Question

(a) The temperature on a warm summer day is \(87^{\circ} \mathrm{F}\) . What is the temperature in \(^{\circ} \mathrm{C} ?\) (b) Many scientific data are reported at \(25^{\circ} \mathrm{C}\) . What is this temperature in kelvins and in degrees Fahrenheit? (c) Suppose that a recipe calls for an oven temperature of \(400^{\circ} \mathrm{F}\) . Convert this temperature to degrees Celsius and to kelvins. (d) Liquid nitrogen boils at 77 \(\mathrm{K}\) . Convert this temperature to degrees Fahrenheit and to degrees Celsius.

Short answer

(a) 87°F is approximately \(30.56^{\circ} \mathrm{C}\). (b) 25°C is equivalent to 298.15K and \(77^{\circ} \mathrm{F}\). (c) 400°F is approximately \(204.44^{\circ} \mathrm{C}\) and 477.59K. (d) 77K is equivalent to \(-196.15^{\circ} \mathrm{C}\) and approximately \(-320.87^{\circ} \mathrm{F}\).

Step-by-step solution

(a) Convert 87°F to Celsius

First, apply the Fahrenheit to Celsius formula: \(C = (F - 32) \times \frac{5}{9}\) Substitute F with 87: \(C = (87 - 32) \times \frac{5}{9}\) Now, calculate the Celsius temperature: \(C = 55 \times \frac{5}{9} \approx 30.56^{\circ}\)

(b) Convert 25°C to Kelvin and Fahrenheit

Converting Celsius to Kelvin: \(K = C + 273.15\) Substitute C with 25: \(K = 25 + 273.15 = 298.15K\) Converting Celsius to Fahrenheit: \(F = (C \times \frac{9}{5}) + 32\) Substitute C with 25: \(F = (25 \times \frac{9}{5}) + 32 \approx 77^{\circ}\)

(c) Convert 400°F to Celsius and Kelvin

Converting Fahrenheit to Celsius: \(C = (F - 32) \times \frac{5}{9}\) Substitute F with 400: \(C = (400 - 32) \times \frac{5}{9} \approx 204.44^{\circ}\) Now convert Celsius to Kelvin: \(K = C + 273.15\) Substitute C with 204.44: \(K = 204.44 + 273.15 = 477.59K\)

(d) Convert 77K to Fahrenheit and Celsius

Converting Kelvin to Celsius: \(C = K - 273.15\) Substitute K with 77: \(C = 77 - 273.15 = -196.15^{\circ}\) Now convert Celsius to Fahrenheit: \(F = (C \times \frac{9}{5}) + 32\) Substitute C with -196.15: \(F = (-196.15 \times \frac{9}{5}) + 32 \approx -320.87^{\circ}\)

Key concepts

Celsius to Fahrenheit

Temperature conversion from Celsius to Fahrenheit is a simple yet important concept. When converting from Celsius to Fahrenheit, you multiply the Celsius temperature by \(\frac{9}{5}\), then add 32 to the result. This formula helps determine how warm or cold a temperature in Celsius would feel in Fahrenheit.

Here is the formula in a straightforward way:

• Start with your Celsius temperature.
• Multiply by 9, then divide by 5.
• Add 32. This will give you the temperature in Fahrenheit.

For example, converting \(25^{\circ} \mathrm{C}\) to Fahrenheit would look like this:- Multiply 25 by \(9/5\), which equals 45.- Add 32, resulting in \(77^{\circ} \mathrm{F}\).

This conversion is useful for everything from understanding the weather in different countries to cooking recipes that use Fahrenheit.

Fahrenheit to Celsius

The process of converting Fahrenheit to Celsius is the reverse of converting Celsius to Fahrenheit. You start by subtracting 32 from the Fahrenheit temperature. This is because the two scales have different starting points for freezing. Next, you multiply the result by \(\frac{5}{9}\) to adjust for the scale difference.

The formula for converting Fahrenheit to Celsius is:

• Subtract 32 from the Fahrenheit temperature.
• Multiply the result by \(\frac{5}{9}\).

Consider converting \(87^{\circ} \mathrm{F}\) to Celsius:- Subtract 32 from 87, which gives 55.- Multiply 55 by \(\frac{5}{9}\) to get approximately \(30.56^{\circ} \mathrm{C}\).

This conversion is especially useful in scientific contexts where Celsius is more commonly used.

Celsius to Kelvin

Converting Celsius to Kelvin is quite intuitive since both scales increment linearly, meaning each degree change in Celsius is equal to a degree change in Kelvin. The only adjustment needed is accounted by the offset factor between the two, which is 273.15. The Kelvin scale starts at absolute zero, which is the lowest possible temperature.

Follow these steps for the conversion:

• Add 273.15 to the Celsius temperature.

For instance, converting \(25^{\circ} \mathrm{C}\) to Kelvin involves simply adding 273.15, arriving at \(298.15 \mathrm{K}\).

This method is particularly crucial in scientific experiments where temperatures in Kelvin are often required for calculations involving gas laws and thermodynamics.

Kelvin to Fahrenheit

To convert Kelvin to Fahrenheit, two steps are needed since there isn't a direct conversion formula. First, convert Kelvin to Celsius by subtracting 273.15. Then, use the Celsius to Fahrenheit formula to complete the conversion.

Here's the two-step process:

• Convert Kelvin to Celsius: subtract 273.15 from the Kelvin temperature.
• Convert Celsius to Fahrenheit: multiply the Celsius temperature by \(\frac{9}{5}\) and add 32.

For example, to convert \(77 \mathrm{K}\) to Fahrenheit:- First, convert to Celsius: \(77 - 273.15 = -196.15^{\circ} \mathrm{C}\).- Then convert to Fahrenheit: \(-196.15 \times \frac{9}{5} + 32 \approx -320.87^{\circ} \mathrm{F}\).

This conversion is helpful in fields like cryogenics where Kelvin is commonly used, and understanding temperatures in Fahrenheit can provide more context.