Question

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 coffee has a density of 0.997 g/cm³ in a 7.00-cm-diameter cup and decreases in temperature from 95.0ºC to 20.0ºC. (Most of the drop in level is actually due to escaping bubbles of air.)

a) (2.10 mm)
b) (3.00 mm)
c) (3.90 mm)
d) (4.80 mm)

Answer 1

The calculation of the thermal contraction of coffee as it cools from 95.0°C to 45.0°C in a 7.00-cm-diameter cup indicates that the observed 3.00 mm drop in level is much greater than what thermal contraction would cause, suggesting that thermal contraction is not the primary reason for the coffee level drop.

Step-by-step explanation:

A physicist makes a cup of instant coffee and notices that as the coffee cools from 95.0°C to 45.0°C, its level drops 3.00 mm in the glass cup. The initial volume of the coffee is given as 350 cm³. To determine if this decrease in level can be attributed to thermal contraction, we must calculate the thermal contraction of the coffee and compare it to the observed decrease.

The formula for the change in volume due to thermal contraction is ΔV = βVΔT, where Β is the volume coefficient of expansion for liquid water, V is the initial volume of the coffee, and ΔT is the change in temperature. The average volume coefficient of expansion for water is approximately
`210 x 10^-^6 \degree C^-^1`.

Substituting the known values, we get:

`\triangle V = \beta V\triangle T`

=
`(210 x 10^-^6 \degree C^-^1)(350 cm^3)(45.0\degree C - 95.0\degree C)`

This results in a decrease in volume, but to find the decrease in height h, we need to use the formula for the volume of a cylinder:
`V = \pi r^2h`.

With a cup diameter of 7.00 cm, the radius r is 3.50 cm. Thus, we can solve for Δh:

`\triangle V = \pi r^2\triangle h`.

Substituting the values and solving for Δh, we find that the thermal contraction corresponds to a much smaller decrease in level than the observed 3.00 mm, indicating that thermal contraction is not the sole reason for the observed decrease in coffee level. The calculations show that the actual decrease due to thermal contraction would be substantially less, thus most of the level drop is likely due to other factors such as escaping bubbles of air.