Question

How many iron atoms are present in a stainless steel ball bearing having a radius of $ 0.1 $ inch $ \,\left( 1\text{ }inch=2.54cm \right) $ ? They stainless steel contains $ 85-6%\text{ }Fe $ by weight and has a density of $ 7.75\text{ }g/cc. $

Answer 1

Hint: We know that first of all we need to calculate the density of steel using the formula for specific gravity. Then using density and volume of the sphere we will get the value of mass. The mass of iron balls will be the same as the mass of steel balls.

Complete answer:
We know that the 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 unit less quantity. 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. The density thus calculated is used to get the atoms present by using volume and moles as well.
Here we have radius $ =0.254\text{ }cm $ and Volume of ball $ =\dfrac{4}{3}\pi {{r}^{3}} $
On substituting the values in above equation we get; $ \dfrac{4}{3}\times 3.14\times {{\left( 0.254 \right)}^{3}}=0.068\text{ }c{{m}^{3}}. $
Thus, mass of $ 0.068\text{ }c{{m}^{3}}=7.75\times 0.068=0.532\text{ }g $
Therefore, the percentage of iron present in the ball $ =\dfrac{85.6}{100}\times 0.532=0.455g $
The number of moles of iron that can be accommodated in one steel ball can be calculated by dividing the mass of the ball to the molar mass of iron. The molar mass of iron is $ 56. $ Number of moles present in $ 0.455\text{ }g\text{ }Fe=\dfrac{0.455}{56}=0.0082\text{ }mol. $
Therefore, number atoms in $ 0.0082\text{ }mol=0.0082\times 6.022\times {{10}^{23}}=4.89\times {{10}^{21}}\text{ }atoms. $ .

Note:
Remember that the specific gravity is defined as the ratio of density of the substance to the ration of the density of water. It is also known as the relative density as it is not absolute. It is calculated with respect to the density of water. Specific gravity tells us whether an object will float or sink. If the specific gravity of an element is greater than that of water.