At 298K, a certain buffer solution contains equal concentrations of X⁻ and HX. If K_b for X⁻ is 10⁻¹⁰, what is the pH of this buffer solution?

We are given a buffer that contains equal concentrations of the base X⁻ and its conjugate acid HX at 298 K. The base dissociation constant is

$K$_b = 10^-10

For a base‑acid conjugate pair the expression for K_b is

$K$_b = \frac{[OH⁻][HX]}{[X⁻]}

Because the buffer has equal concentrations, the ratio \(\frac{[HX]}{[X⁻]}\) is 1, so the expression simplifies to

$[OH⁻]\; =\; K$_b = 10^-10 \; \text{M}

At 298 K the water ion‑product constant is

$K\_\{\backslash mathrm\; w\}\; =\; [H⁺][OH⁻]\; =\; 1.0\; \backslash times\; 10^\{-14\}$

The pOH is obtained from the hydroxide concentration:

$\backslash mathrm\{pOH\}=\; -\backslash log[OH⁻]\; =\; -\backslash log(10^\{-10\})\; =\; 10$

Finally, the pH is related to the pOH by

$\backslash mathrm\{pH\}=\; 14\; -\; \backslash mathrm\{pOH\}\; =\; 14\; -\; 10\; =\; 4$

Thus the pH of the buffer solution is 4.