$\text{In a metal deficient oxide sample, }{M}_x{Y}_2{O}_4\left(M\text{ and}\mathrm{ Y}\text{ are metals}\right).M\text{ is present in both}+2\text{ and}+3$

$\text{oxidation states and}\mathrm{ Y}\text{ is in}+3\text{ oxidation state.}\text{If the fraction of}{\mathrm{ M}}^{2+}\text{ions present in}\mathrm{ M}\text{ is }\frac{1}{\mathrm{ 3}},\text{the}$

value of X is ________.

The correct answer is 0.75.

Step 1: Analyze the chemical formula and oxidation states
We are given a metal-deficient oxide sample with the formula:

$M_x{Y}_2{O}_4$

Here, the oxidation states of the components are:
• Oxide ion ($O^{2-}$) has an oxidation state of -2.
• Metal $Y$ is present in the +3 oxidation state.
• Metal $M$ is present in both +2 and +3 oxidation states, i.e., as $M^{2+}$ and $M^{3+}$.

Step 2: Determine the average oxidation state of metal M
We are given that the fraction of $M^{2+}$ ions present in $M$ is $\frac{1}{3}$.
Therefore, the fraction of $M^{3+}$ ions in $M$ is:

$1-\frac{1}{3}=\frac{2}{3}$

Now, we can calculate the average oxidation state of metal $M$:

$\text{Average Oxidation State of}M=\left(\frac{1}{3}\times (+2)\right)+\left(\frac{2}{3}\times (+3)\right)$

$\text{Average Oxidation State of}M=\frac{2}{3}+2=\frac{8}{3}$

Step 3: Apply the principle of electrical neutrality
For the neutral compound $M_x{Y}_2{O}_4$, the total positive charge must equal the total negative charge:

$x\times \left(\text{Average Oxidation State of},M\right)+2\times \left(\text{Oxidation State of},Y\right)+4\times \left(\text{Oxidation State of},O\right)=0$

Substitute the respective values into the equation:

$x\left(\frac{8}{3}\right)+2(+3)+4(-2)=0$

Simplify the equation step-by-step:

$\frac{8x}{3}+6-8=0$

$\frac{8x}{3}-2=0$

$\frac{8x}{3}=2$

$8x=6$

$x=\frac{6}{8}=0.75$

Thus, the value of x is 0.75.