Question

(a) The temperature on a warm summer day is \(87^{\circ} \mathrm{F}\). What is the temperature in \({ }^{\circ} \mathrm{C}\) ? (b) Many scientific data are reported at \(25{ }^{\circ} \mathrm{C}\). What is this temperature in kelvins and in degrees Fahrenheit? (c) Suppose that a recipe calls for an oven temperature of \(175^{\circ} \mathrm{F}\). Convert this temperature to degrees Celsius and to kelvins.(d) The melting point of sodium bromide (a salt) is \(755^{\circ} \mathrm{C}\). Calculate this temperature in \({ }^{\circ} \mathrm{F}\) and in kelvins. (e) Neon, a gaseous element at room temperature, is used to make electronic signs. Neon has a melting point of \(-248.6^{\circ} \mathrm{C}\) and a boiling point of \(-246.1^{\circ} \mathrm{C}\). Convert these temperatures to kelvins.

Short answer

(a) The temperature in Celsius is approximately \(30.56^\circ C\). (b) The temperature is \(298.15 \, \text{K}\) and \(77^\circ F\). (c) The oven temperature is approximately \(79.44^\circ C\) and \(352.59 \, \text{K}\). (d) The melting point of sodium bromide is approximately \(1391^\circ F\) and \(1028.15 \, \text{K}\). (e) The melting and boiling points of neon are approximately \(24.55 \, \text{K}\) and \(27.05 \, \text{K}\), respectively.

Step-by-step solution

Fahrenheit to Celsius formula

Use the formula to convert Fahrenheit to Celsius: \(C = \frac{5}{9}(F - 32)\).

Apply the formula

Plug the given temperature in Fahrenheit, \(87^\circ F\), into the formula: \(C = \frac{5}{9}(87 - 32)\).

Calculate Celsius temperature

Calculate the temperature in Celsius: \(C = \frac{5}{9}(55) \approx 30.56^\circ C\). #b. Celsius to Kelvin and Fahrenheit conversions#

Celsius to Kelvin formula

Use the formula to convert Celsius to Kelvin: \(K = C + 273.15\).

Apply Celsius to Kelvin formula

Plug the given temperature in Celsius, \(25^\circ C\), into the formula: \(K = 25 + 273.15\).

Calculate Kelvin temperature

Calculate the temperature in Kelvin: \(K = 298.15 \, \text{K}\).

Celsius to Fahrenheit formula

Use the formula to convert Celsius to Fahrenheit: \(F = \frac{9}{5}C + 32\).

Apply Celsius to Fahrenheit formula

Plug the given temperature in Celsius, \(25^\circ C\), into the formula: \(F = \frac{9}{5}(25) + 32\).

Calculate Fahrenheit temperature

Calculate the temperature in Fahrenheit: \(F = 77^\circ F\). #c. Fahrenheit to Celsius and Kelvin conversions# Repeat the process used in parts a and b for given Fahrenheit temperature: Fahrenheit to Celsius: \(C = \frac{5}{9}(175 - 32) \approx 79.44^\circ C\) Celsius to Kelvin: \(K = 79.44 + 273.15 \approx 352.59 \, \text{K}\) #d. Celsius to Fahrenheit and Kelvin conversions# Repeat the process used in part b for given Celsius temperature: Celsius to Fahrenheit: \(F = \frac{9}{5}(755) + 32 \approx 1391^\circ F\) Celsius to Kelvin: \(K = 755 + 273.15 = 1028.15 \, \text{K}\) #e. Celsius to Kelvin conversions#

Apply Celsius to Kelvin formula for melting point

Plug the given melting point in Celsius, \(-248.6^\circ C\), into the formula: \(K = -248.6 + 273.15\).

Calculate Kelvin melting point

Calculate the melting point in Kelvin: \(K = 24.55 \, \text{K}\).

Apply Celsius to Kelvin formula for boiling point

Plug the given boiling point in Celsius, \(-246.1^\circ C\), into the formula: \(K = -246.1 + 273.15\).

Calculate Kelvin boiling point

Calculate the boiling point in Kelvin: \(K = 27.05 \, \text{K}\).

Key concepts

Fahrenheit to Celsius

Converting temperatures from Fahrenheit to Celsius is a common practice in science and everyday life. The formula used for this conversion is:
\[ C = \frac{5}{9}(F - 32) \]
where:

• \( C \) is the temperature in degrees Celsius
• \( F \) is the temperature in degrees Fahrenheit

This formula originates from the need to align the freezing and boiling points of water between different scales. In Fahrenheit, water freezes at \(32^\circ F\) and boils at \(212^\circ F\), while in Celsius, water freezes at \(0^\circ C\) and boils at \(100^\circ C\). The conversion formula accounts for this difference.
To illustrate, if you have a Fahrenheit temperature of \(87^\circ F\), you first subtract \(32\) from \(87\), then multiply by \(\frac{5}{9}\) to find \(C \approx 30.56^\circ C\).
It's beneficial to memorize this simple formula if you find yourself needing to convert temperatures often.

Celsius to Kelvin

The transition from Celsius to Kelvin is quite straightforward, as both scales are metric-based. To convert Celsius to Kelvin, the formula is:
\[ K = C + 273.15 \]
where:

• \( K \) is the temperature in Kelvin
• \( C \) is the temperature in degrees Celsius

Kelvin is the absolute temperature scale often used in scientific contexts because it starts at absolute zero, the point at which all motion stops. The Celsius scale is offset by exactly \(273.15\) from the Kelvin scale.
For instance, converting \(25^\circ C\) to Kelvin involves simply adding \(273.15\), resulting in \( K = 298.15 \, \text{K} \).
This method provides a seamless transition between the two scales and is advantageous for calculations in thermodynamics and other scientific areas.

Celsius to Fahrenheit

To convert Celsius to Fahrenheit, a different formula is used, designed to re-align these systems:
\[ F = \frac{9}{5}C + 32 \]
where:

• \( F \) is the temperature in degrees Fahrenheit
• \( C \) is the temperature in degrees Celsius

This conversion is valuable for understanding temperatures in contexts where Fahrenheit is customary, such as everyday weather forecasts in the United States.
For example, converting a typical room temperature of \(25^\circ C\) into Fahrenheit means multiplying \(25\) by \(\frac{9}{5}\) and then adding \(32\), resulting in \( F = 77^\circ F\).
Understanding this formula helps in maintaining fluency between different measurement systems, which can often arise in fields like culinary arts, meteorology, and international travel.