The pK₆ of dimethylamine and pKₐ of acetic acid are 3.27 and 4.77 respectively at T (K). The correct option for the pH of dimethylammonium acetate solution is :

The correct option for the pH of the dimethylammonium acetate solution is 7.75.

Step-by-Step Explanation:

Dimethylammonium acetate is a salt formed by the reaction of a weak base, dimethylamine ($({\mathrm{CH}}_3{)}_2\mathrm{NH}$), and a weak acid, acetic acid (${\mathrm{CH}}_3\mathrm{COOH}$).

For a salt of a weak acid (WA) and a weak base (WB), the pH of the aqueous solution is independent of its concentration (at moderate concentrations) and is given by the standard formula:
$\mathrm{pH}=7+\frac{1}{2}({\mathrm{pK}}_a-{\mathrm{pK}}_b)$

Here, we are given:
- The ${\mathrm{pK}}_b$ of dimethylamine = 3.27
- The ${\mathrm{pK}}_a$ of acetic acid = 4.77

Now, substitute these values directly into the pH formula:
$\mathrm{pH}=7+\frac{1}{2}(4.77-3.27)$

Calculate the difference inside the parentheses:
$4.77-3.27=1.50$

Substitute this difference back into the equation:
$\mathrm{pH}=7+\frac{1}{2}\left(1.50\right)$
$\mathrm{pH}=7+0.75$
$\mathrm{pH}=7.75$

Thus, the pH of the dimethylammonium acetate solution is 7.75, which matches the third option.