Mercury Thermometer Calibration Explained

This problem involves determining the temperature of tap water using a mercury thermometer. We are given the length of the mercury column at known temperatures (melting ice and steam) and at the unknown temperature (tap water).

Understanding Thermometer Scales

Thermometers work based on the principle that physical properties of substances change predictably with temperature. For a mercury thermometer, the length of the mercury column is used. We use the melting point of ice and the boiling point of water as standard reference points.

• Lower Fixed Point (LFP): The temperature of melting ice, which is defined as 0°C.
• Upper Fixed Point (UFP): The temperature of steam from boiling water at standard atmospheric pressure, defined as 100°C.

Given Information

Let's list the data provided:

• Length of mercury column at melting ice (0°C): $L_{0^\circ C} = 10$ mm
• Length of mercury column at steam (100°C): $L_{100^\circ C} = 250$ mm
• Length of mercury column in tap water: $L_{tap} = 58$ mm

Calculating Tap Water Temperature

We assume a linear relationship between the length of the mercury column ($L$) and the temperature ($T$). The formula to find the unknown temperature is:

$$ \frac{T - LFP}{UFP - LFP} = \frac{L - L_{LFP}}{L_{UFP} - L_{LFP}} $$

Substituting the given values:

• $LFP = 0^\circ C$
• $UFP = 100^\circ C$
• $L_{LFP} = 10$ mm
• $L_{UFP} = 250$ mm
• $L = 58$ mm

The equation becomes:

$$ \frac{T - 0^\circ C}{100^\circ C - 0^\circ C} = \frac{58 \text{ mm} - 10 \text{ mm}}{250 \text{ mm} - 10 \text{ mm}} $$

Step-by-Step Calculation

1. Calculate the difference in length at the fixed points: $$ L_{100^\circ C} - L_{0^\circ C} = 250 \text{ mm} - 10 \text{ mm} = 240 \text{ mm} $$
2. Calculate the difference in length for the tap water reading from the lower fixed point: $$ L_{tap} - L_{0^\circ C} = 58 \text{ mm} - 10 \text{ mm} = 48 \text{ mm} $$
3. Set up the proportion: $$ \frac{T}{100^\circ C} = \frac{48 \text{ mm}}{240 \text{ mm}} $$
4. Simplify the fraction: $$ \frac{48}{240} = \frac{1}{5} = 0.2 $$
5. Solve for the temperature $T$: $$ \frac{T}{100^\circ C} = 0.2 $$ $$ T = 0.2 \times 100^\circ C $$ $$ T = 20^\circ C $$

Conclusion

The temperature of the tap water is calculated to be 20°C.

Thermometer Readings and Calculation Summary

Condition	Temperature (°C)	Mercury Column Length (mm)
Melting Ice (LFP)	0	10
Steam (UFP)	100	250
Tap Water	T	58

Final Temperature Determination

Based on the linear relationship and the measured lengths, the temperature of the tap water corresponds to 20°C.