Question

An oxide of uranium contains 84.8% of uranium. The empirical formula of the oxide is ? ( atomic mass of U = 238 , O = 16)

(a) UO₂
(b) U₂O₃
(c) U₃O₄
(d) U₃O₈

Answer 1

see the solution in the picture

Answer 2

Answer: $(d) \, \, \bold{U_3O_8}$


An Empirical Formula is the formula of simplest ratios of elements in a compound.


For example, for Ethane $C_2H_6$, the empirical formula is $CH_3$.


This is the ratio of moles. That is for one mole of Carbon, there are three moles of Oxygen. 

So, we see that in order to calculate Empirical Formula, we must know the ratio of moles of elements in the compounds.

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We have a compound which consists of 84.8 % Uranium-238.

That means (100-84.8) = 15.2 % is Oxygen.

Consider that we have 100 grams of the compound. Then this sample consists of 84.8 grams of Uranium-238 and 15.2 grams of Oxygen-16

We can now find the number of moles in this 100 gram sample. 


$\text{No. of moles} = \frac{\text{Given Mass}}{\text{Molar Mass}}$

So, we have:

$\begin{array}{ccl}\text{No. of moles of U}& =& \frac{84.8 \, \, g}{238 \, \, g/mol} \\ \\& \approx & 0.356 \, \, moles \end{array}$


And similarly

$\begin{array}{ccl}\text{No. of moles of O}& =& \frac{15.2 \, \, g}{16 \, \, g/mol} \\ \\& = & 0.95 \, \, moles \end{array}$


So, we see that the ratio of Uranium and Oxygen is:

$U : O = 0.356 : 0.95$

This ratio is not so useful. We need ratios in integer form to make conclusions. In order to get integer (or near integer ratios) we divide both by the smallest number available.


$\begin{array}{ccl} U : O & = & 0.356 : 0.95 \\ \\ & = & \frac{0.356}{0.356} : \frac{0.95}{0.356} \\ \\ & \approx & 1 : 2.668 \end{array}$


2.668 is no close to an integer. So we know that U and O are not in 1 : something ratio. 

In order to bring both numbers into an integer, we now multiply them we larger integers.

[For example, if our ratio was 1 : 0.5, then we would multiply both by 2, so our ratio would become 2 : 1]

Here, we need to multiply 2.668 with some integer that would bring it close to an integer.
We see:

$2.668 \times 2 = 5.336 \quad \text{Not Useful} \\ \\ 2.668 \times 3 = 8.004 \approx 8\quad {Useful}$

So, we can multiply the ratios by 3 to bring them to whole numbers:

$U : O = 1 : 2.668 \\ \\ \implies U : O = (1\times 3) : (2.668 \times 3) \\ \\ \implies \boxed{U : O = 3 : 8}$

Finally, we have integral ratios. We see that for every three moles of Uranium, eight moles of Oxygen are required. 

Thus, our Empirical Formula is: $\bold{U_3O_8}$


So, the Answer is Option (d)