There are two compounds of titanium and chlorine. One compound contains 31.04% titanium by mass, and the other contains 74.76% chlorine by mass. What are the ratios of titanium and chlorine atoms in the two compounds?

The idea here is that you need to use the percent composition of the two elements in the two compounds to find their mole ratios.

Once you have their mole ratios, you also have the ratio of the number of atoms each element contributes to the one mole of compound.

So, the first compound is known to contain #"31.04%"# titanium by mass, which implies that it also contains

#100 - 31.04 = 68.96%#

chlorine by mass.

To make the calculations as simple as possible, pick a #"100-g"# sample of the compound. The mass of titanium and the mass of chlorine in this sample will be

#100color(red)(cancel(color(black)("g compound"))) * "31.04 g Ti"/(100color(red)(cancel(color(black)("g compound")))) = "31.04 g Ti"#

and

#100color(red)(cancel(color(black)("g compound"))) * "68.96 g Cl"/(100color(red)(cancel(color(black)("g compound")))) = "68.96 g Ti"#

Use titanium and chlorine's molar masses to get how many moles of each you get

#31.04color(red)(cancel(color(black)("g Ti"))) * "1 mole Ti"/(47.867color(red)(cancel(color(black)("g Ti")))) = "0.6485 moles Ti"#

and

#68.96color(red)(cancel(color(black)("g Cl"))) * "1 mole Cl"/(47.867color(red)(cancel(color(black)("g Cl")))) = "1.945 moles Cl"#

Divide both numbers by the smallest one to get mole ratio of the two elements in the compound

#"For Ti: " (0.6485color(red)(cancel(color(black)("moles"))))/(0.6485color(red)(cancel(color(black)("moles")))) = 1#

#"For Cl: " (1.945color(red)(cancel(color(black)("moles"))))/(0.6485color(red)(cancel(color(black)("moles")))) = 2.999 ~=3#

The empirical formula of the compound, which tells you the mole ratio of the two elements in the compound, is

#"TiCl"""_3#

Now, this will also be the ratio of atoms, because you know that one mole is equal to #6.022 * 10^(23)# atoms - this is known as Avogadro's number.

This means that you have

#"atoms of Ti"/"atoms o Cl" = (1 * color(red)(cancel(color(black)(6.022 * 10^(23)"atoms"))))/(3 * color(red)(cancel(color(black)(6.022 * 10^(23)"atoms")))) = color(green)(1/3)#

The exact same approach can be used to find the atoms ratio for the second compound, so I'll skip the detailed calculations.

Once again, pick an #"100-g"# sample to get #"25.24 g Ti"# and #"74.76 g Cl"#.

Use their molar masses to get

#"0.5273 moles Ti "# and #" 2.109 moles Cl"#

Their mole ratio will now be

#"For Ti: "1" "# and #" For Cl: "3.999 ~= 4#

The ratio of their atoms will thus be

#"atoms of Ti"/"atoms o Cl" = (1 * color(red)(cancel(color(black)(6.022 * 10^(23)"atoms"))))/(4 * color(red)(cancel(color(black)(6.022 * 10^(23)"atoms")))) = color(green)(1/4)#