1. Write the reaction conceptually

Dimethyldichlorosilane is: $\mathrm{(CH_3)_2SiCl_2}$

On hydrolysis with water, it first forms silanol and then condenses to give a tetrameric cyclic siloxane: $\left[(\mathrm{CH_3})_2\mathrm{SiO}\right]_4$

So, the cyclic tetramer $X$ has formula: $\mathrm{[(CH_3)_2SiO]_4}$

2. Find molar mass of dimethyldichlorosilane

For $\mathrm{(CH_3)_2SiCl_2}$:

• $2\times \mathrm{CH_3} = 2\times(12+3\times1)=2\times15=30$
• $\mathrm{Si}=28$
• $2\mathrm{Cl}=2\times35.5=71$

Therefore, $M\big(\mathrm{(CH_3)_2SiCl_2}\big)=30+28+71=129\ \mathrm{g\,mol^{-1}}$

3. Moles of dimethyldichlorosilane used

Given mass = $516\,\mathrm{g}$ $n=\frac{516}{129}=4\ \mathrm{mol}$

4. Stoichiometry for tetramer formation

Each tetramer contains 4 units of $\mathrm{(CH_3)_2SiO}$, so: $4\ \text{mol of } \mathrm{(CH_3)_2SiCl_2} \rightarrow 1\ \text{mol of } X$

Hence, from $4$ mol starting compound, theoretical moles of $X$ formed: $n_X=1\ \mathrm{mol}$

5. Molar mass of tetrameric cyclic product $X$

One repeat unit $\mathrm{(CH_3)_2SiO}$ has mass:

• $2\times \mathrm{CH_3}=30$
• $\mathrm{Si}=28$
• $\mathrm{O}=16$

So, $M\big(\mathrm{(CH_3)_2SiO}\big)=30+28+16=74$

For tetramer: $M(X)=4\times74=296\ \mathrm{g\,mol^{-1}}$

6. Theoretical mass of $X$

Since theoretical moles of $X=1$ mol, $m_{\text{theoretical}}=1\times296=296\ \mathrm{g}$

7. Actual mass at 75% yield

$m_{\text{actual}}=0.75\times296=222\ \mathrm{g}$

8. Final answer

$\boxed{222}$

The derived answer matches the stored correct answer.