# Mercury is often used as an expansion medium in a thermometer. The mercury sits in a bulb on the bottom of the thermometer and rises up a thin capillary as the temperature rises. Suppose a mercury thermometer contains 3.250 g of mercury and has a capillary that is 0.180 mm in diameter.     How far does the mercury rise in the capillary when the temperature changes from 0.0 ∘C to 25.0 ∘C? The density of mercury at these temperatures is 13.596 g/cm3 and 13.534 g/cm3, respectively.

Let's first obtain the volume at both temperatures using the relationship Volume = Mass/Density.

At 0 degrees C (`V_1`e have:

`V_1` = `(3.250 g)/(13.596 g/(cm^3))` = .2390 `cm^3` ` `

At 25 degrees C (`V_2`e have:

`V_2`  = `(3.250 g)/(13.534 g/(cm^3))` = .2401 `cm^3`

We will think of the column of mercury inside the capillary as a cylinder. The height of the cylinder is given by the formula

`h = V/(pi * r^2)`

The radius, r, in centimeters is `(0.180 mm) / 2 * (1 cm)/(10 mm)` or .009 cm. Therefore, at 0 degrees, the height is:

`h_1` =`(.2390 cm^3)/(pi*(.009 cm)^2)`  = 939.2 `cm`

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At 25 degrees C, the height is:

`h_2 =``(.2401 cm^3)/(pi*(.009 cm)^2) = 943.5 cm`

Subtracting these two, 943.5 cm - 939.2 cm, we obtain a rise in mercury height of 4.3 cm, the final answer.

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