Problem 103 - Chemistry Principles and Practice

Question

The average body temperature of a cow is about \(101.5^{\circ} \mathrm{F}\). Express this in degrees Celsius and in kelvins, using the correct number of significant figures.

Step-by-Step Solution

Answer

38.61 °C, 311.8 K.

Step 1: Convert Fahrenheit to Celsius

To convert from Fahrenheit to Celsius, use the formula: \( C = \frac{5}{9}(F - 32) \), where \( F \) is the temperature in Fahrenheit. Substitute \( F = 101.5 \) into the formula: \( C = \frac{5}{9}(101.5 - 32) \). Calculate to find \( C = \frac{5}{9}(69.5) \). This simplifies to \( C = 38.6 \).

Step 2: Convert Celsius to Kelvins

Once you have the temperature in Celsius, convert to Kelvins using the formula: \( K = C + 273.15 \). Substitute \( C = 38.6 \) into the formula: \( K = 38.6 + 273.15 \). Calculate to find \( K = 311.75 \).

Step 3: Apply Significant Figures

The original temperature (101.5 °F) has four significant figures. Therefore, ensure the temperatures in Celsius and Kelvin are both expressed with four significant figures. Thus, the Celsius temperature is 38.61 °C, and the Kelvin temperature is 311.8 K.

Key Concepts

Understanding Significant Figures

Significant figures are crucial for expressing precision in numerical measurements and calculations. They represent all the digits that contribute to a measurement's accuracy, including certain digits and the first uncertain digit. Using significant figures correctly ensures that the precision of a result reflects the precision of the input data.

Here are some key tips on recognizing and applying significant figures:

• Non-zero digits are always significant. For example, in 101.5, all digits are significant.
• Any zeros between significant digits are also significant, like in 101.5.
• Leading zeros, or zeros before non-zero digits, are not significant.
• Trailing zeros in a decimal number are significant, such as the "0" in 38.60 when expressing four significant figures.

When converting units in problems, make sure to maintain the same number of significant figures to ensure accuracy. This concept ensures consistency and precision, making calculations more reliable.

Fahrenheit to Celsius Conversion

Converting temperatures from Fahrenheit to Celsius involves a straightforward formula. The formula is:

• \( C = \frac{5}{9}(F - 32) \)

Where \( C \) is the temperature in Celsius, and \( F \) is the temperature in Fahrenheit.

For instance, to convert 101.5°F to Celsius, you would subtract 32 from 101.5, giving 69.5. Then, multiply by \( \frac{5}{9} \), achieving 38.61 when rounded to four significant figures.
This process provides an accurate Celsius temperature in line with the original data's precision. It's a common conversion in science and everyday life, useful for understanding weather patterns or even cooking temperatures.

Celsius to Kelvin Conversion

Converting from Celsius to Kelvin is incredibly simple yet vital for scientific calculations. The Kelvin scale is the absolute temperature scale, starting from absolute zero, the theoretical point where all molecular motion ceases.

The conversion formula is:

• \( K = C + 273.15 \)

Where \( K \) is the temperature in Kelvin, and \( C \) is the temperature in Celsius.

For example, after converting 101.5°F to 38.61°C, the conversion to Kelvin would be:

• \( K = 38.61 + 273.15 \)

This equates to 311.76 K. It's essential to adjust to the same number of significant figures, thus reporting it as 311.8 K. Understanding this conversion allows for easy temperature transitions between Celsius and Kelvin, crucial for scientific studies and laboratory practices.