At 0^ ^ ^ C , a glass vessel has exact dimensions of 10 cm × 10 cm...

The initial volume of the glass vessel and mercury is $V_0 = 10 \times 10 \times 10 = 1000 \text{ cm}^3$.

The change in temperature is $\Delta T = 10^{{}^{{}^{\circ}}}\text{C} - 0^{{}^{{}^{\circ}}}\text{C} = 10^{{}^{{}^{\circ}}}\text{C}$.

The coefficient of volume expansion of glass is \gamma_g = 3\alpha_g = 3 \times 6.5 \times 10^{-6} = 19.5 \times 10^{-6} ^{{}^{{}^{\circ}}}\textC^{-1}.

The volume of mercury that overflows is given by the apparent expansion of mercury:

$\Delta V = V_0 \gamma_{app} \Delta T$

Substituting the given values:

$1.6 = 1000 \times \gamma_{app} \times 10$

\gamma_{app} = \dfrac{1.6}{10000} = 1.6 \times 10^{-4} ^{{}^{{}^{\circ}}}\textC^{-1}

The apparent coefficient of volume expansion is related to the absolute coefficient of volume expansion of mercury ($\gamma_{Hg}$) and that of glass ($\gamma_g$) by:

$\gamma_{app} = \gamma_{Hg} - \gamma_g$

$\gamma_{Hg} = \gamma_{app} + \gamma_g$

$\gamma_{Hg} = 1.6 \times 10^{-4} + 19.5 \times 10^{-6}$

\gamma_{Hg} = 1.6 \times 10^{-4} + 0.195 \times 10^{-4} = 1.795 \times 10^{-4} ^{{}^{{}^{\circ}}}\textC^{-1}