Question

Suppose you are given a cube made of magnesium (mg) metal of edge length 2.15 cm. (a) calculate the number of mg atoms in the cube. (b) atoms are spherical in shape. therefore, the mg atoms in the cube cannot fill all the available space. if only 74 percent of the space inside the cube is taken up by mg atoms, calculate the radius in picometers of an mg atom. (the density of mg is 1.74 g/cm3, and the volume of a sphere of radius r is 4/3π r3.) enter your answers in scientific notation.

Answer 1

solution:

a) To calculate the volume of cube of edge 1.0cm

Volume= (edge)³

=(100cm) ³

= 1cm³

To find the mass of MG atoms  

Density = mass/volume

Mass of mg = 1.74g/cm³ x 1cm³

       = 1.74g

To find the number of mg atoms,

Molar mass of mg = 24.31g

24.31g mg contain 6.022 x 10²³ atoms of mg  

1.74g mg contain = 6.022 x 10²³mg atoms/24.31g mg x 1.74g mg

    = 0.4310 x 10²³mg atoms

    = 4.31 x 10²² mg atoms

B) to find the volume occupied by mg atoms  

    Total volume of cube = 1 cm³

    Volume occupied by mg atoms  

  = 74% of 1cm³

=74/100 x 1cm³= 0.74cm³

To find the volume of 1 mg atom  

4.31 x 10²² mg atoms occupy 0.74 cm³

1 atom will occupy 0.74cm³

1 atom will occupy = 0.74cm³/4.31 x 10²²mg atoms x 1mg atom

volume of 1 mg atom = 0.1716 x 10⁻²²cm²

Volume of 1 mg atom = 4/3 π³ in the form of sphere.

4/3πr³ = 0.1716 x 10⁻²²cm³

R³ = 0.1716 x 10⁻²²cm³x 4/3 x 7/22

R³ = 0.04095 x 10⁻²²cm³

R = 0.1599 x 10⁻⁷cm

1 = 10⁻¹⁰cm

 = 0.1599 x 10⁻⁷cm

 = 1pm/1 x 10⁻¹⁰cm  x 0.1599 x 10-7cm

= 1.599 x 10² pm  

R= 1.6 x 10²pm