Question

Consider an alcohol and a mercury thermometer that read exactly \(0^{\circ} \mathrm{C}\) at the ice point and \(100^{\circ} \mathrm{C}\) at the steam point. The distance between the two points is divided into 100 equal parts in both thermometers. Do you think these thermometers will give exactly the same reading at a temperature of, say, \(60^{\circ} \mathrm{C} ?\) Explain.

Short answer

Answer: The readings of an alcohol thermometer and a mercury thermometer might differ at 60°C due to their different thermal expansion coefficients, which cause them to expand at different rates when heated. Both thermometers are accurate at 0°C and 100°C, but their readings at other temperatures like 60°C can vary because of these differing expansion properties.

Step-by-step solution

Understand the concept of linear expansion

Linear expansion is the increase in length or volume of a material when its temperature increases. Different materials expand at different rates when heated. This is an important aspect of thermometers - they rely on the expansion of the material inside them to measure temperature.

Know the materials used in thermometers

In this exercise, we are comparing an alcohol thermometer and a mercury thermometer. Alcohol (usually a mixture of ethanol or methanol) and mercury are two common materials used in liquid-in-glass thermometers because of their desirable properties, such as a relatively wide temperature range and consistent thermal expansion.

Recognize different expansion rates

Alcohol and mercury do not have the same thermal expansion coefficient, which means they expand at different rates when heated. While both thermometers are calibrated to be accurate at 0°C and 100°C, their readings at other temperatures might differ because of these differing expansion rates.

Comparing readings at 60°C

When comparing the readings of an alcohol and a mercury thermometer at 60°C, it is likely that the two will not give the exact same reading. This is due to the fact that both materials have different expansion coefficients and will expand at different rates between 0°C and 100°C. In conclusion, the alcohol and mercury thermometers may not give the exact same reading at a temperature of 60°C due to the different expansion rates of the respective materials.

Key concepts

Linear Expansion

Linear expansion is a fundamental concept in understanding how materials behave under the influence of temperature change. At its core, linear expansion refers to the increase in length (linear dimension) of an object when heated. It is a proportionate increase, which means that for a given material, the change in length is directly related to the original length and the change in temperature.

In the context of the exercise, the idea behind linear expansion is crucial because it affects how thermometers work. The thermometer operates on the principle that a liquid (alcohol or mercury) contained within will expand linearly in response to temperature increase. Therefore, knowing the linear expansion coefficient of the liquid allows us to predict how much it will expand or contract due to temperature changes.

The mathematical expression for linear expansion is: \( \Delta L = \alpha L_0 \Delta T \) where \(\Delta L\) is the change in length, \(\alpha\) is the linear expansion coefficient, \(L_0\) is the original length, and \(\Delta T\) is the change in temperature. This formula is a linear approximation and is most accurate over small temperature ranges. However, at larger ranges or extreme temperatures, other factors may influence expansion rates. Remember, clarity of measurement is key, and minute errors can have bigger implications in precision tasks.

Thermometer Calibration

Calibration is the process of setting or correcting a measuring device or instrument, comparing it against a standard reference. For thermometers, calibration is vital to ensure accurate temperature readings across a range of temperatures. During calibration, key reference points, typically the freezing point and boiling point of water (0°C and 100°C, respectively), are used to ensure the thermometer measures accurately at these temperatures.

However, the accuracy of calibration can only be guaranteed at these fixed points. As the exercise highlights, between these points - say, at 60°C - the readings could vary due to the unequal thermal expansion coefficients of alcohol and mercury. Each liquid responds to heat differently, expanding more or less per degree change in temperature. This is why well-calibrated thermometers still require a margin of understanding for the nuances of the material's expansion properties.

Applying the knowledge of calibration, we come to understand that the linear expansion differences between two substances can lead to variations in readings that, while still within a functional range, may not always align perfectly with the standard scales devised for uniform materials.

Temperature Measurement

The ultimate purpose of any thermometer is to provide an accurate measurement of temperature. This is achieved through the physical properties of a substance reacting predictably to temperature changes. For both alcohol and mercury thermometers, temperature measurement involves the expansion and contraction of the liquid inside the glass tube, which then corresponds to a marked scale that we read the temperature from.

While the concept may seem straightforward, the challenge comes from the fact that different substances have unique thermal properties. Our exercise demonstrates that assuming equal performance at non-calibrated temperatures could lead to errors. Since mercury and alcohol expand at different rates, their level of accuracy when measuring intermediate temperatures will differ. For precise applications, selecting the appropriate thermometer with the correct thermal expansion coefficient is crucial for ensuring that temperature measurements are both accurate and reliable.

Ultimately, when using a thermometer for any critical process, the user needs to consider the nonlinear nature of the material's response to temperature. The importance of understanding the science of measuring temperature lies in recognizing these subtle differences that impact readability and accuracy, which is essential for fields ranging from meteorology to medicine.