A cube of gold that is 1.00 cm on a side has a mass of 19.3 g. A single gold atom has a mass of 197.0 amu. (a) How many gold atoms are in the cube? (b) From the information given, estimate the diameter in Å o ? f a single gold atom. (c) What assumptions did you make in arriving at your answer for part (b)?
Solution : Step 1 : Gold is a chemical element having symbol Au. It has an atomic number of 79. Here, in the problem given : Side of gold cube = 1.00cm. Mass of gold cube = 19.3g. Mass of gold atom = 197.0 amu. Step 2: (a) How many gold atoms are in the cube Number of gold atoms = Mass of Cube Mass of Atom Here, ass of gold atom is given in amu and ass of gold cube is given in grams. So, let’s convert amu to grams a nd proceed with further calculation. 1 amu = 1.6605 × 10 -24g Therefore, mass of gold atom in amu is: -22 197amu = 3.271 × 10 g. Hence, number of gold atoms : -22 = 5.90 × 10 atoms. Step 3: (b) From the information given, estimate the diameter in Å of a single gold atom. To find the diameter of one single gold atom, first we need to find how many atoms are there in one edge of the cube. Diameter is 2 times the radius. So, we find radius of the atom using the formula to find volume of the sphere as atom is spherical in shape. Formula for volume of sphere : V = r where, v = volume, r = radius and = 22/7 or 3.1415 3 To proceed with further calculation we need to find the volume of cube : Volume of cube = a . 3 Where, a = side We have given the side of gold cube. 3 3 Therefore, volume of cube is = (1.0cm) = 1.0cm . Now, let’s find volume of one atom. volume of cube Volume of one atom = Number of atoms = 1.694 × 10 -23cm / atom.