Problem — Volume expansion

Problem

A glass vessel contains $V_0 = 500$ ml of mercury at $20\,^{\circ}$C. The volumetric expansion coefficients are $\beta_{\text{Hg}} = 1{,}8 \cdot 10^{-4}$ K$^{-1}$ for mercury and $\beta_{\text{vetro}} \approx 1{,}0 \cdot 10^{-5}$ K$^{-1}$ for glass. Heating the whole to $70\,^{\circ}$C, how much mercury spills out, assuming the vessel was initially filled to the brim?

Solution

The temperature change is $\Delta T = 70\,^{\circ}\text{C} - 20\,^{\circ}\text{C} = 50$ K.

The mercury expands by: $\Delta V_{\text{Hg}} = V_0 \, \beta_{\text{Hg}} \, \Delta T = 500 \cdot 1{,}8 \cdot 10^{-4} \cdot 50 = 4{,}5 \ \text{ml}$

But the glass cavity also increases its own volume (a container that expands “grows” as if it were filled with the material of its walls): $\Delta V_{\text{vetro}} = V_0 \, \beta_{\text{vetro}} \, \Delta T = 500 \cdot 1{,}0 \cdot 10^{-5} \cdot 50 = 0{,}25 \ \text{ml}$

The mercury that actually overflows is the difference between the expansion of the liquid and that of the cavity containing it: $\Delta V = \Delta V_{\text{Hg}} - \Delta V_{\text{vetro}} = 4{,}5 - 0{,}25 = 4{,}25 \ \text{ml}$

$\ev{\Delta V \approx 4{,}25 \ \text{ml}}$

Collegamenti

Argomenti: Termologia Concetti: Dilatazione termica