Problem 70 - Algebra 2

Question

For Exercises 69 and \(70,\) use the following information. There are three temperature scales: Fahrenheit \(\left(^{\circ} \mathrm{F}\right),\) Celsius \(\left(^{\circ} \mathrm{C}\right)\) and Kelvin (K). The function \(K(C)=C+273\) can be used to convert Celsius temperatures to Kelvin. The function \(C(F)=\frac{5}{9}(F-32)\) can be used to convert Fahrenheit temperatures to Celsius. Find the temperature in Kelvin for the boiling point of water and the freezing point of water boils at \(212^{\circ} \mathrm{F}\) and freezes at \(32^{\circ} \mathrm{F} .\)

Step-by-Step Solution

Answer

Boiling point: 373 K, Freezing point: 273 K.

Step 1: Convert boiling point from Fahrenheit to Celsius

The boiling point of water is given as \( 212^{\circ} \mathrm{F} \). To convert this to Celsius, we'll use the function \( C(F) = \frac{5}{9}(F - 32) \). Substitute \( F = 212 \) into the formula: \[ C(212) = \frac{5}{9} \times (212 - 32) = \frac{5}{9} \times 180 = 100^{\circ} \mathrm{C} \].

Step 2: Convert boiling point from Celsius to Kelvin

Now that we have the boiling point in Celsius (\( 100^{\circ} \mathrm{C} \)), we'll convert it to Kelvin using the function \( K(C) = C + 273 \). Substitute \( C = 100 \): \[ K(100) = 100 + 273 = 373 \; \mathrm{K} \].

Step 3: Convert freezing point from Fahrenheit to Celsius

The freezing point of water is given as \( 32^{\circ} \mathrm{F} \). We'll use the same function \( C(F) = \frac{5}{9}(F - 32) \) to convert this to Celsius. Substitute \( F = 32 \) into the formula: \[ C(32) = \frac{5}{9} \times (32 - 32) = \frac{5}{9} \times 0 = 0^{\circ} \mathrm{C} \].

Step 4: Convert freezing point from Celsius to Kelvin

Now we'll convert the freezing point in Celsius (\( 0^{\circ} \mathrm{C} \)) to Kelvin using the function \( K(C) = C + 273 \). Substitute \( C = 0 \): \[ K(0) = 0 + 273 = 273 \; \mathrm{K} \].

Key Concepts

Fahrenheit to Celsius conversion

Converting temperatures from Fahrenheit to Celsius is a common task in scientific fields and everyday life. The conversion formula is \( C(F) = \frac{5}{9}(F - 32) \). This formula helps translate the temperature from the Fahrenheit scale, which is often used in the United States, to the Celsius scale, used in most other parts of the world. Let's take an example: If the temperature is noted as \( 212^{\circ} \mathrm{F} \), you start by subtracting \( 32 \) from \( 212 \), giving you \( 180 \). Multiply this by \( \frac{5}{9} \) to convert to Celsius, resulting in \( 100^{\circ} \mathrm{C} \). Therefore, the boiling point of water in Celsius is \( 100^{\circ} \mathrm{C} \).

• Subtract \( 32 \) from Fahrenheit value.
• Multiply the result by \( \frac{5}{9} \).
• The result is the temperature in Celsius.

Celsius to Kelvin conversion

In scientific terms, converting from Celsius to Kelvin is crucial as Kelvin is the standard unit of measurement for temperature in the physical sciences. The conversion is straightforward: \( K(C) = C + 273 \). This conversion is simple because the Kelvin scale is directly aligned with Celsius and starts at absolute zero, which is equivalent to \(-273^{\circ} \mathrm{C} \). For example, if the boiling point of water is \( 100^{\circ} \mathrm{C} \), you can convert it to Kelvin by adding \( 273 \). This gives you \( 373 \; \mathrm{K} \). The Kelvin scale does not use the degree symbol, and supports a wider range of scientific studies due to its absolute scale.

• Add \( 273 \) to the Celsius temperature.
• The result is the temperature in Kelvin.

Boiling point of water

The boiling point of water is a well-known reference point in temperature measurement. It occurs when water transitions from liquid to gas at \( 212^{\circ} \mathrm{F} \) or \( 100^{\circ} \mathrm{C} \) under normal atmospheric pressure. This is an important constant in chemistry and physics as it helps define the transition states of water and is used in calibrating thermometers. Once converted from Fahrenheit to Celsius with our formula, it can also be brought to Kelvin using the Celsius to Kelvin conversion, resulting in \( 373 \; \mathrm{K} \). Knowing the boiling point helps in applications like cooking, understanding atmospheric pressure effects, and even in designing heating equipment.

• The boiling point in Fahrenheit is \( 212^{\circ} \mathrm{F} \).
• Convert to Celsius as \( 100^{\circ} \mathrm{C} \).
• Convert to Kelvin as \( 373 \; \mathrm{K} \).

Freezing point of water

The freezing point of water marks the transition from liquid to solid for H₂O. It is one of the standard reference points for temperature scales. Water freezes at \( 32^{\circ} \mathrm{F} \) or \( 0^{\circ} \mathrm{C} \). This fundamental temperature change is critical for areas like meteorology, environmental sciences, and general climate studies. Upon converting from Fahrenheit using \( C(F) = \frac{5}{9}(F - 32) \), we see that the Celsius freezing point is \( 0^{\circ} \mathrm{C} \). Switching from Celsius to Kelvin, this becomes \( 273 \; \mathrm{K} \). Understanding this point is vital for studying physical properties of water, influences on ecosystems, and engineering applications in cold environments.

• Freezing in Fahrenheit is \( 32^{\circ} \mathrm{F} \).
• Convert to Celsius yields \( 0^{\circ} \mathrm{C} \).
• Convert to Kelvin turns into \( 273 \; \mathrm{K} \).