The Correcting of Temperature Readings on a Mismarked Mercury Thermometer: A Mathematical Analysis

Imagine a mercury thermometer where the steam point and ice point are incorrectly marked as 95°C and 5°C, respectively. This raises a fundamental question: how would the temperature be correctly read on such a mismarked thermometer?

Understanding the Mismarked Thermometer

In a standard mercury thermometer, the temperature range from the ice point (0°C) to the steam point (100°C) is divided into 100 equal parts. However, in this mismarked thermometer, the range is incorrectly divided into 90 parts. The division between 0°C and 100°C in a correct thermometer can be mapped to:

0°C to 100°C (original) 2°C to 90°C (mismarked)

This means that each degree in the standard thermometer is represented by more than one division on the mismarked thermometer. To find the actual conversion factor, we can use the following equation:

1° in original thermometer 10/9° in mismarked thermometer.

Mathematical Calculation

To find the correct temperature that would be read by the mismarked thermometer, let's denote the temperature by T. If both thermometers show the same temperature at T, the relationship can be written as:

2T x 9/10 T

Solving for T in this equation:

T/10 2

T 20°C

Thus, at 20°C, both the thermometers, whether correctly or incorrectly marked, would show the same temperature.

Implications for Readings

Given the mismarkings, the readings on the thermometer will be incorrect for several reasons:

The ice point (0°C) is incorrectly marked as 5°C, indicating a zero error. The steam point (100°C) is marked as 95°C, compressing the temperature range into fewer divisions.

For instance, if the correct temperature is 20°C, the mismarked thermometer would show a value based on the incorrect divisions. If we use the provided equation:

x - 2 / 95 - 5 C / 100

If we assume x 20°C as a test value, the equation becomes:

20 - 2 / 95 - 5 20 / 100

This simplifies to:

18 / 90 0.2

Which is true because 20°C in a correctly calibrated thermometer would indeed correspond to this result.

Conclusion

In the absence of a standard thermometer, the correct temperature reading can be calculated using the given information and mathematical relationships. However, it is important to understand that the mismarked thermometer will give inaccurate readings for all other temperatures, except at the point where the correct and mismarked values align, which in this case is 20°C.

This analysis highlights the importance of proper calibration and the implications of incorrect markings on thermometers.