A capillary glass tube of uniform bore contains a thread of mercury 1 m long at 0℃. When temperature is raised to 100℃, the thread is found to be 16.5 mm longer. If the coefficient of real expansion of mercury is 18.2 × 10^-5 /℃, then what will be the coefficient of linear expansion of glass?

To find the coefficient of linear expansion of glass based on the given information about mercury and the capillary tube, we first need to understand the relationship between the expansions of the two materials. The thread of mercury expands when heated, and this expansion will affect the mercury's length as well as the constraints imposed by the glass tube. Let's work through the problem step by step.

Understanding the Problem

We have a mercury thread that is initially 1 meter long. When the temperature rises from 0℃ to 100℃, this thread lengthens by 16.5 mm. The coefficient of real expansion for mercury is provided as 18.2 × 10^-5 /℃. Our goal is to find the coefficient of linear expansion for the glass tube.

Formulas Involved

First, we should recall the formulas for linear expansion. The change in length (ΔL) of a material due to temperature change (ΔT) can be expressed as:

ΔL = L₀ × α × ΔT

Where:

• ΔL = change in length
• L₀ = original length
• α = coefficient of linear expansion
• ΔT = change in temperature

Calculating the Change in Length of Mercury

For mercury, we have:

• L₀ = 1 m = 1000 mm
• ΔT = 100℃ - 0℃ = 100℃
• α (mercury) = 18.2 × 10^-5 /℃

Now, substituting these values into the linear expansion formula for mercury:

ΔL (mercury) = 1000 mm × 18.2 × 10^-5 /℃ × 100℃

ΔL (mercury) = 1000 mm × 0.00182 = 1.82 mm

Relating Glass and Mercury Expansions

The total length increase of the mercury thread due to temperature rise is 16.5 mm. However, part of this increase is due to the expansion of the glass tube itself. Therefore, we can express the total change as:

ΔL (total) = ΔL (mercury) + ΔL (glass)

Now, we know that:

ΔL (glass) = ΔL (total) - ΔL (mercury)

Substituting in the values:

ΔL (glass) = 16.5 mm - 1.82 mm = 14.68 mm

Calculating the Coefficient of Linear Expansion for Glass

Using the length change of the glass and the same temperature change (100℃), we can now find the coefficient of linear expansion for glass (α glass):

ΔL (glass) = L₀ (glass) × α (glass) × ΔT

Assuming the original length of the glass tube (L₀) is also 1000 mm, we rearrange the formula to solve for α (glass):

α (glass) = ΔL (glass) / (L₀ × ΔT)

Substituting in our known values:

α (glass) = 14.68 mm / (1000 mm × 100℃)

α (glass) = 14.68 mm / 100000 mm⋅℃ = 0.0001468 /℃

Final Result

To express this in scientific notation, we have:

α (glass) = 1.468 × 10^-4 /℃

Thus, the coefficient of linear expansion of the glass tube is approximately 1.468 × 10^-4 /℃. This value indicates how much the glass expands per degree Celsius, which is essential for understanding thermal effects in engineering and materials science.