Problem 14 - Chemistry: Principles and Reactions

Question

A recipe for apple pie calls for a preheated \(350^{\circ} \mathrm{F}\) (three significant figures) oven. Express this temperature setting in \({ }^{\circ} \mathrm{C}\) and in \(\mathrm{K}\).

Step-by-Step Solution

Answer

Answer: The temperature settings for the apple pie recipe are \(176.7^{\circ}\mathrm{C}\) in Celsius and \(449.8\mathrm{K}\) in Kelvin.

Step 1: Convert Fahrenheit to Celsius

To convert the given temperature from Fahrenheit to Celsius, use the formula: \({ }^{\circ}\mathrm{C} = \frac{5}{9}({ }^{\circ}\mathrm{F} - 32)\). Plug in the given temperature \(350^{\circ}\mathrm{F}\): \({ }^{\circ}\mathrm{C} = \frac{5}{9}(350^{\circ}\mathrm{F} - 32^{\circ}\mathrm{F})\)

Step 2: Calculate the Celsius temperature

Now, simplify the expression and calculate the temperature in Celsius: \({ }^{\circ}\mathrm{C} = \frac{5}{9}(318^{\circ}\mathrm{F})\) \({ }^{\circ}\mathrm{C} = 176.7^{\circ}\mathrm{C}\) (rounded to three significant figures)

Step 3: Convert Celsius to Kelvin

To convert the temperature from Celsius to Kelvin, use the formula: \(\mathrm{K} = { }^{\circ}\mathrm{C} + 273.15\). Plug in the calculated Celsius temperature \(176.7^{\circ}\mathrm{C}\): \(\mathrm{K} = 176.7^{\circ}\mathrm{C} + 273.15\)

Step 4: Calculate the Kelvin temperature

Now, simplify the expression and calculate the temperature in Kelvin: \(\mathrm{K} = 449.8\ \mathrm{K}\) (rounded to three significant figures) The temperature setting for the apple pie recipe is \(176.7^{\circ}\mathrm{C}\) in Celsius and \(449.8\mathrm{K}\) in Kelvin.

Key Concepts

Fahrenheit to Celsius Conversion

Understanding how to convert temperatures from Fahrenheit to Celsius is crucial in chemistry, especially for international students who might not be familiar with the Fahrenheit scale. The formula used for conversion is straightforward:
\[ { }^{\textdegree}\mathrm{C} = \frac{5}{9}({ }^{\textdegree}\mathrm{F} - 32) \].
To apply this formula, subtract 32 from the Fahrenheit temperature, then multiply by the fraction \( \frac{5}{9} \). For example, to convert a preheated oven temperature of \(350^{\textdegree}\mathrm{F}\) to Celsius, we first deduct 32, resulting in \(318^{\textdegree}\mathrm{F}\). We then multiply by \(\frac{5}{9}\) to obtain \(176.7^{\textdegree}\mathrm{C}\), rounded to three significant figures. This simple conversion process is mandatory knowledge for various scientific and culinary tasks worldwide.

Celsius to Kelvin Conversion

When converting from Celsius to Kelvin, which is essential in many scientific fields, the formula is even simpler:
\[ \mathrm{K} = { }^{\textdegree}\mathrm{C} + 273.15 \].
Kelvin is the SI unit for temperature and does not use degrees; instead, it's an absolute temperature scale starting from absolute zero. To convert our preheated oven temperature originally in Fahrenheit, first converted to Celsius (\(176.7^{\textdegree}\mathrm{C}\)), we simply add 273.15 to arrive at a temperature of \(449.85\mathrm{K}\). However, due to significant figures, we round this to \(449.8\mathrm{K}\) to maintain the precision given in the original Fahrenheit temperature.

Significant Figures

In chemistry and other sciences, the concept of significant figures is used to indicate the precision of a measured or calculated quantity. The rules for determining the number of significant figures include noting all non-zero digits, zeroes between non-zero digits, trailing zeroes in a decimal part, and all digits in a coefficient of a number written in scientific notation.
The crucial aspect is to maintain the integrity of significant figures throughout the calculation steps. For example, in our temperature conversion exercise, we are given the Fahrenheit temperature with three significant figures (\(350^{\textdegree}\mathrm{F}\)). Thus, our final Celsius (\(176.7^{\textdegree}\mathrm{C}\)) and Kelvin (\(449.8\mathrm{K}\)) results are also expressed with three significant figures. This ensures that our final answer reflects the precision of our initial measurements or specifications. Keeping track of significant figures is vital, as it affects the final reported answer and helps to avoid overestimating the precision of the derived values.