Problem 33 - Chemistry: The Central Science in SI Units, Global Edition

Question

Make the following conversions: (a) \(83^{\circ} \mathrm{F}\) to \({ }^{\circ} \mathrm{C}\) (b) \(29^{\circ} \mathrm{C}\) to \({ }^{\circ} \mathrm{F}\) (c) \(294^{\circ} \mathrm{C}\) to \(\mathrm{K}\) (d) \(832 \mathrm{~K}\) to \({ }^{\circ} \mathrm{C}\) (f) \(35^{\circ} \mathrm{F}\) to \(\mathrm{K}\). (e) \(721 \mathrm{~K}\) to \({ }^{\circ} \mathrm{F}\)

Step-by-Step Solution

Answer

(a) \(83^{\circ} \mathrm{F} \approx 28.33^{\circ} \mathrm{C}\) (b) \(29^{\circ} \mathrm{C} = 84.2^{\circ} \mathrm{F}\) (c) \(294^{\circ} \mathrm{C} = 567.15 \mathrm{~K}\) (d) \(832 \mathrm{~K} = 558.85^{\circ} \mathrm{C}\) (e) \(35^{\circ} \mathrm{F} \approx 274.82 \mathrm{~K}\) (f) \(721 \mathrm{~K} \approx 838.13^{\circ} \mathrm{F}\)

Step 1: a) 83°F to °C

Use the Fahrenheit to Celsius conversion formula: \(C = \frac{5}{9} (F - 32)\) Let F = 83, then \(C = \frac{5}{9} (83 - 32)\) Calculate the result: \(C = \frac{5}{9} (51) = 28.333\) So, \(83^{\circ} \mathrm{F} \approx 28.33^{\circ} \mathrm{C}\)

Step 2: b) 29°C to °F

Use the Celsius to Fahrenheit conversion formula: \(F = \frac{9}{5} C + 32\) Let C = 29, then \(F = \frac{9}{5} (29) + 32\) Calculate the result: \(F = \frac{9}{5} (29) + 32 = 84.2\) So, \(29^{\circ} \mathrm{C} = 84.2^{\circ} \mathrm{F}\)

Step 3: c) 294°C to K

Use the Celsius to Kelvin conversion formula: \(K = C + 273.15\) Let C = 294, then \(K = 294 + 273.15\) Calculate the result: \(K = 294 + 273.15 = 567.15\) So, \(294^{\circ} \mathrm{C} = 567.15 \mathrm{~K}\)

Step 4: d) 832 K to °C

Use the Kelvin to Celsius conversion formula: \(C = K - 273.15\) Let K = 832, then \(C = 832 - 273.15\) Calculate the result:\(C = 832 - 273.15 = 558.85\) So, \(832 \mathrm{~K} = 558.85^{\circ} \mathrm{C}\)

Step 5: e) 35°F to K

First, convert Fahrenheit to Celsius using the formula: \(C = \frac{5}{9} (F - 32)\) Let F = 35, then \(C = \frac{5}{9} (35 - 32) = 1.666\) Now, convert Celsius to Kelvin using the formula: \(K = C + 273.15\) Let C = 1.666, then \(K = 1.666 + 273.15 = 274.816\) So, \(35^{\circ} \mathrm{F} \approx 274.82 \mathrm{~K}\)

Step 6: f) 721 K to °F

First, convert Kelvin to Celsius using the formula: \(C = K - 273.15\) Let K = 721, then \(C = 721 - 273.15 = 447.85\) Now, convert Celsius to Fahrenheit using the formula: \(F = \frac{9}{5} C + 32\) Let C = 447.85, then \(F = \frac{9}{5} (447.85) + 32 = 838.13\) So, \(721 \mathrm{~K} \approx 838.13^{\circ} \mathrm{F}\)

Key Concepts

Fahrenheit to Celsius conversion

Converting temperatures from Fahrenheit to Celsius is a common task in both everyday life and science. To make this conversion, we use the formula: \[C = \frac{5}{9} (F - 32)\] Here, \(C\) represents the temperature in Celsius and \(F\) stands for the temperature in Fahrenheit. The formula subtracts 32 from the Fahrenheit value to account for the freezing point difference, then multiplies the result by \(\frac{5}{9}\) because there are 180 degrees between the freezing and boiling points of water in Fahrenheit, compared to 100 degrees in Celsius. Let's say we want to convert \(83^{\circ} F\) to Celsius. We plug the numbers into the formula like this:\[C = \frac{5}{9} (83 - 32) = 28.333\]So, \(83^{\circ} F\) is approximately \(28.33^{\circ} C\). This method is crucial for converting weather data or cooking instructions, ensuring accurate and consistent temperature readings.

Celsius to Kelvin conversion

Kelvin is often used in scientific calculations where absolute temperature is needed. It's a thermodynamic temperature scale where 0 K is absolute zero. To convert from Celsius to Kelvin, we use the straightforward formula: \[K = C + 273.15\] Here's why: the Kelvin scale ensures that zero reflects the absence of thermal energy. Unlike Celsius, which sets 0°C at the freezing point of water, Kelvin sets 0K as absolute zero. This makes it very useful in physics.Suppose we need to convert \(294^{\circ} C\) to Kelvin. Simply add 273.15: \[K = 294 + 273.15 = 567.15\]Therefore, \(294^{\circ} C\) equals \(567.15 K\). This addition shifts the entire temperature scale, preserving intervals but aligning it with absolute energy, useful for scientific measurements.

Kelvin to Fahrenheit conversion

Converting Kelvin directly to Fahrenheit involves a two-step process. First, we need to transform the temperature from Kelvin to Celsius, and then from Celsius to Fahrenheit. Let's break this down:1. **Kelvin to Celsius**: Subtract 273.15 from the Kelvin temperature since Celsius starts at the freezing point of water which is 0, whereas Kelvin starts at absolute zero. Use this formula: \[ C = K - 273.15 \]2. **Celsius to Fahrenheit**: Once in Celsius, convert to Fahrenheit using this formula: \[ F = \frac{9}{5} C + 32 \]For example, to convert \(721 K\) to Fahrenheit:- First, convert to Celsius: \[ C = 721 - 273.15 = 447.85 \]- Then convert Celsius to Fahrenheit: \[ F = \frac{9}{5} (447.85) + 32 = 838.13 \]Thus, \(721 K\) is approximately \(838.13^{\circ} F\). This two-step method ensures you understand the relation between all three temperature scales.