Problem 46 - Chemistry The Molecular Nature of Matter

Question

Perform the following conversions. (a) \(98^{\circ} \mathrm{F}\) to \({ }^{\circ} \mathrm{C}\) (c) \(299 \mathrm{~K}\) to \({ }^{\circ} \mathrm{C}\) (b) \(-55^{\circ} \mathrm{C}\) to \({ }^{\circ} \mathrm{F}\) (d) \(40.0^{\circ} \mathrm{C}\) to \(\mathrm{K}\)

Step-by-Step Solution

Answer

\(98^\circ F = 36.67^\circ C, 299 K = 25.85^\circ C, -55^\circ C = -67^\circ F, 40.0^\circ C = 313.15 K\)

Step 1 - Convert Fahrenheit to Celsius

Use the formula \(C = (F - 32) \times \frac{5}{9}\) to convert Fahrenheit to Celsius. Substitute 98 for F: \((C = (98 - 32) \times \frac{5}{9} = 66 \times \frac{5}{9} = 36.67^\circ C\).

Step 2 - Convert Kelvin to Celsius

Use the formula \(C = K - 273.15\) for the conversion. Substitute 299 for K: \((C = 299 - 273.15 = 25.85^\circ C\).

Step 3 - Convert Celsius to Fahrenheit

Use the formula \(F = C \times \frac{9}{5} + 32\) to convert Celsius to Fahrenheit. Substitute -55 for C: \((F = -55 \times \frac{9}{5} + 32 = -99 + 32 = -67^\circ F\).

Step 4 - Convert Celsius to Kelvin

Use the formula \(K = C + 273.15\) for the conversion. Substitute 40.0 for C: \((K = 40.0 + 273.15 = 313.15 K\).

Key Concepts

Fahrenheit to Celsius Conversion

Understanding how to convert temperature from Fahrenheit to Celsius is an essential skill in chemistry and daily life. The formula used for this conversion is relatively simple:
\[ C = (F - 32) \times \frac{5}{9} \]
To apply this formula, start by subtracting 32 from the Fahrenheit temperature. Next, multiply this number by five-ninths. For example, if you have a temperature of 98 degrees Fahrenheit and want to convert it to Celsius, you first subtract 32, leaving you with 66. When you then multiply 66 by five-ninths (or 5/9), you get approximately 36.67 degrees Celsius.

• Start with the Fahrenheit temperature.
• Subtract 32 from this value.
• Multiply the result by five-ninths (or 5/9).

Remember to pay attention to significant figures in scientific contexts, rounding your answer to maintain the precision of the original measurement.

Kelvin to Celsius Conversion

When dealing with temperature in scientific settings, Kelvin is often used as the standard unit. Converting Kelvin to Celsius is straightforward since both scales increment evenly. The Celsius scale is simply offset from the Kelvin scale by 273.15 units. Thus, to convert a temperature from Kelvin to Celsius, you simply subtract 273.15 from the Kelvin value:
\[ C = K - 273.15 \]
For instance, to convert 299 K to Celsius, you subtract 273.15 from 299, yielding 25.85 degrees Celsius.

• Identify the Kelvin temperature.
• Subtract 273.15 from the Kelvin temperature.
• The result is the temperature in Celsius.

This conversion is vital in many scientific calculations, such as those involving thermodynamics or reaction kinetics.

Celsius to Fahrenheit Conversion

Converting Celsius temperatures to Fahrenheit, often required when traveling between countries that use different temperature scales, involves two steps: multiplying the Celsius temperature by nine-fifths (or 1.8) and then adding 32. The formula for this conversion looks like this:
\[ F = C \times \frac{9}{5} + 32 \]
If you have a temperature of -55 degrees Celsius and need to convert it to Fahrenheit, multiply -55 by nine-fifths to get -99, and then add 32, resulting in -67 degrees Fahrenheit.

• Multiply the Celsius temperature by nine-fifths (or by 1.8).
• Add 32 to this product.
• The resulting value is the temperature in Fahrenheit.

This calculation is often encountered in everyday life, especially by individuals dealing with international temperature reports or data.

Celsius to Kelvin Conversion

Scientists frequently use the Kelvin scale as it is absolute and does not have negative values, making it ideal for expressing thermodynamic temperatures. For converting Celsius to Kelvin, which is a common task in scientific fields, the formula is very simple:
\[ K = C + 273.15 \]
This formula indicates that you need to add 273.15 to the Celsius temperature to get the Kelvin equivalent. For example, if you have a temperature of 40.0 degrees Celsius, you simply add 273.15 to find that it is equal to 313.15 Kelvin.

• Take the Celsius temperature.
• Add 273.15 to it.
• The sum is the temperature in Kelvin.

Remember to keep in mind that because the Kelvin scale is an absolute scale, temperatures on this scale are not referred to in degrees; they are simply Kelvin (K).