To find the pH of a dimethylammonium acetate solution, we need to consider the properties of the salts formed from a weak acid and a weak base.$\\$Given:$\\$• $\mathrm{pK}_b$ of dimethylamine $= 3.27 \\$• $\mathrm{pK}_a$ of acetic acid $= 4.77 \\$Dimethylammonium acetate is a salt formed by the reaction of dimethylamine (a weak base) and acetic acid (a weak acid).$\\$The pH of a solution of such a salt can be calculated using the formula:$\\$pH$= \frac{1}{2} (\mathrm{pK}_a + \mathrm{pK}_b) \\$Substitute the given values:$\\$pH$= \frac{1}{2} (4.77 + 3.27) \\$Calculate the sum:$\\ 4.77 + 3.27 = 8.04 \\$Now, calculate the average:$\\$pH$= \frac{8.04}{2} \\$Simplify:$\\$pH$= 4.02 \\$However, this calculation seems incorrect based on the options provided. Let's re-evaluate the approach:$\\$For a salt of a weak acid and a weak base, the correct formula to use is:$\\$pH$= 7 + \frac{1}{2} (\mathrm{pK}_a - \mathrm{pK}_b) \\$Substitute the given values:$\\$pH$= 7 + \frac{1}{2} (4.77 - 3.27) \\$Calculate the difference:$\\ 4.77 - 3.27 = 1.50 \\$Now, calculate the adjustment:$\\$pH$= 7 + \frac{1.50}{2} \\$Simplify:$\\$pH$= 7 + 0.75 \\$Final result:$\\$pH$= 7.75 \\$Therefore, the correct option for the pH of the dimethylammonium acetate solution is 7.75, which corresponds to Option 3.