If a mass of #"16.26 mg"# contains #1.66 xx 10^20# atoms, what is the molar mass of that atom? If you have #"1 million"# silver atoms constituting a total of #1.79 xx 10^(-16) "g"#, what is the molar mass? #"1 amu"# #=# #1.660599 xx 10^(-24) "g"#.

Both questions ask the same thing in opposite ways.

One gives you #"16.26 mg"# containing #1.66xx10^20# #"atoms"#. That gives you a mass and a quantity, which is enough to convert to the atomic units of #"g/mol"#.

#("16.26" cancel"mg")/(1.66xx10^20 cancel"atoms") xx "1 g"/("1000" cancel"mg") xx (6.0221413xx10^23 cancel"atoms")/"mol"#

#=# #color(blue)("58.988 g/mol")#,

which is close enough to cobalt, whose accepted value is #"58.933 g/mol"#, and our value is off by only #0.093%#.

The other gives you #"1 million"# silver atoms weighing #1.79xx10^(-16) "g"#. That gives you a quantity and a mass, which is enough to convert to the atomic units of #"g/mol"#, again.

#(1.79xx10^(-16) "g")/(10^6 cancel"Ag atoms")xx(6.0221413xx10^23 cancel"atoms")/"mol"#

#=# #color(green)("107.796 g/mol")#

which is close enough to the accepted value of #"107.868 g/mol"# (#0.066%# difference).

In either case you still get #\mathbf("g/mol")#.

WHAT IS AMU?

You are not required to use the conversion

#"1 g"/(6.0221413xx10^(23) "atoms") = "1 amu"#

#= 1.660599xx10^(-24) "g",#

unless you want to. It just depends on whether you are talking about #"1 mol"# of atoms or #"1 atom"#, and you need to choose which one is more convenient for you.

CONVERTING BETWEEN G/MOL and AMU

You can still interconvert between #"g/mol"# and #"amu"#...

#(58.988 cancel"g")/cancel"mol"xxcancel"1 mol"/(6.0221413xx10^23 cancel"atoms") xx "1 amu"/(1.660599xx10^(-24) cancel"g")xxcancel("1 atom")#

#~~ color(blue)(58.988)# #color(blue)("amu for one atom")#

i.e. The experimental atomic mass of #"1 atom"# of cobalt according to the given information is #"58.988 amu"#, while the molar mass calculated at the top of the answer is #"58.988 g/mol"#.

So, #"amu"# is just a way of giving the mass of one atom in atomic units (#"1 amu"# #ne# #"1 g"#), while #"g/mol"# gives you the mass of #\mathbf("1 mol")# of atoms.