Question

Convert: (a) \(23.0{ }^{\circ} \mathrm{C}\) to \(\mathrm{K}\) (b) \(98.6^{\circ} \mathrm{F}\) to \({ }^{\circ} \mathrm{C}\) (c) \(296 \mathrm{~K}\) to \({ }^{\circ} \mathrm{F}\) (d) \(32^{\circ} \mathrm{F}\) to \(\mathrm{K}\) (e) \(523 \mathrm{~K}\) to \({ }^{\circ} \mathrm{C}\) (f) \(38^{\circ} \mathrm{C}\) to \({ }^{\circ} \mathrm{F}\)

Short answer

(a) \(296.15 \mathrm{K}\) (b) \(37.0^{\circ} \mathrm{C}\) (c) \(73.0^{\circ} \mathrm{F}\) (d) \(273.15 \mathrm{K}\) (e) \(249.85^{\circ} \mathrm{C}\) (f) \(100.4^{\circ} \mathrm{F}\)

Step-by-step solution

(a) Convert 23.0°C to K

To convert 23.0°C to Kelvin, use the Celsius to Kelvin formula: K = °C + 273.15. So, K = 23.0 + 273.15 = 296.15 K.

(b) Convert 98.6°F to °C

To convert 98.6°F to Celsius, use the Fahrenheit to Celsius formula: °C = (°F - 32) × 5/9. So, °C = (98.6 - 32) × 5/9 ≈ 37.0°C.

(c) Convert 296 K to °F

To convert 296 K to Fahrenheit, use the Kelvin to Fahrenheit formula: °F = (K - 273.15) × 9/5 + 32. So, °F = (296 - 273.15) × 9/5 + 32 ≈ 73.0°F.

(d) Convert 32°F to K

First, convert 32°F to Celsius using the Fahrenheit to Celsius formula: °C = (°F - 32) × 5/9. So, °C = (32 - 32) × 5/9 = 0°C. Then, convert 0°C to Kelvin using the Celsius to Kelvin formula: K = °C + 273.15. So, K = 0 + 273.15 = 273.15 K.

(e) Convert 523 K to °C

To convert 523 K to Celsius, use the Kelvin to Celsius formula: °C = K - 273.15. So, °C = 523 - 273.15 = 249.85°C.

(f) Convert 38°C to °F

To convert 38°C to Fahrenheit, use the Celsius to Fahrenheit formula: °F = °C × 9/5 + 32. So, °F = 38 × 9/5 + 32 ≈ 100.4°F.

Key concepts

Celsius to Kelvin

Understanding how to convert temperatures from Celsius to Kelvin is a fundamental skill in scientific disciplines, particularly in chemistry and physics. This conversion is beautifully simple because the two scales have the same size degree, meaning the step from one degree to the next is the same magnitude. The Kelvin scale was designed to have a null point, known as absolute zero, which is the point at which particles have no thermal energy.

To convert a temperature from Celsius to Kelvin, you simply add 273.15 to your Celsius temperature. It's important to remember this exact value, as approximating it to 273 can lead to inaccuracies, especially in scientific calculations.

For example:

• If you have a temperature of 23.0°C, adding 273.15 will result in a Kelvin temperature of 296.15K.

The reason for the seemingly arbitrary addition of 273.15 is that absolute zero in Celsius is -273.15°C. Thus, 0K equates to -273.15°C, and you're effectively moving the scale up by that amount to convert to Kelvin.

Fahrenheit to Celsius

When converting Fahrenheit to Celsius, a bit more math is involved compared to the Celsius to Kelvin conversion. This necessary step is crucial for understanding many everyday temperature comparisons, such as weather forecasts or cooking temperatures.

The formula to convert Fahrenheit to Celsius is: \[ °C = (°F - 32) \times \frac{5}{9} \]
Why these numbers? The formula represents the differences in where each scale starts (freezing point of water) and how they increment (size of a degree). Fahrenheit starts at 32 for the freezing point of water, whereas Celsius starts at 0. The fraction \(5/9\) compensates for the difference in the size of each degree.

As an example:

• Converting 98.6°F, you would first subtract 32 (yielding 66.6), then multiply by \(5/9\), resulting in approximately 37.0°C.

This approach means we can compare temperatures easily between the two scales. It’s also an essential conversion for sciences such as biology and environmental science, where temperatures are often reported in Celsius.

Kelvin to Fahrenheit

The conversion from Kelvin to Fahrenheit involves understanding both of the previous conversions as it's essentially a two-step process: converting Kelvin to Celsius, then Celsius to Fahrenheit. This conversion might be less common in everyday contexts but is certainly vital in areas of science and engineering that work with thermodynamic systems, among others.

To convert Kelvin directly to Fahrenheit, use the formula: \[ °F = (K - 273.15) \times \frac{9}{5} + 32 \]
This equation combines the two steps of converting Kelvin to Celsius first (\( K - 273.15 \)) and then Celsius to Fahrenheit by multiplying by \(9/5\) and adding 32.

For instance:

• For a temperature of 296K, subtracting 273.15 gives us 22.85°C, which when converted using the \(9/5\) ratio and added to 32, results in approximately 73°F.

Remembering these formulas and the reasons behind the numbers can help reinforce your understanding of temperature scales and their conversions, which can be invaluable in academic and professional settings.