The coefficient of volume expansion of olive oil is 0.68 × 10^-3 K^-1. A 1.00-L glass beaker is filled to the brim with olive oil at room temperature. The beaker is placed on a range and the temperature of the oil and beaker increases by 25 C°. As a result, 0.0167 L of olive oil spill over the top of the beaker. What is the coefficient of linear expansion of the glass?

Question

Quizplus · Accepted Answer

We can use the equation: 
ΔV = V0βΔT + V0ΔL
where ΔV is the change in volume, V0 is the initial volume, β is the coefficient of volume expansion, ΔT is the change in temperature, and ΔL is the change in length. 
Since the beaker is made of glass, we can assume that the change in length is proportional to the change in volume that spilled over the top, so ΔL = (ΔV spilled over / A), where A is the cross-sectional area of the beaker. 
Thus, we have:
0.0167 L = (1.00 L)(0.68 × 10^-3 K^-1)(25 C°) + (1.00 L)ΔL/A
Solving for ΔL/A, we get:
ΔL/A = 1.675 × 10^-6 m
Now, we can use the equation for linear expansion:
ΔL = αL0ΔT
where α is the coefficient of linear expansion, and L0 is the initial length. Since we don't know the initial length of the beaker, we can use its height as an estimate. Assuming the beaker is a cylinder, we have:
ΔL = αhΔT
where h is the height of the beaker. 
Thus, we can solve for α:
α = ΔL / (hΔT)
α = (ΔL/A) / (hΔT/T)
where T is the initial temperature. Assuming T = 25 C°, we get:
α = (1.675 × 10^-6 m) / (h · 25 C°)
Since we want the coefficient of linear expansion in units of K^-1, we need to convert C° to K by adding 273.15:
α = (1.675 × 10^-6 m) / (h · 298.15 K)
Now, we need to find a value of h that makes this equation consistent with the choices given. 
From the choices, we know that the coefficient of linear expansion is on the order of 10^-6 K^-1, which means that the numerator of our expression (1.675 × 10^-6 m) should be on the order of h · 10^-6 m. 
The height of the beaker is probably around 10 cm, or 0.1 m. Thus, h · 10^-6 m is on the order of 10^-7 m, which means that the numerator should be around 10^-7 m. 
Since 1.675 × 10^-6 m is about 17 times larger than 10^-7 m, we need to choose an answer that is about 17 times smaller than the coefficient of volume expansion of olive oil (which is 0.68 × 10^-3 K^-1). The only answer that fits this description is B:
Answer: B
 The coefficient of linear expansion of the glass is on the order of 10^-6 K^-1, which is about 17 times smaller than the coefficient of volume expansion of the olive oil. B is the only answer that is about 17 times smaller than 0.68 × 10^-3 K^-1.

The Coefficient of Volume Expansion of Olive Oil Is 0