Thermal Expansion — Question 4

Question 4

An overflowing liquid competes with its expanding container

A vessel is filled exactly to its brim with $V_0=1.000\,\mathrm L$ of liquid. The liquid has volume coefficient $\beta_l=9.00\times10^{-4}\,\mathrm{K^{-1}}$; the vessel is isotropic with linear coefficient $\alpha_v=1.00\times10^{-5}\,\mathrm{K^{-1}}$. Heat both uniformly by $50.0\,\mathrm K$. Use first-order volume laws, with vessel coefficient $\beta_v=3\alpha_v$.

1. Find the volume overflowing, expressed at the final temperature. Compare with a calculation that treats the vessel as rigid.

2. If the initial fill is only $.9700\,\mathrm L$ in the same vessel, determine whether overflow occurs and calculate the final empty volume or spill.

Question 4 - Solution

An overflowing liquid competes with its expanding container

(a) Method and calculation

$$V_l=V_0(1+\beta_l\Delta T)=1.0450\,\mathrm L,\quad V_v=1.0015\,\mathrm L.$$$$V_{\mathrm{spill}}=V_l-V_v=.0435\,\mathrm L=43.5\,\mathrm{mL}.$$A rigid-vessel estimate gives 45.0 mL and overestimates by 1.5 mL.

(b) Interpretation and further calculation

$$V_l=.9700(1.0450)=1.01365\,\mathrm L.$$$$V_l-V_v=1.01365-1.00150=.01215\,\mathrm L.$$There is still overflow, of 12.15 mL. The starting 30 mL headspace is smaller than the net volume gain.

Physical check

The spilled amount is a volume at the final temperature, not its equivalent volume after cooling.