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 = 18.5 mm) 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"(Figure 1), and is typically H = 1.5 cm for a 750-mL bottle filled at 20 ∘C. 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.

A) Estimate H if the bottle is kept at 11 ∘C.

B) Estimate H if the bottle is kept at 28 ∘C.

Answer 1

Solution: (A) Liquid expansion can happen only through volume. The expression for change in volume is, AV =V.BAT Here, V, is the initial volume, AT is the change in temperature, B is the coefficient of volume expansion, and AV is the change in volume. The expression for change in volume in terms of height is, AV =-nr-Ah Radius is equal to half of the diameter. av--- Jan Substitute -T(d/2)* Ah for AV in equation (1). --() sh=V,BAT An - - V.BAT V.B(1,-1) Substitute 750 mL for V., 420x109 /°C for B, 11°C for T., 20°C for T., and 18.5 mm for d.

(750ml (10627" ) (420x10^ rc)(11°C-20°C) Ah=-- 10 m 18.5 mm I mm (750ml ( 19 min) (420x1049 C)(11°C-20°C) (9.25*10*m) =1.054 x10 m =1.054cm The new height is, H=h+Ah = 1.5cm+1.054cm = 2.554cm (B) When the temperature raises to 28°C. Ahz-V.BAT V.B (77'-T2) Substitute 750 mL for V, 420x109 /°C for B, 28 °C for T, 20°C for T., and 18.5 mm for d.

*PC)(28°C-20°C) Ah'=- 18.5mm 1 mm 06/°C)(28°C -20°C) (9.25x10 m =-9.37x10 m = -0.937 cm The new height is, H'=h+Ah =1.5 cm +(-0.937 cm) -0.563 cm