Question

Wine bottles are never completely filled: a small volume of air is left in the glass bottle's cylindrically shaped neck (inner diameter dd = 18.5 mmmm) 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 cmcm for a 750-mLmL bottle filled at 20 ∘C∘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 15 ∘C

b.) Estimate H if the bottle is kept at 29 ∘C

Answer 1

Part A.

We know that relation between change in volume and temperature is given by:

dV = V0**dT

V0 = Initial Volume of wine inside bottle = 750 mL

Assuming Ti = room temperature = 20 C (Assuming at room temperature, If this temperature is different then let me know through comments)

Tf = final temperature at which bottle is kept = 15 C

dT = Change in temperature = Tf - Ti = 15 - 20 = -5 C

= thermal expansion coefficient of wine = 4.20*10^-4 /C (Using Standard value, check your reference book for this, if different then let me know)

So,

dV = 750*4.20*10^-4*(-5) = -1.575 mL = -1.575 cm^3

Now that means volume of wine is reduced by 1.575 cm^3.

Now Height by which wine is reduced in bottle will be:

dV = A*h1

h1 = Volume reduced/Area of reduced part = dV/A

A = pi*d^2/4

d = diameter of neck = 18.5 mm = 18.5*10^-3 m, So

h1 = dV*4/(pi*d^2)

h1 = 1.575*10^-6*4/(pi*(18.5*10^-3)^2)

h1 = 0.586*10^-2 m = 0.586 cm

So Now in this case total height of headspace will be:

H1 = H + h1 = 1.5 cm + 0.586 cm = 2.086 cm

In two significant figures

H1 = 2.1 cm

(Remember that since Volume of wine was reduced in this case, So headspace height will increase by h1 height)

Part B.

V0 = Initial Volume of wine inside bottle = 750 mL

Assuming Ti = room temperature = 20 C (Assuming at room temperature, If this temperature is different then let me know through comments)

Tf = final temperature at which bottle is kept = 29 C

dT = Change in temperature = Tf - Ti = 29 - 20 = 9 C

= thermal expansion coefficient of wine = 4.20*10^-4 /C (Using Standard value, check your reference book for this, if different then let me know)

So,

dV = 750*4.20*10^-4*(9) = 2.835 mL = 2.835 cm^3

Now that means volume of wine is reduced by 2.835 cm^3.

Now Height by which wine is reduced in bottle will be:

dV = A*h1

h1 = Volume reduced/Area of reduced part = dV/A

A = pi*d^2/4

d = diameter of neck = 18.5 mm = 18.5*10^-3 m, So

h1 = dV*4/(pi*d^2)

h1 = 2.835*10^-6*4/(pi*(18.5*10^-3)^2)

h1 = 1.055*10^-2 m = 1.055 cm

So Now in this case total height of headspace will be:

H1 = H + h1 = 1.5 cm - 1.055 cm = 0.445 cm

In two significant figures

H1 = 0.45 cm

(Remember that since Volume of wine was increased in this case, So headspace height will decrease by h1 height)

Let me know if you've any query.