A physicist makes a cup of instant coffee and notices that, as the coffee cools, its level drops 3.00 mm in the glass cup. Show that this decrease cannot be due to thermal contraction by calculating the decrease in level if the 350cm3 of coffee is in a 7.00-cm-diameter cup and decreases in temperature from 95.0°C to 45.0°C. (Most of the drop in level is actually due to escaping bubbles of air.)

Decrease in level by thermal contraction = 0.0076 mm

Drops in level = 3 mm

Therefore The decrease in level is not due to thermal contraction

Explanation:

From linear expansivity,

L₂ = L₁ + (1 + αΔT) ... equation 1

Where L₂ = final length (m), L₁ = initial length, α = linear expansivity of liquid, ΔT = change in temperature.

Where L₁ = level of coffee in the glass cup = V / (πd²) / 4 ... equation 2

L₁ = 4V/πd², where V = Volume of coffee, d = diameter of the cup,

L₁ = 350cm³, d = 7.00cm, and π = 22/7 = 3.143

applying these values in equation 2 above,

∴ L ₁ = (4*350) / (3.143*7²)

L₁ = 9.09

α = γ/3 Where γ = Volumetric expansion coefficient of liquid. = 0.00005/°C

α = 0.00005/3 = 0.0000167/°C, ΔT = L₂ - L₁ = 45 - 95 = - 50 °C.

Applying these values in equation 1 above,

L₂ = 9.09 [1+{0.0000167 * (-50) }]

L₂ = 9.09 [1 - 0.000833]

L₂ = 9.09 (0.999167)

L₂ = 9.082cm

∴ ΔL = Decrease in length = L₂ - L₁ = 9.082 - 9.09 = 0.3 cm

ΔL = 0.0076

Decrease in level = 0.076 mm

From the above, the decrease in level is not due to thermal contraction.

As Thermal contraction is less than Drops in level.