Which of the following statement (s) is/are correct?
(A)The pH of 1 x 10^-8M HCl solution is 8
(B)The conjugate base of H₃PO₄^- is HPO₄^2-
(C) K_w increases with increase in temperature
(D) When a solution of a weak monoprotic acid is titrated against a strong base At half
neutralization point, $\mathrm{pH}=\frac{1}{2}{\mathrm{pK}}_{\mathrm{a}}$

(A) The pH of   1×10⁻⁸ M HCl solution is 8.

This statement is incorrect. For a strong acid like HCl, the concentration of H⁺ ions will be the same as the concentration of the acid, i.e.,  1×10⁻⁸ M. The pH can be calculated using the formula:

pH=−log⁡[H⁺]=−log⁡(1×10⁻⁸)=8

However, because the concentration is so low, it approaches the range where water auto-ionization becomes significant. In this case, the solution pH will be slightly higher than 7, but not exactly 8.

(B) The conjugate base of ${\mathrm{H}}_2{\mathrm{PO}}_4^{-}$ is ${\mathrm{HPO}}_4^{2-}$.

This statement is correct. The conjugate base of an acid is formed when it loses one H⁺ ion:

${\mathrm{H}}_2{\mathrm{PO}}_4^{-}$→${\mathrm{HPO}}_4^{2-}$ + H⁺

(C) K_w increases with an increase in temperature.

This statement is correct. The ion product of water, K_w, increases with increasing temperature. This is because the auto-ionization of water is an endothermic process, meaning it absorbs heat:

H₂O⇌H⁺ + OH⁻

As the temperature increases, the equilibrium shifts towards the formation of more H⁺ and OH⁻ ions, leading to an increase in K_w.

(D) When a solution of a weak monoprotic acid is titrated against a strong base at the half-neutralization point, pH=$\frac{1}{2}$pK_a

This statement is incorrect. At the half-neutralization point, the concentration of the weak acid ([HA]) is equal to the concentration of its conjugate base ([A^-]). According to the Henderson-Hasselbalch equation:

pH=pK_a+log⁡$\frac{\left[{\mathrm{A}}^{-}\right]}{\left[\mathrm{HA}\right]}$

At the half-neutralization point, the ratio of [A^-] to [HA] is 1, so the equation becomes:

pH=pK_a + log(1) =pK_a

Therefore, the correct answer is:

(B) and (C) are correct.