity gives information about the weight and density of the object by comparing the weight, mass and density of the given object with water of the same amount at \[{4^0}C\]. The density thus calculated is used to get the atoms present by using volume and moles as well.

Complete step-by-step answer:
Specific gravity, also known as relative gravity is a dimensionless quantity which is defined as the ratio of the density of a substance to the density of a substance to the density of water at a specified pressure and temperature. It is a unitless quantity.
Since, \[Specific{\text{ }}gravity{\text{ }}of{\text{ }}iron = \dfrac{{Density{\text{ }}of{\text{ }}iron}}{{Density{\text{ }}of{\text{ }}water}}\]
Putting the value of specific gravity of iron and density of water (1 g/ml) in it, we get the density of iron,
\[10.2\] = \[\dfrac{\rm{Density \space of \space iron}}{1}\]
\[\therefore \]density of iron is \[10.2\] g/ml
As the diameter given is 1cm, the radius of the sphere is \[0.5\]cm. So,
Volume of the sphere = \[\dfrac{4}{3}\pi {r^3}\]
= \[\dfrac{4}{3} \times 3.14 \times {(0.5)^3}\]
= \[0.52\]\[c{m^3}\]
From the formula of density, we get
Mass = Density \[ \times \]Volume
= \[10.2\]\[ \times \]\[0.52\] = \[5.34\] g
Provided that atomic mass of iron is 56, so number of moles of iron can be determined by
\[Moles = \dfrac{{given{\text{ }}mass}}{{{\text{molar }}mass}}\]
= \[\dfrac{{5.34}}{{56}}\]= \[0.095\]
Given that the balls contain 84 percent iron by mass, the iron atoms present in it are
\[ = 0.84 \times 0.095 \times 6.022 \times {10^{23}}\]
=\[4.82 \times {10^{22}}\] iron atoms

Hence, the correct option is (B).

Note: Specific gravity tells us whether an object will float or sink.If the specific gravity of an element is greater than that of water i.e. 1, it will sink in the water. And if it is lower than 1, it will float on the water.