Problem 34 - Chemistry: The Central Science in SI Units, Global Edition

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 \({ }^{\circ} \mathrm{F} ?\) (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) The child's fever is approximately \(38.3^{\circ}C\). (b) The desert temperature is \(113^{\circ}F\). (c) The Arctic winter temperature is \(-58^{\circ}F\) and \(223.15 \mathrm{K}\). (d) The dry ice sublimation temperature is approximately \(-109.3^{\circ}F\) and \(194.65 \mathrm{K}\). (e) The boiling point of ethanol is approximately \(77.85^{\circ}C\) and \(172.13^{\circ}F\).

Step 1: Use Fahrenheit to Celsius formula

To convert the fever from Fahrenheit to Celsius, use the formula: \(C = \frac{5}{9} (F - 32)\). Substitute the given temperature (101°F) for F and solve for C.

Step 2: Calculate Celsius temperature

Using the formula, we have: \(C = \frac{5}{9} (101 - 32)\) \(C = \frac{5}{9} \times 69\) \(C\approx 38.3^{\circ}C\) (b) Convert a desert high of 45°C to Fahrenheit

Step 1: Use Celsius to Fahrenheit formula

To convert the desert high temperature from Celsius to Fahrenheit, use the formula: \(F = \frac{9}{5} C + 32\). Substitute the given temperature (45°C) for C and solve for F.

Step 2: Calculate Fahrenheit temperature

Using the formula, we have: \(F = \frac{9}{5} \times 45 + 32\) \(F = 81 + 32\) \(F = 113^{\circ}F\) (c) Convert an Arctic winter low of -50°C to Fahrenheit and Kelvin

Step 1: Use Celsius to Fahrenheit formula to find Fahrenheit temperature

Substitute the given temperature (-50°C) for C in the formula: \(F = \frac{9}{5} C + 32\), and solve for F.

Step 2: Calculate Fahrenheit temperature

Using the formula, we have: \(F = \frac{9}{5} \times (-50) + 32\) \(F = -90 + 32\) \(F = -58^{\circ}F\)

Step 3: Use Celsius to Kelvin formula to find Kelvin temperature

Substitute the given temperature (-50°C) for C in the formula: \(K = C + 273.15\), and solve for K.

Step 4: Calculate Kelvin temperature

Using the formula, we have: \(K = -50 + 273.15\) \(K = 223.15 \mathrm{K}\) (d) Convert dry ice sublimation temperature -78.5°C to Fahrenheit and Kelvin

Step 1: Use Celsius to Fahrenheit formula to find Fahrenheit temperature

Substitute the given temperature (-78.5°C) for C in the formula, \(F = \frac{9}{5} C + 32\), and solve for F.

Step 2: Calculate Fahrenheit temperature

Using the formula, we have: \(F = \frac{9}{5} \times (-78.5) + 32\) \(F \approx -109.3^{\circ}F\)

Step 3: Use Celsius to Kelvin formula to find Kelvin temperature

Substitute the given temperature (-78.5°C) for C in the formula, \(K = C + 273.15\), and solve for K.

Step 4: Calculate Kelvin temperature

Using the formula, we have: \(K = -78.5 + 273.15\) \(K \approx 194.65 \mathrm{K}\) (e) Convert boiling point of ethanol 351K to Fahrenheit and Celsius

Step 1: Use Kelvin to Celsius formula to find Celsius temperature

Substitute the given temperature (351K) for K in the formula, \(C = K - 273.15\), and solve for C.

Step 2: Calculate Celsius temperature

Using the formula, we have: \(C = 351 - 273.15\) \(C \approx 77.85^{\circ}C\)

Step 3: Use Celsius to Fahrenheit formula to find Fahrenheit temperature

Substitute the calculated Celsius temperature (77.85°C) for C in the formula, \(F = \frac{9}{5} C + 32\), and solve for F.

Step 4: Calculate Fahrenheit temperature

Using the formula, we have: \(F = \frac{9}{5} \times 77.85 + 32\) \(F \approx 172.13^{\circ}F\)

Key Concepts

Celsius to Fahrenheit formula

When you need to convert a temperature from Celsius to Fahrenheit, you use the formula:

• \( F = \frac{9}{5} C + 32 \)

This formula helps you translate the more familiar Celsius scale, used widely around the world, into the Fahrenheit scale, often used in the United States. Let's break it down with an example:
Suppose you're standing in a desert, and the temperature is a scorching 45°C. To find out what this is in Fahrenheit, plug into our formula:

• \( F = \frac{9}{5} \times 45 + 32 \)
• \( F = 81 + 32 \)
• \( F = 113^{\circ}F \)

So, a temperature of 45°C is the same as 113°F. Whether for travel plans or scientific experiments, knowing this conversion formula is extremely useful.

Fahrenheit to Celsius formula

Converting Fahrenheit to Celsius is vital for understanding temperatures in a different context. The formula to use here is:

• \( C = \frac{5}{9}(F - 32) \)

This formula allows you to move from the Fahrenheit scale back to Celsius. Say, a child's fever registers at 101°F. To convert this to Celsius:

• \( C = \frac{5}{9} (101 - 32) \)
• \( C = \frac{5}{9} \times 69 \)
• \( C \approx 38.3^{\circ}C \)

With this calculation, you see the fever translates to about 38.3°C. This tool can help when you need to judge a condition's severity through temperature readings.

Celsius to Kelvin formula

The Kelvin scale is particularly useful in scientific contexts as it starts from absolute zero, the theoretical coldest temperature possible. To convert from Celsius to Kelvin, you use this formula:

• \( K = C + 273.15 \)

Let’s imagine it's extremely cold, like in the Arctic where the temperature might dip to -50°C. To find the equivalent in Kelvin:

• \( K = -50 + 273.15 \)
• \( K = 223.15 \text{ K} \)

No negative numbers exist on the Kelvin scale, representing a complete halt in thermal motion at zero. Temperature in Kelvin is crucial for studies in physics and chemistry, providing an absolute perspective on thermal energy.