Problem 34 - Chemistry The Central Science In Si Units

Question

(a) A child has a fever of \(101^{\circ} \mathrm{F}\). What is the temperature in \({ }^{\circ} \mathrm{C} ?\) (b) In a desert, the temperature can be as high as \(45^{\circ} \mathrm{C},\) what is the temperature in \({ }^{\circ}\) ? (c) During winter, the temperature of the Arctic region can drop below \(-50^{\circ} \mathrm{C},\) what is the temperature in degree Fahrenheit and in Kelvin? (d) The sublimation temperature of dry ice is \(-78.5^{\circ} \mathrm{C}\). Convert this temperature to degree Fahrenheit and Kelvin. (e) Ethanol boils at \(351 \mathrm{~K}\). Convert this temperature to degree Fahrenheit and degree Celsius.

Step-by-Step Solution

Answer

(a) About 38.3°C, (b) 113°F, (c) -58°F and 223.15 K, (d) -109.3°F and 194.65 K, (e) 77.85°C and 172.13°F.

Step 1: Convert Fahrenheit to Celsius

To convert the temperature from Fahrenheit (6F) to Celsius (6C), use the formula:\[6C = \frac{5}{9} ( 6F - 32 )\]For part (a), substitute 6F = 101:\[6C = \frac{5}{9} (101 - 32) = \frac{5}{9} (69) \approx 38.3^{6}C\]

Step 2: Convert Celsius to Fahrenheit

To convert the temperature from Celsius (6C) to Fahrenheit (6F), use the formula:\[6F = \frac{9}{5} 6C + 32\]For part (b), substitute 6C = 45:\[6F = \frac{9}{5} (45) + 32 = 81 + 32 = 113^{6}F\]

Step 3: Convert Celsius to Fahrenheit and Kelvin (Sub-zero temperatures)

For part (c), first convert the temperature from Celsius to Fahrenheit using the same formula as in Step 2, where 6C = -50:\[6F = \frac{9}{5} (-50) + 32 = -90 + 32 = -58^{6}F\]Next, convert Celsius to Kelvin using the formula:\[K = 6C + 273.15\]Substitute 6C = -50:\[K = -50 + 273.15 = 223.15 6K\]

Step 4: Convert Celsius to Fahrenheit and Kelvin (Specific temperature)

For part (d), follow the same formulas. Convert 6C = -78.5 to Fahrenheit:\[6F = \frac{9}{5} (-78.5) + 32 = -109.3^{6}F\]To convert to Kelvin:\[K = -78.5 + 273.15 = 194.65 6K\]

Step 5: Convert Kelvin to Fahrenheit and Celsius

For part (e), to convert from Kelvin to Celsius, use the formula:\[6C = K - 273.15\]Substitute K = 351:\[6C = 351 - 273.15 = 77.85^{6}C\]Now, convert this Celsius temperature to Fahrenheit as done in Step 2:\[6F = \frac{9}{5} (77.85) + 32 \approx 172.13^{6}F\]

Key Concepts

Celsius to Fahrenheit

Understanding how to convert Celsius to Fahrenheit is crucial for interpreting temperature readings in regions using different temperature scales. The formula used is:\[6F = \frac{9}{5} 6C + 32\]Here's a step-by-step explanation of the conversion process:

• Multiply the Celsius temperature by \(\frac{9}{5}\). This step translates the heavier scale of degrees Celsius to the lighter scale of degrees Fahrenheit, which has a larger range between the freezing and boiling points of water.
• Add 32 to the result. This accounts for the offset in the starting point between the two scales, as water freezes at 32°F but at 0°C.

Let's use an example: If the temperature in a desert is 45°C, you can convert it as follows:\[6F = \frac{9}{5} \times 45 + 32 = 113^{\circ}\mathrm{F}\]This formula is straightforward once you understand the need for multiplicative and additive adjustments between Celsius and Fahrenheit scales.

Fahrenheit to Celsius

To switch from Fahrenheit to Celsius, you need to reverse the conversion process with a formula:\[6C = \frac{5}{9} (6F - 32)\]Here's how it works:

• Subtract 32 from the Fahrenheit temperature. This removes the scale offset, aligning the starting points of the Celsius and Fahrenheit scales.
• Multiply the result by \(\frac{5}{9}\). This adjustment resizes the measured temperature to fit the Celsius scale, which uses smaller intervals compared to Fahrenheit.

Take an example: To convert 101°F to Celsius for a child's fever reading:\[6C = \frac{5}{9} \times (101 - 32) \approx 38.3^{\circ}\mathrm{C}\]This conversion is essential when you need precise temperature readings, such as for medical purposes, where Celsius is often the standard.

Celsius to Kelvin

Converting from Celsius to Kelvin is a simpler process compared to Fahrenheit conversions, since both Celsius and Kelvin are metric units. Here's the formula you need:\[K = 6C + 273.15\]Here's a detailed breakdown:

• Simply add 273.15 to the Celsius temperature. This accounts for the difference between the zero points in both scales, with absolute zero (0 K) being equivalent to -273.15°C.

For instance, when converting a temperature of -50°C (as can be in the Arctic during winter) to Kelvin:\[K = -50 + 273.15 = 223.15 6K\]Kelvin is often used in scientific contexts, as it starts from absolute zero, providing a clear measure for temperature-dependent research.

Kelvin to Celsius

The Kelvin to Celsius conversion is perhaps the most straightforward as it involves a simple addition or subtraction due to their corresponding zero points. To convert Kelvin to Celsius:\[6C = K - 273.15\]To convert a temperature of 351K (the boiling point of ethanol) back to Celsius, you would:\[6C = 351 - 273.15 = 77.85^{\circ}\mathrm{C}\]Here's what you need to remember:

• The subtracting of 273.15 realigns the temperatures to a familiar Celsius scale.
• This conversion is useful in everyday applications as well as scientific research, as it helps portray temperature changes intuitively.

The Kelvin scale's zero is absolute zero, making it efficient for scientific matters, while Celsius provides better practical understanding for daily temperature experiences.