Question

The diameter of a copper atom is approximately$\mathbf{2}\mathbf{.}\mathbf{28}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{10}}\mathbf{m}$. The mass of one mole of copper is 64g. Assume that the atoms are arranged in a simple cubic array. Remember to convert to SI units. (a) What is the mass of one copper atom, in kg? (b) How many copper atoms are there in a cubical block of copper that is 4.6 cmon each side? (c) What is the mass of the cubical block of copper, in kg?

Short answer

1. The mass of one copper atom is$1.06\times {10}^{-25}\mathrm{kg}/\mathrm{atom}$
2. Number of copper atoms in the cubical block$8.12\times {10}^{24}\mathrm{kg}/\mathrm{atom}$
3. Mass of the copper cubical block 0.680 kg

Step-by-step solution

Identification of given data

The given data can be listed below as:

• Diameter of the copper atom is$\mathrm{d}=2.28\times {10}^{-10}\mathrm{m}$
• Mass of one mole of copper is${\mathrm{m}}_1=64\mathrm{kg}\times \left(\frac{{10}^{-3}\mathrm{kg}}{1\mathrm{g}}\right)=0.064\mathrm{kg}$
• The Avogadro’s number is$N=6.023\times {10}^{23}$
• The side of the cube is, $l=4.6\mathrm{cm}\times \left(\frac{{10}^{-3}\mathrm{m}}{1\mathrm{cm}}\right)=0.046\mathrm{cm}$$l=4.6\mathrm{cm}\times \left(\frac{{10}^{-3}\mathrm{m}}{1\mathrm{cm}}\right)=0.046\mathrm{cm}$

Significance of the mass of the cubical block

The mass of one atom of an object is described as dividing the mass of one mole of that object and the Avogadro’s number. Moreover, the mass of a cubical block is expressed as the product of the number of atoms inside the cubical block and the mass of one atom.

(a) Calculation for the mass of one copper atom

The equation of the mass of one copper is expressed as:

$M=\frac{{\mathrm{m}}_1}{\mathrm{N}}$

Here, ${\mathit{m}}_{\mathbf{1}}$is the mass of one mole of copper and N is the Avogadro’s number.

Substitute all the values in the above equation.

$$\begin{gathered} M=\frac{0.064\mathrm{kg}}{6.023\times {10}^{23}} \\ =1.06\times {10}^{23}\mathrm{kg}/\mathrm{atom} \end{gathered}$$

Thus, the mass of one copper atom is $1.06\times {10}^{-25}\mathrm{kg}/\mathrm{atom}$.

(b) Calculation for the copper atoms in the cubical block

The equation of the number of the atoms in one cubical row is expressed as:

$$\begin{gathered} l=zd \\ z=\frac{l}{d} \end{gathered}$$

Here, l is the side of the cube, z is the number of atoms in one cubical row, and d is the diameter of the cube.

Substitute the values in the above equation.

role="math" localid="1656669222590" $$\begin{gathered} z=\frac{0.046\mathrm{m}}{2.28\times {10}^{-10}\mathrm{m}} \\ =2.01\times {10}^8 \end{gathered}$$

As the copper atom consists of three cubical blocks, hence the equation of the number of atoms in the cubical block is expressed as:

$a=z^3$

Here, a is the number of atoms in the cubical block and z is the number of atoms in one cubical row.

Substitute the value in the above equation.

role="math" localid="1656669407115" $$\begin{gathered} a={(2.01\times {10}^8\mathrm{atom})}^3 \\ =8.12\times {10}^{24}\mathrm{atom} \end{gathered}$$

Thus, the number of atoms in the cubical block is $8.12\times {10}^{24}\mathrm{atom}$.

(c) Calculation for the mass of the copper cubical block

The equation of the mass of the copper cubical block is:

m = aM

Here, m is the mass of the copper cubical block, a is the number of atoms in the cubical block, and M is the mass of one copper atom.

Substitute the values in the above equation.

$$\begin{gathered} m=8.12\times {10}^{24}\mathrm{atom}\times 1.06\times {10}^{-25}\mathrm{kg}/\mathrm{atom} \\ =0.860\mathrm{kg} \end{gathered}$$

Thus, the total mass of the cubical block is 0.860 kg.