Problem 136 - Introductory Chemistry Atoms First

Question

Convert: (a) \(72{ }^{\circ} \mathrm{F}\) to \({ }^{\circ} \mathrm{C}\) and \(\mathrm{K}\) (b) \(-12{ }^{\circ} \mathrm{C}\) to \({ }^{\circ} \mathrm{F}\) and \(\mathrm{K}\) (c) \(178 \mathrm{~K}\) to \({ }^{\circ} \mathrm{F}\) and \({ }^{\circ} \mathrm{C}\)

Step-by-Step Solution

Answer

The short answer version of the solution is as follows: (a) \( 72^{\circ} \mathrm{F} \) is equivalent to approximately \( 22.22^{\circ} \mathrm{C} \) and \( 295.37 \mathrm{K} \). (b) \( -12^{\circ} \mathrm{C} \) is equivalent to approximately \({-10.4}^{\circ} \mathrm{F} \) and \( 261.15 \mathrm{K} \). (c) \( 178 \mathrm{~K} \) is equivalent to approximately \({-95.15}^{\circ} \mathrm{C} \) and \({-139.27}^{\circ} \mathrm{F} \).

Step 1: Convert Fahrenheit to Celsius

First, use the Fahrenheit to Celsius formula to convert the temperature: \( C = \frac{5}{9} (F - 32) \) \( C = \frac{5}{9} (72 - 32) \) \( C \approx 22.22^{\circ} \mathrm{C} \)

Step 2: Convert Celsius to Kelvin

Next, use the Celsius to Kelvin formula to convert the temperature: \( K = C + 273.15 \) \( K = 22.22 + 273.15 \) \( K \approx 295.37 \: \mathrm{K} \) The temperature in Celsius is \( 22.22^{\circ} \mathrm{C} \) and in Kelvin, it is \( 295.37 \mathrm{K} \). (b) Convert \( -12^{\circ} \mathrm{C} \) to \(^{\circ} \mathrm{F} \) and \( \mathrm{K} \)

Step 3: Convert Celsius to Fahrenheit

This time, use the Celsius to Fahrenheit formula to convert the temperature: \( F = \frac{9}{5} C + 32 \) \( F = \frac{9}{5} (-12) + 32 \) \( F = {-10.4}^{\circ} \mathrm{F} \)

Step 4: Convert Celsius to Kelvin

Use the Celsius to Kelvin formula again: \( K = C + 273.15 \) \( K = -12 + 273.15 \) \( K = 261.15 \: \mathrm{K} \) The temperature in Fahrenheit is \({-10.4}^{\circ} \mathrm{F} \) and in Kelvin, it is \( 261.15 \mathrm{K} \). (c) Convert \( 178 \mathrm{~K} \) to \(^{\circ} \mathrm{F} \) and \(^{\circ} \mathrm{C} \)

Step 5: Convert Kelvin to Celsius

First, use the Kelvin to Celsius formula to convert the temperature: \( C = K - 273.15 \) \( C = 178 - 273.15 \) \( C \approx -95.15^{\circ} \mathrm{C} \)

Step 6: Convert Celsius to Fahrenheit

Finally, use the Celsius to Fahrenheit formula: \( F = \frac{9}{5} C + 32 \) \( F = \frac{9}{5} (-95.15) + 32 \) \( F \approx -139.27^{\circ} \mathrm{F} \) The temperature in Celsius is \({-95.15}^{\circ} \mathrm{C} \) and in Fahrenheit, it is \({-139.27}^{\circ} \mathrm{F} \).

Key Concepts

Fahrenheit to Celsius

Understanding how to convert temperatures from Fahrenheit to Celsius is an essential skill for science classes and everyday life situations, like when traveling to a country that uses a different temperature scale. The formula for conversion is rooted in the fact that the Celsius and Fahrenheit scales have different starting points (freezing point of water) and scale increments per degree.

To convert from Fahrenheit to Celsius (\textdegree C), we use the following formula:
\[ C = \frac{5}{9} \times (F - 32) \]
Let's break down the formula. The subtraction of 32 is due to the freezing point of water being 0\textdegree C or 32\textdegree F. The multiplication by \frac{5}{9}\ is because a Celsius degree is larger than a Fahrenheit degree—so for every 5 degrees of temperature change in Celsius, there is a corresponding change of 9 degrees in the Fahrenheit scale. Applying this formula, a temperature of 72\textdegree F will translate into approximately 22.22\textdegree C.

Remember, in conversions, precision counts. The decimal places matter, especially in scientific contexts, and an approximate answer might be inappropriate. Always check whether an exact or an estimated value is acceptable in your situation.

Celsius to Kelvin

The Kelvin scale is vital in the scientific community because it is the SI unit of temperature and begins at absolute zero, the theoretically coldest possible temperature. Converting from Celsius to Kelvin is quite simple and does not involve the complexity of fractions as seen in the Fahrenheit to Celsius conversion.

The direct formula to convert Celsius to Kelvin (K) is:
\[ K = C + 273.15 \]
In this formula, you're essentially shifting the Celsius scale so that 0\textdegree C matches 273.15 K—the absolute zero on the Celsius scale. As a result, a temperature of -12\textdegree C is equal to 261.15 K. It's important to note that the 'degree' symbol is not used with the Kelvin scale, as it represents an 'absolute' temperature scale.

Precision is again important here. The '.15' in the formula is critical for accuracy, especially in scientific experiments where accurate temperature readings can significantly affect results. While some may use 273 as a quick estimate, keep in mind that this can introduce errors in precise calculations.

Kelvin to Fahrenheit

The conversion from Kelvin to Fahrenheit (\textdegree F) merges the complexities of both the conversions we've looked at so far. This is because we first convert Kelvin to Celsius and then Celsius to Fahrenheit. Here's the process in two main steps:

First, we convert Kelvin to Celsius using the inverse of the formula used for Celsius to Kelvin conversion.
\[ C = K - 273.15 \]
Once we have the temperature in Celsius, we then apply the Celsius to Fahrenheit formula:
\[ F = \frac{9}{5} \times C + 32 \]
This means for a temperature of 178 K, we end up with a temperature of around -95.15\textdegree C, which we then convert to -139.27\textdegree F. Each step must be followed exactly to ensure accuracy—keeping the order of operations in mind is crucial, as failing to subtract before multiplying or dividing could give an incorrect result.

Understanding how to seamlessly navigate between these scales is not only about memorizing formulas—it's about recognizing how temperature is represented in different parts of the world and in different scientific contexts. Being comfortable with these conversions facilitates better comprehension of global weather reports, scientific data, and even cooking recipes.