Question

Wine bottles are never completely filled: a small volume of air is left in the glass bottle's cylindrically shaped neck (inner diameter d = 18.5mm) to allow for wine's fairly large coefficient of thermal expansion. The distance H between the surface of the liquid contents and the bottom of the cork is called the "headspace height" , and is typically H = 1.5cm for a 750mL bottle filled at 20degC. Due to its alcoholic content, wine's coefficient of volume expansion is about double that of water; in comparison, the thermal expansion of glass can be neglected.

Estimate H if the bottle is kept at 11degC.

Estimate H if the bottle is kept at 32degC.

Answer 1

Begin by looking up the thermal expansion coefficient of water.
You can find it at
http://en.wikipedia.org/wiki/Coefficient_of_thermal_expansion
and many other places.

I get 2.07*10^-4 C^-1 at 20 °C

For wine, it would then be
beta = 4.14*10^-4 C^-1 at 20 °C
That coefficient applies to the volume, not the linear length of the wine in the neck of the bottle. At 11 K (a 9 C reduction from 20), the volume will shrink from 750 ml to
750 * (1 - 4.14*10^-4*9) = 747.2 ml

Now assume the area A of the neck of the bottle stays the same. The volume occupied by wine shrinks 2.8 cm^3.

Divide that by the cross sectional area of the neck of the bottle to see how much the liquid level drops.
A = (pi/4)(1.85)^2 = 2.69 cm^2

The liquid level drops and the "head space" lengthens by 2.8/2.69 = 1.04 cm

Perform a similar calculation for the higher temperature.