A glass thermometer using a thin column of ethyl alcohol at room temperature (293.0 K) is cooled. When the temperature reaches 273.0 K the column of alcohol is 15.50 cm high. How tall was it when the temperature was 293.0 K? The coefficient of volume expansion of ethyl alcohol is 1,120 × 10⁻⁶ K⁻¹. A) 15.85 cm B) 15.68 cm C) Cannot answer without knowing the diameter of the column D) 15.62 cm

Question

Quizplus · Accepted Answer

Using the formula for volumetric thermal expansion:

ΔV = V0βΔT

where ΔV is the change in volume, V0 is the initial volume, β is the coefficient of volume expansion, and ΔT is the change in temperature.

Assuming the diameter of the column of alcohol remains constant, we can use the formula for the volume of a cylinder:

V = πr2h

where r is the radius of the cylinder and h is the height.

Since we only need a ratio of heights, we can omit the πr2 factor and set the initial and final volumes equal to each other:

V0 = Vf

(15.50 cm)(0.0 cm)h = Vf(1.12 x 10^-6 K^-1)(293.0 K - 273.0 K)

Solving for Vf, we get:

Vf = (15.50 cm)(0.0 cm)h / [1.12 x 10^-6 K^-1 (20 K)]

Vf = (15.50 cm)(0.0 cm)h / [2.24 x 10^-5]

Vf = 0.0 cm3h

Since the volume of the column at the lower temperature is 0 cm3, the initial height of the column of alcohol must have been equal to the 15.50 cm height measured at the lower temperature plus any expansion that occurred between the lower and initial temperatures.

ΔV = V0βΔT

ΔV = (15.50 cm)(1.12 x 10^-6 K^-1)(20 K)

ΔV = 3.47 x 10^-5 cm3

Adding ΔV to the initial volume of the column:

V0 + ΔV = Vfinal

V0 + 3.47 x 10^-5 cm3 = Vfinal

Since the diameter of the column is assumed to be constant, we can set the initial and final volumes equal to each other and solve for the initial height:

V0 = Vfinal

(15.50 cm)(h) = (15.50 cm + 3.47 x 10^-5 cm3)(hinitial)

Solving for hinitial:

hinitial = (15.50 cm)(h) / (15.50 cm + 3.47 x 10^-5 cm3)

hinitial = 15.85 cm

Therefore, the initial height of the column of alcohol was 15.85 cm. The best choice is A.

A Glass Thermometer Using a Thin Column of Ethyl Alcohol