Obtain the moles of C from CO₂ and the moles of H atoms from H₂O. Then use the STP molar volume to determine the gas molar mass.
\[\begin{aligned}n(C) &= n(CO_2)\\&= \frac{3.38}{44.0}\\&= 0.0768\ \mathrm{mol}\end{aligned}\]
\[\begin{aligned}n(H) &= 2n(H_2O)\\&= 2\left(\frac{0.690}{18.0}\right)\\&= 0.0767\ \mathrm{mol}\end{aligned}\]
\[C:H\approx1:1\Rightarrow\text{empirical formula }CH\]
\[\begin{aligned}\rho &= \frac{11.6}{10.0}\\&= 1.16\ \mathrm{g\,L^{-1}}\end{aligned}\]
\[\begin{aligned}M &= \rho V_m\\&= (1.16)(22.4)\\&= 26.0\ \mathrm{g\,mol^{-1}}\end{aligned}\]
\[\begin{aligned}n &= \frac{26.0}{13.0}\\&= 2\Rightarrow(CH)_2\\&= C_2H_2\end{aligned}\]
Final answers by part(a)\[\text{Empirical formula }CH
(b)\quad M=26.0\ \mathrm{g\,mol^{-1}}
(c)\quad\text{molecular formula }C_2H_2\]