Question

A column of liquid is found to expand linearly on heating. Assume the column rises \(5.25 \mathrm{~cm}\) for a \(10.0^{\circ} \mathrm{F}\) rise in temperature. If the initial temperature of the liquid is \(98.6^{\circ} \mathrm{F}\), what will the final temperature be in \({ }^{\circ} \mathrm{C}\) if the liquid has expanded by \(18.5 \mathrm{~cm} ?\)

Short answer

The final temperature of the liquid in Celsius is approximately \(38.47^{\circ}\mathrm{C}\).

Step-by-step solution

Calculate the temperature increase in Fahrenheit

First, we need to find out the temperature increase in Fahrenheit. We know that the column of liquid rises by 5.25 cm when the temperature increases by 10.0 °F. So, we can set up a proportion: \(\frac{5.25 \mathrm{~cm}}{10.0^{\circ}\mathrm{F}} = \frac{18.5 \mathrm{~cm}}{x}\) We need to solve for x, which represents the temperature increase in Fahrenheit.

Solve the proportion

Cross-multiply and divide to find the value of x: \(x = \frac{10.0^{\circ}\mathrm{F} \times 18.5 \mathrm{~cm}}{5.25 \mathrm{~cm}} = \ 35.2381... \,^{\circ}\mathrm{F}\) Therefore, the temperature increases by approximately 35.24 °F.

Convert the initial temperature to Celsius

We need to convert the initial temperature of 98.6 °F to Celsius. The conversion formula is: \(C = \frac{5}{9}(F - 32)\) So, the initial temperature in Celsius is: \(C = \frac{5}{9}(98.6 - 32) = 36.78^{\circ}\mathrm{C}\)

Convert the temperature increase to Celsius

Now, we need to convert the temperature increase of 35.24 °F to Celsius. We can use the same conversion formula: \(C = \frac{5}{9}(35.24 - 32) = 1.69^{\circ}\mathrm{C}\) Therefore, the temperature increase in Celsius is approximately 1.69 °C.

Calculate the final temperature in Celsius

Finally, we can find the final temperature in Celsius by adding the initial temperature and the temperature increase: Final Temperature = Initial Temperature + Temperature Increase Final Temperature = \(36.78^{\circ}\mathrm{C} + 1.69^{\circ}\mathrm{C} = 38.47^{\circ}\mathrm{C}\) So, the final temperature of the liquid in Celsius is approximately \(38.47^{\circ}\mathrm{C}\).

Key concepts

Temperature Conversion

Converting temperatures between Fahrenheit and Celsius is a very useful skill, especially when working with thermal expansion problems like the one we have here. The formula to convert from Fahrenheit to Celsius is:

• \( C = \frac{5}{9}(F - 32) \)

This equation helps us calculate what a temperature in Celsius would be, given its Fahrenheit equivalent. In the context of the given problem, this conversion is necessary because the initial temperature is provided in Fahrenheit (98.6 °F), but the solution requires the final temperature to be in Celsius.
Conversion such as this is crucial for understanding measurements globally, as different regions use different temperature scales. Being adept at quickly switching between scales ensures you can work with data from any source and contributes to precise calculations in scientific contexts.

Celsius to Fahrenheit

Understanding how changes in temperature affect systems is important. Often, information is presented in one temperature scale, but needs to be understood in another. To convert from Celsius back to Fahrenheit, the formula is:

• \( F = \frac{9}{5}C + 32 \)

In the step-by-step solution given for the problem, the conversion from Fahrenheit to Celsius was needed to interpret temperature changes effectively. Recognizing when and why to convert between these units aids in comprehensive, accurate interpretations of temperature changes in scientific problems.
Temperatures are given frequently in Celsius in scientific reports because it's part of the metric system, which is used widely around the world. Thus, fluency in the conversion between these two temperature units ensures better understanding and communication in different scientific fields.

Linear Expansion

Linear expansion refers to how a material's dimensions change when the temperature changes. Specifically, it describes the proportional increase in length per degree change in temperature. In our problem, a column of liquid expands when the temperature goes up, measured by how much taller the column becomes.
The linear expansion can be described by comparing how the column height increased with a small temperature rise. Mathematically, this can be expressed:

• \( \Delta L = \alpha L_0 \Delta T \)

Here, \(\Delta L\) is the change in length, \(\alpha\) represents the coefficient of linear expansion, \(L_0\) is the original length, and \(\Delta T\) is the change in temperature.
In many problems like the one described, understanding the properties of linear expansion helps predict how much a substance will expand or contract with temperature changes. It's a foundational concept in thermodynamics and material science, crucial for applications ranging from engineering to daily life appliances.