The mass % of [tex]Gd[/tex] = 86.76 % (given)
The mass % of [tex]O[/tex] = 100 - 86.76 = 13.24 %
Considering the mass of Gadolinium oxide as 100 g.
Then 86.76 mass % [tex]Gd[/tex] means the compound contains 86.76 g of [tex]Gd[/tex] in 100 g of Gadolinium oxide.
So, 13.24 mass % [tex]O[/tex] means the compound contains 13.24 g of [tex]O[/tex] in 100 g of Gadolinium oxide.
Using formula of number of moles:
[tex]Number of moles = \frac{mass}{Atomic mass}[/tex]
The atomic mass of [tex]Gd = 157.25 u[/tex]
[tex]Number of moles = \frac{86.76}{157.25} = 0.5517 mole[/tex]
The atomic mass of [tex]O = 16 u[/tex]
[tex]Number of moles = \frac{13.24}{16} = 0.8275 mole[/tex]
In order to get the empirical formula, the moles are divide by the lowest value that is 0.5517.
[tex]Gd_{\frac{0.5517}{0.5517}} = Gd[/tex]
[tex]O_{\frac{0.8275}{0.5517}} = O_{1.499}\simeq O_{1.5}[/tex]
So, the formula is [tex]GdO_{1.5}[/tex], will multiply by 2 to get the whole integer.
Hence, the empirical formula is [tex]Gd_2O_{3}[/tex].