Problem 85 - Chemistry The Science in Context

Question

A person has a fever of \(102.5^{\circ} \mathrm{F}\). What is this temperature in \(^{\circ} \mathrm{C} ?\)

Step-by-Step Solution

Answer

Answer: The temperature in Celsius is approximately \(39.2^\circ \mathrm{C}\).

Step 1: Write down the temperature to convert

The given temperature is \(102.5^\circ \mathrm{F}\).

Step 2: Recall the conversion formula from Fahrenheit to Celsius

The formula to convert temperature from Fahrenheit to Celsius is: \(^\circ \mathrm{C} = \dfrac{5}{9}(^\circ \mathrm{F} - 32).\)

Step 3: Substitute the given temperature into the formula

Replace \(^\circ \mathrm{F}\) in the formula with the given temperature, \(102.5^\circ \mathrm{F}\): \(^\circ \mathrm{C} = \dfrac{5}{9}(102.5^\circ \mathrm{F} - 32).\)

Step 4: Calculate the value inside the parentheses

Subtract 32 from the given temperature (102.5) to get the result within the parentheses: \(^\circ \mathrm{F} - 32 = 102.5 - 32 = 70.5.\)

Step 5: Multiply the result by the conversion factor

Multiply the result inside the parentheses by the conversion factor \(\dfrac{5}{9}\): \(^\circ \mathrm{C} = \dfrac{5}{9}(70.5) = 39.17 \approx 39.2.\)

Step 6: Write the final answer in Celsius

The temperature in Celsius is approximately \(39.2^\circ \mathrm{C}\).

Key Concepts

Fahrenheit to Celsius Conversion

Temperature conversion between different scales, specifically from Fahrenheit to Celsius, is a common task in both academic and everyday settings. Understanding how to perform this conversion is helpful, especially when traveling between countries that use different temperature scales. The formula to convert Fahrenheit (\(^\circ \mathrm{F}\)) to Celsius (\(^\circ \mathrm{C}\)) is:

• \[^\circ \mathrm{C} = \dfrac{5}{9}(^\circ \mathrm{F} - 32)\]

This formula works because it adjusts for the starting point differences between the scales (32°F is equivalent to 0°C), and accounts for the different scaling factor between the two systems. By subtracting 32, you remove the offset specific to the Fahrenheit scale. The scaling factor of \(\dfrac{5}{9}\) comes from the fact that a degree change in Celsius represents a larger physical temperature change than one in Fahrenheit. By multiplying the adjusted Fahrenheit by this factor, you align it accurately with the Celsius scale.

Temperature Measurement

Temperature is a crucial physical quantity that measures the hotness or coldness of an object. Different systems use different units to measure temperature, with Fahrenheit and Celsius being two of the most common. The Fahrenheit scale is primarily used in the United States, whereas most of the world uses Celsius.

The concept of absolute zero is important in understanding temperature. Absolute zero is the point at which there is no thermal energy at all, and it's a base point for the Kelvin scale, another major temperature scale.

• Fahrenheit and Celsius share no thermal measurement at zero as it varies with the freezing point of certain materials like water.
• Comparison and conversion between different scales become essential for various applications such as scientific research, cooking, and medical diagnostics.

Accurate measurement tools, such as thermometers, are essential in obtaining precise temperature readings. The standard for measuring temperature has evolved, but understanding the differences and connections between various units helps in interpreting and converting these measurements.

Mathematical Calculations

Mathematical calculations are integral to temperature conversions, ensuring precision and accuracy. When converting temperatures, always follow a step-by-step approach to avoid errors and obtain the correct result. Here’s a brief look at the process to solve such conversions:

• Define the problem: Identify the given temperature and what it needs to be converted to.
• Use the correct formula: Apply the correct conversion formula, such as \(^\circ \mathrm{C} = \dfrac{5}{9}(^\circ \mathrm{F} - 32)\) for Fahrenheit to Celsius.
• Substitute values: Insert the known values into the equation.
• Perform arithmetic operations: Begin with operations inside parentheses, then apply multiplication or division.

Error-checking steps like revisiting arithmetic or comparing predicted results can save time and avoid inaccuracies. Different calculation methods, including mental math, calculators, or software, can support these conversions.

Practicing these calculations enhances comprehension of both mathematics and scientific principles involved in temperature differences.