We're asked to find the total number of atoms in 145 "mg" of caffeine, with its given chemical formula.
Let's first do a simple conversion from milligrams to grams, because we'll be using molar mass calculations in a bit:
145cancel("mg")((1"g")/(10^3cancel("mg"))) = 0.145"g"
Now, let's calculate the molar mass of caffeine, using the molar masses of the individual elements and how many of each element is in the compound:
overbrace((8)(12.01"g/mol"))^"C" + overbrace((10)(1.01"g/mol"))^"H" + overbrace((4)(14.01"g/mol"))^"N" + overbrace((2)(16.00"g/mol"))^"O"
= color(red)(194.22 color(red)("g/mol"
Now, let's convert the given mass (0.145 "g") to moles using this molar mass:
0.145cancel("g C"_8"H"_10"N"_4"O"_2)((1"mol C"_8"H"_10"N"_4"O"_2)/(color(red)(194.22)cancel(color(red)("g C"_8"H"_10"N"_4"O"_2))))
= 7.466 xx 10^-4 "mol C"_8"H"_10"N"_4"O"_2"
Now, using Avogadro's number, let's convert this mole number to the number of caffeine molecules:
7.466 xx 10^-4
cancel("mol C"_8"H"_10"N"_4"O"_2)((6.022xx10^23"molecules C"_8"H"_10"N"_4"O"_2)/(1cancel("mol C"_8"H"_10"N"_4"O"_2")))
= 4.496 xx 10^20 "molecules C"_8"H"_10"N"_4"O"_2
In one molecule of caffeine, there are 8 ("C") + 10 ("H") + 4 ("N") + 2 ("O") = color(green)(24 sfcolor(green)("atoms", so therefore,
4.496 xx 10^20 cancel("molecules C"_8"H"_10"N"_4"O"_2)((color(green)(24"atoms"))/(1cancel("molecule")))
= color(blue)(1.08 xx 10^22 color(blue)("atoms"
rounded to 3 significant figures, the amount given in the problem.
Therefore, there are color(blue)(1.08 xx 10^22 atoms in 145 "mg" of caffeine.
