What is the change in length of a 3.00-cm-long column of mercury if its temperature changes from 37.0°C to 40.0°C, assuming the mercury is constrained to a cylinder but unconstrained in length?
Your answer will show why thermometers contain bulbs at the bottom instead of simple columns of liquid.
The Correct Answer and Explanation is:
To calculate the change in length of a 3.00-cm-long column of mercury due to a temperature change from 37.0°C to 40.0°C, we can use the coefficient of linear expansion for mercury, which is approximately ( \alpha = 0.000018 \, \text{°C}^{-1} ).
Step 1: Calculate the change in temperature.
[
\Delta T = T_f – T_i = 40.0°C – 37.0°C = 3.0°C
]
Step 2: Use the formula for linear expansion.
The formula for the change in length (( \Delta L )) due to thermal expansion is given by:
[
\Delta L = L_0 \cdot \alpha \cdot \Delta T
]
where:
- ( L_0 ) = initial length of the mercury column = 3.00 cm
- ( \alpha ) = coefficient of linear expansion for mercury = ( 0.000018 \, \text{°C}^{-1} )
- ( \Delta T ) = change in temperature = ( 3.0°C )
Step 3: Plug in the values.
[
\Delta L = 3.00 \, \text{cm} \cdot 0.000018 \, \text{°C}^{-1} \cdot 3.0°C
]
[
\Delta L = 3.00 \cdot 0.000018 \cdot 3.0 = 0.000162 \, \text{cm}
]
Step 4: Interpret the results.
Thus, the change in length of the mercury column is approximately 0.000162 cm (or 0.00162 mm).
Explanation
The minuscule change in length of the mercury column demonstrates the principle behind thermometers. If a thermometer were merely a long column of mercury without a bulb, the response to temperature changes would be inefficient, as the mercury would expand equally in both directions and the rise in the column would be minimal. The bulb at the bottom of a thermometer serves a crucial function: it provides a reservoir that accommodates the expansion of mercury.
When the temperature increases, the mercury expands and pushes up into the narrower capillary tube, which allows a more significant and visible change in height. This mechanism ensures that the thermometer can quickly and accurately reflect temperature changes. The design optimizes sensitivity and visibility, allowing users to read temperatures easily. Therefore, bulbs are essential for enhancing the responsiveness and usability of liquid-in-glass thermometers.