Question

Select the warmer temperature in each pair. a. \(10^{\circ} \mathrm{C}\) or \(10^{\circ} \mathrm{F}\) b. \(30^{\circ} \mathrm{C}\) or \(15^{\circ} \mathrm{F}\) c. \(-10^{\circ} \mathrm{C}\) or \(32^{\circ} \mathrm{F}\) d. \(200^{\circ} \mathrm{C}\) or \(200 \mathrm{~K}\)

Short answer

A) 10°C, B) 30°C, C) 0°C, D) 200°C.

Step-by-step solution

- Convert Fahrenheit to Celsius for part (a)

Use the formula: \ \ C = \frac{5}{9}(F - 32) \ \ \(10^{\rm{o}}F\) becomes \ \ C = \frac{5}{9}(10 - 32)\ \ C= \frac{5}{9}(-22)\= -12.22^\rm{o} C \ \ Therefore, the warmer temperature between \(10^{\rm{o}}C \ and -12.22^\rm{o}C\) is \ \(10^{\rm{o}} C\).

- Convert Fahrenheit to Celsius for part (b)

Use the formula: \ \ C = \frac{5}{9}(F - 32) \ \(15^{\rm{o}} F\) becomes: \ C = \frac{5}{9}(15-32) \ \ C = \frac{5}{9}(-17) \text{C}\ \ C = -9.44^{\rm{o}}C \ \ Therefore, the warmer temperature between \(30^{\rm{o}}C\ and -9.44^{\rm{o}}C\) is \ \ \(30^{\rm{o}}C\).

- Convert Fahrenheit to Celsius for part (c)

Use the formula: \ \ C = \frac{5}{9}(F - 32) \ \(32^{\rm{o}}F \) becomes \ \ C = \frac{5}{9}(32-32)\ \ C=0^{\rm{o}}C\ \ Therefore, the warmer temperature between \ \-10^{\rm{o}}C and 0^\rm{o} C \ is \ \ \ 0^{\rm{o}} C\.

- Convert Kelvin to Celsius for part (d)

Use the formula: \ \C = K - 273.15\ \ For \(200 \rm{K} \) \ \ \ \C = 200 - 273.15 \ \ C = -73.15^{\rm{o}}C\ \ Therefore, the warmer temperature between \(200^{\rm{o}}C\ and -73.15^{\rm{o}}\C\) is \(200^{\rm{o}}C\).

Key concepts

Celsius to Fahrenheit

Converting Celsius to Fahrenheit is a common need in both everyday life and scientific contexts. The conversion is quite simple and uses the formula \( F = \frac{9}{5}C + 32 \). Here, \( F \) represents degrees Fahrenheit and \( C \) represents degrees Celsius.

For example, if you have a temperature of \(25^\rm{o}C \), you can convert it to Fahrenheit using the formula:
\( F = \frac{9}{5}(25) + 32\)
\( F = 45 + 32 = 77^\rm{o}F \)

This formula is helpful when traveling to countries using different temperature scales or when reading scientific data presented in Celsius.

Kelvin to Celsius

Kelvin (K) is a temperature scale primarily used in scientific contexts because it starts at absolute zero. Converting Kelvin to Celsius is simple: you subtract 273.15 from the temperature in Kelvin using the formula \( C = K - 273.15 \).

For instance, if you have a temperature of 300 K, you'd convert it to Celsius as follows:
\( C = 300 - 273.15 = 26.85^\rm{o}C\)

Kelvin temperatures are commonly used in physics and chemistry, making this conversion essential for basic chemistry calculations and scientific experiments.

Temperature Comparison

When comparing temperatures in different units, it is crucial to first convert them to the same unit. For example, let's compare \(10^\rm{o}C \) and \(10^\rm{o}F\). First, convert \(10^\rm{o}F \) to Celsius using the formula:
\ C = \frac{5}{9}(F - 32) \
\ C = \frac{5}{9}(10 - 32) = \frac{5}{9}(-22) = -12.22^\rm{o}C\

Now it becomes clear that \(10^\rm{o}C \) is warmer than \(-12.22^\rm{o}C\).

Using this step-by-step comparison approach helps avoid mistakes and ensures accurate temperature assessments across different scales.

Basic Chemistry Calculations

Temperature conversions are fundamental in basic chemistry calculations, affecting how reactions and processes are understood. For instance, in thermodynamics, knowing the exact temperature in Kelvin or Celsius can determine the outcome of a calculation related to reaction rates or gas laws.

Many formulas used in chemistry require temperatures in Kelvin. One example is the Ideal Gas Law:
\ PV = nRT \
where \( P \) is pressure, \( V \) is volume, \( n \) is the amount of gas, \( R \) is the gas constant, and \( T \) is temperature in Kelvin.

Accurate temperature conversion ensures reliable results in such calculations, underscoring their importance in both academic and practical chemistry applications.