The first step we have to follow is to convert the units of each of the measurements (mL to L, mmHg to atm and °C to K):
[tex]\begin{gathered} 100mL\cdot\frac{1L}{1000mL}=0.1L \\ 688mmHg\cdot\frac{1atm}{760mmHg}=0.91atm \\ K=565+273.15=838.15K \end{gathered}[/tex]Now, use the ideal gases law to find the number of moles of carbon dioxide at these conditions:
[tex]\begin{gathered} P\cdot V=n\cdot R\cdot T \\ n=\frac{P\cdot V_{}}{R\cdot T} \\ n=\frac{0.91atm\cdot0.1L}{0.082\frac{atmL}{molK}\cdot838.15K} \\ n=0.00132mol \end{gathered}[/tex]Now, use the chemical equation to find how many moles of acetic acid are needed to produce that amount of carbon dioxide:
[tex]CH_3COOH+NaHCO_3\to CH_3COONa+CO_2+H_2O[/tex]Use the stoichiometric ratio of moles of acetic acid used to moles of carbon dioxide produced:
[tex]0.00132molCO_2\cdot\frac{1molCH_3COOH}{1molCO_2}=0.00132molCH_3COOH[/tex]Use the molar mass of acetic acid to convert this amount of moles to mass:
[tex]0.00132molCH_3COOH\cdot\frac{60.1gCH_3COOH}{1molCH_3COOH}=0.079gCH_3COOH[/tex]Finally use the density of acetic acid to find volume of 0.079g of this compound:
[tex]0.079g\cdot\frac{1ml}{1.05g}=0.08292ml[/tex]It means that 0.08292 would be needed to obtain the sample of carbon dioxide.