Problem 33 - Chemistry The Central Science In Si Units

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 \({ }^{\circ} \mathrm{C}\) (e) \(721 \mathrm{~K}\) to \({ }^{\circ} \mathrm{F}\) (f) \(35^{\circ} \mathrm{F}\) to \(\mathrm{K}\).

Step-by-Step Solution

Answer

(a) 28.33°C, (b) 84.2°F, (c) 567.15 K, (d) 558.85°C, (e) 838.13°F, (f) 274.82 K

Step 1: Fahrenheit to Celsius Formula

To convert degrees Fahrenheit to degrees Celsius, use the formula: \[ C = \frac{5}{9} (F - 32) \]

Step 1a: Convert 83°F to Celsius

Using the formula: \[ C = \frac{5}{9} (83 - 32) \]Calculate: \[ C = \frac{5}{9} (51) = \frac{255}{9} \approx 28.33^{\circ}\mathrm{C}\]

Step 2: Celsius to Fahrenheit Formula

To convert degrees Celsius to degrees Fahrenheit, use the formula: \[ F = \frac{9}{5} C + 32 \]

Step 2a: Convert 29°C to Fahrenheit

Using the formula: \[ F = \frac{9}{5} (29) + 32 \] Calculate: \[ F = 52.2 + 32 = 84.2^{\circ} \mathrm{F} \]

Step 3: Celsius to Kelvin Formula

To convert degrees Celsius to Kelvin, use the formula: \[ K = C + 273.15 \]

Step 3a: Convert 294°C to Kelvin

Using the formula: \[ K = 294 + 273.15 \] Calculate: \[ K = 567.15 \mathrm{~K} \]

Step 4: Kelvin to Celsius Formula

To convert Kelvin to degrees Celsius, use the formula: \[ C = K - 273.15 \]

Step 4a: Convert 832 K to Celsius

Using the formula: \[ C = 832 - 273.15 \] Calculate: \[ C = 558.85^{\circ} \mathrm{C} \]

Step 5: Kelvin to Fahrenheit Process

First, convert Kelvin to degrees Celsius using the formula: \[ C = K - 273.15 \] Then convert Celsius to Fahrenheit using the formula: \[ F = \frac{9}{5} C + 32 \]

Step 5a: Convert 721 K to Fahrenheit

First, convert to Celsius: \[ C = 721 - 273.15 = 447.85^{\circ} \mathrm{C} \] Then convert to Fahrenheit: \[ F = \frac{9}{5}(447.85) + 32 \] Calculate: \[ F \approx 838.13^{\circ} \mathrm{F} \]

Step 6: Fahrenheit to Kelvin Process

First, convert Fahrenheit to degrees Celsius using the formula: \[ C = \frac{5}{9} (F - 32) \]Then convert Celsius to Kelvin using the formula: \[ K = C + 273.15 \]

Step 6a: Convert 35°F to Kelvin

First, convert to Celsius: \[ C = \frac{5}{9}(35 - 32) = \frac{15}{9} = \frac{5}{3}^{\circ} \mathrm{C} \] Then convert to Kelvin: \[ K = \frac{5}{3} + 273.15 \approx 274.82 \mathrm{~K} \]

Key Concepts

Fahrenheit to Celsius Conversion

To grasp the process of converting Fahrenheit to Celsius, imagine you are transforming a temperature expressed in another language to one that you are more familiar with; Celsius. This conversion is common when traveling between countries using different temperature standards. To accurately translate Fahrenheit temperatures to Celsius, use the formula:\[ C = \frac{5}{9} (F - 32) \]Here's a step-by-step guide to conversion:

• Subtract 32 from the Fahrenheit temperature.
• Multiply the result by \( \frac{5}{9} \).

For example, convert \( 83^{\circ} \text{F} \) to Celsius:\[ C = \frac{5}{9} (83 - 32) = \frac{5}{9} \times 51 = \frac{255}{9} \approx 28.33^{\circ} \text{C} \]This means \( 83^{\circ} \text{F} \) translates to approximately \( 28.33^{\circ} \text{C} \). Remember that Celsius centers around the freezing and boiling points of water, making it intuitive for daily use.

Celsius to Kelvin Conversion

Understanding the transition from Celsius to Kelvin can be crucial, particularly in scientific settings where Kelvin is the standard unit. Unlike Celsius, which begins at the freezing point of water, Kelvin starts from absolute zero, the theoretical lowest possible temperature.The formula to convert Celsius to Kelvin is straightforward:\[ K = C + 273.15 \]Here's how you can convert step-by-step:

• Take the Celsius temperature.
• Add 273.15 to it.

For instance, if you want to convert \( 294^{\circ} \text{C} \) to Kelvin:\[ K = 294 + 273.15 = 567.15 \text{ K} \]This process shows that \( 294^{\circ} \text{C} \) is equivalent to \( 567.15 \text{ K} \). Remember that Kelvin doesn’t use the term "degrees," because it measures absolute temperature, keeping scientists precise in their experiments and observations.

Kelvin to Fahrenheit Conversion

The Kelvin to Fahrenheit conversion might seem a bit more complex since it involves a two-step process. This transformation is less typical in everyday life but crucial in specific scientific contexts.To convert from Kelvin to Fahrenheit, follow these steps:

• First, convert Kelvin to Celsius using:
• \[ C = K - 273.15 \]
• Next, convert the Celsius temperature to Fahrenheit using:
• \[ F = \frac{9}{5} C + 32 \]

To illustrate, let's convert \( 721 \text{ K} \) to Fahrenheit:First, convert to Celsius:\[ C = 721 - 273.15 = 447.85^{\circ} \text{C} \]Then, convert to Fahrenheit:\[ F = \frac{9}{5} \times 447.85 + 32 \approx 838.13^{\circ} \text{F} \]This method shows that \( 721 \text{ K} \) is equivalent to \( 838.13^{\circ} \text{F} \). Understanding this conversion is valuable in scientific environments where precise temperature readings across different scales are needed.