Problem 9 - Chemistry for Allied Health

Question

Convert each temperature to the missing one between Celsius and Fahrenheit. a. \(77^{\circ} \mathrm{F}\) b. \(212^{\circ} \mathrm{F}\) c. \(37^{\circ} \mathrm{C}\) d. \(22^{\circ} \mathrm{C}\) e. \(95^{\circ} \mathrm{F}\) f. \(15^{\circ} \mathrm{C}\) g. \(0^{\circ} \mathrm{F}\) h. \(0^{\circ} \mathrm{C}\) i. \(-10^{\circ} \mathrm{C}\) j. \(-10^{\circ} \mathrm{F}\)

Step-by-Step Solution

Answer

77°F = 25°C, 212°F = 100°C, 37°C = 98.6°F, 22°C = 71.6°F, 95°F = 35°C, 15°C = 59°F, 0°F = -17.78°C, 0°C = 32°F, -10°C = 14°F, -10°F = -23.33°C.

Step 1: Conversion Formula from Fahrenheit to Celsius

The formula to convert from Fahrenheit to Celsius is given by \( C = \frac{5}{9}(F - 32) \). We'll use this formula to convert the Fahrenheit temperatures to Celsius.

Step 2: Convert 77°F to Celsius

For 77°F: \(C = \frac{5}{9}(77 - 32) = \frac{5}{9} \times 45 = 25^{\circ} \mathrm{C}\).

Step 3: Convert 212°F to Celsius

For 212°F: \(C = \frac{5}{9}(212 - 32) = \frac{5}{9} \times 180 = 100^{\circ} \mathrm{C}\).

Step 4: Convert 95°F to Celsius

For 95°F: \(C = \frac{5}{9}(95 - 32) = \frac{5}{9} \times 63 = 35^{\circ} \mathrm{C}\).

Step 5: Convert 0°F to Celsius

For 0°F: \(C = \frac{5}{9}(0 - 32) = \frac{5}{9} \times (-32) = -17.78^{\circ} \mathrm{C}\).

Step 6: Convert -10°F to Celsius

For -10°F: \(C = \frac{5}{9}(-10 - 32) = \frac{5}{9} \times (-42) = -23.33^{\circ} \mathrm{C}\).

Step 7: Conversion Formula from Celsius to Fahrenheit

The formula to convert from Celsius to Fahrenheit is given by \( F = \frac{9}{5}C + 32 \). We'll use this formula to convert the Celsius temperatures to Fahrenheit.

Step 8: Convert 37°C to Fahrenheit

For 37°C: \(F = \frac{9}{5}\times 37 + 32 = 66.6 + 32 = 98.6^{\circ} \mathrm{F}\).

Step 9: Convert 22°C to Fahrenheit

For 22°C: \(F = \frac{9}{5}\times 22 + 32 = 39.6 + 32 = 71.6^{\circ} \mathrm{F}\).

Step 10: Convert 15°C to Fahrenheit

For 15°C: \(F = \frac{9}{5} \times 15 + 32 = 27 + 32 = 59^{\circ} \mathrm{F}\).

Step 11: Convert 0°C to Fahrenheit

For 0°C: \(F = \frac{9}{5} \times 0 + 32 = 32^{\circ} \mathrm{F}\).

Step 12: Convert -10°C to Fahrenheit

For -10°C: \(F = \frac{9}{5}\times (-10) + 32 = -18 + 32 = 14^{\circ} \mathrm{F}\).

Key Concepts

Fahrenheit to Celsius

Converting a temperature from Fahrenheit to Celsius might seem challenging, but it's straightforward once you familiarize yourself with the formula.
Begin with this: when converting, think of the number 32 as your starting point. Subtract 32 from the Fahrenheit temperature.
This adjustment sets the stage for conversion. Next, multiply the result by \(\frac{5}{9}\). This step accounts for the difference between the Fahrenheit and Celsius scales. For example, to convert \(77^{\circ} \mathrm{F}\) to Celsius: subtract 32, leaving you with 45, and then multiply by \(\frac{5}{9}\). The conversion results in \(25^{\circ} \mathrm{C}\). Remember, the formula is:

• \(C = \frac{5}{9}(F - 32)\)

Celsius to Fahrenheit

Turning a Celsius temperature into Fahrenheit is equally simple when you get a handle on the respective formula.
The process begins by multiplying the Celsius temperature by \(\frac{9}{5}\).
This accounts for the broader range of degrees on the Fahrenheit scale compared to Celsius. After multiplying, add 32 to complete the conversion.
Take, for example, \(37^{\circ} \mathrm{C}\): multiply by \(\frac{9}{5}\) to get 66.6, then add 32, arriving at \(98.6^{\circ} \mathrm{F}\). Always stick to this formula:

• \(F = \frac{9}{5}C + 32\)

temperature conversion formula

Understanding temperature conversion formulas is essential for accurately switching between Fahrenheit and Celsius.
These formulas are key tools that make conversions easy and prevent common mistakes like switching symbols or forgetting crucial steps.
Here's a refreshing note: both formulas are derived from the fixed points known between water’s freezing and boiling points in each scale.
This mathematical relationship ensures that each conversion is precise and consistent. Simply put:- **Fahrenheit to Celsius**: \[C = \frac{5}{9}(F - 32)\] - **Celsius to Fahrenheit**: \[F = \frac{9}{5}C + 32\] As part of learning these formulas, practice repeatedly. It speeds up the process, strengthens your understanding, and builds confidence.

step by step solution

Applying conversion step by step helps solidify understanding, ensuring accuracy every time.
A systematic approach can unravel the process, making it feel less daunting.First, decide which scale you're converting from, and identify the formula you'll use.
Next, substitute the given temperature into the appropriate conversion formula.
Carefully follow each algebraic step: subtraction, multiplication, division, or addition, depending on the formula.For instance:

• To convert \(95^{\circ} \mathrm{F}\) to Celsius: \(C = \frac{5}{9}(95 - 32)\), so you would subtract 32 from 95, resulting in 63, and then multiply by \(\frac{5}{9}\) to get \(35^{\circ} \mathrm{C}\).
• For \(-10^{\circ} \mathrm{C}\) to Fahrenheit: \(F = \frac{9}{5}(-10) + 32\), multiply \(-10\) by \(\frac{9}{5}\) to get \(-18\), then add 32 to reach \(14^{\circ} \mathrm{F}\).

Following steps systematically clarifies your understanding, making these everyday conversions feel natural and intuitive.