Question

A column of liquid is found to expand linearly on heating \(5.25\) \(\mathrm{cm}\) 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 after expanding by 18.5 cm will be approximately \(56.58^{\circ}\textrm{C}\).

Step-by-step solution

Find the relationship between expansion and temperature

For every 10.0°F increase in temperature, the column of liquid expands by 5.25 cm. Therefore, we can write the relationship between the increase in temperature (ΔT°F) and the expansion in length as: ΔT°F = (ΔL cm) * (10.0°F / 5.25 cm) where ΔT°F - Increase in temperature in Fahrenheit ΔL cm - Increase in length in centimeters

Find the total increase in temperature

The liquid expands by 18.5 cm, so we can plug this value into the relationship derived in Step 1 to find the total increase in temperature: ΔT°F = (18.5 cm) * (10.0°F / 5.25 cm) ΔT°F ≈ 35.2381°F

Find the final temperature in Fahrenheit

Now that we have found the total increase in temperature, we can add this value to the initial temperature to find the final temperature in Fahrenheit: T_final°F = T_initial°F + ΔT°F T_final°F = 98.6°F + 35.2381°F T_final°F ≈ 133.8381°F

Convert the final temperature to Celsius

To convert the final temperature from Fahrenheit to Celsius, we can use the following formula: T°C = (T°F - 32) * (5/9) Plugging in the final temperature in Fahrenheit we found in Step 3: T_final°C = (133.8381°F - 32) * (5/9) T_final°C ≈ 56.5767°C So, the final temperature of the liquid will be approximately 56.58°C.

Key concepts

Linear Expansion

When substances are heated, they tend to expand; this phenomenon is known as linear expansion. The basics of linear expansion can be understood as the proportional increase in a material's dimensions owing to an increase in temperature. In the context of our problem, a liquid column expands as its temperature increases. This expansion is directly proportional to the temperature change—meaning that for a known temperature increase, the expansion can be predicted, and vice versa.

Understanding linear expansion is crucial in various applications, from designing thermometers to constructing bridges, where allowances must be made for changes in length with temperature. The formula representing this relationship, typically written as ΔL = αL₀ΔT, is simplified in the exercise, correlating expansion directly with temperature change without explicitly mentioning the coefficient of linear expansion (α) or the original length (L₀).

Temperature Scales

Temperature scales provide a way to quantify the warmth or coldness of an object. The most commonly used temperature scales are Celsius (°C), Fahrenheit (°F), and Kelvin (K). Understanding these scales is fundamental in chemistry, as reactions are often dependent on specific temperatures.

Celsius is based on the freezing and boiling points of water, set at 0°C and 100°C respectively. The Fahrenheit scale, primarily used in the United States, sets the freezing point of water at 32°F and the boiling point at 212°F. The Kelvin scale is the standard unit of temperature in the scientific community and starts at absolute zero, the point at which particles theoretically stop moving. There is no degree symbol used with the Kelvin scale; temperatures are simply stated in kelvins.

Converting between these scales requires understanding their relationships and the mathematical formulas that link them. For instance, the conversions between Celsius and Fahrenheit are important when dealing with different scientific and cultural norms.

Fahrenheit to Celsius Conversion

Converting from Fahrenheit to Celsius is a key skill in chemistry, especially for students who are accustomed to one temperature scale and need to work with the other. The conversion between these two scales can be achieved through a simple formula: \( T(°C) = \left(T(°F) - 32\right) \times \frac{5}{9} \).

This formula was used in the step-by-step solution to convert the final temperature of the liquid from Fahrenheit to Celsius after determining its expansion. When engaging in temperature conversion, it is essential to pay close attention to detail and ensure any addition or subtraction is done prior to the application of the multiplication or division step, preventing potential errors in the final result.