Question

The edge length of an FCC crystal is defined as $100pm$ and its density is given as $10gc{{m}^{-3}}$. Find the number of atoms present in $2g$ of this crystal.

Answer 1

Recall the formula that is used to calculate the density and relates the number of atoms per unit cell, the molar mass, the Avogadro’s number and the edge length of the unit cell.

Complete answer:
To calculate the number of atoms, in any given mass of crystals, we will need to know the number of moles of the crystal present and the Avogadro’s number. The generalized formula for that will be:
$N=n\times {{N}_{A}}$
Here, $N$ is the number of atoms present, $n$ is the number of moles of the crystal, and ${{N}_{A}}$ is the Avogadro’s number which is $6.022\times {{10}^{23}}$.
We know that the number of moles of any substance is given by the given weight of the substance divided by its molecular mass. So, we can replace the number of moles of substance in this formula by those variables. So, the formula will be:
$i)N=\dfrac{m}{M}\times {{N}_{A}}$
Where, $m$ is the given mass of the crystal, which is $2g$, and $M$ is the molar mass of the crystal.
Now, we know the formula that we use to calculate the density of a unit cell, it is as follows:
$d=\dfrac{z\times M}{{{a}^{3}}\times {{N}_{A}}}$
Where, $z$ is the number of atoms per unit cell - which is defined as 4 for an FCC crystal, $a$ is the edge length of the unit cell which is given to us as $100pm$, and $d$ is the density which is given as $10gc{{m}^{-3}}$.
Now we will rearrange this equation for the molar mass of the molecules of element and we get the formula:
$ii)M=\dfrac{d\times {{a}^{3}}\times {{N}_{A}}}{z}$
Now, we will put ii) in i), substitute the values and solve for $N$. The process will be as follows:

& N=\dfrac{m\times {{N}_{A}}}{\dfrac{d\times {{a}^{3}}\times {{N}_{A}}}{z}} \\\ & N=\dfrac{m\times z}{d\times {{a}^{3}}} \\\ & N=\dfrac{2g\times 4}{10gc{{m}^{-3}}\times {{\left( {{10}^{-8}}cm \right)}^{3}}} \\\ & N=\dfrac{8}{{{10}^{-23}}} \\\ & N=8\times {{10}^{23}} \\\ \end{aligned}$$ Hence, we can say that the number of atoms in $2g$ of the given crystal are $8\times {{10}^{23}}$. **Note:** Here, while putting the value of $a$ in the final equation, we have converted the value from picometers to centimeters and the factor of ${{10}^{-8}}$ has appeared because of that. 1 picometer is equal to ${{10}^{-10}}$ centimeters. We have converted the value to centimeters since the density is also given in centimeters. But we can easily convert all the units to SI units and solve the equation so that there is no room for error. Always make sure that the units match before solving any equation.