Question

Make the following conversions: (a) \(72^{\circ} \mathrm{F}\) to \(^{\circ} \mathrm{C},\) (b) \(216.7^{\circ} \mathrm{C}\) to \(^{\circ} \mathrm{F},(\mathbf{c}) 233^{\circ} \mathrm{C}\) to \(\mathrm{K},(\mathbf{d}) 315 \mathrm{K} \mathrm{to}^{\circ} \mathrm{F},(\mathbf{e}) 2500^{\circ} \mathrm{Fto} \mathrm{K},(\mathbf{f}) 0 \mathrm{K}\) to \(^{\circ} \mathrm{F}\)

Short answer

The short answer for the temperature conversions is as follows: (a) \(22.2^{\circ}\mathrm{C}\) (b) \(422.06^{\circ}\mathrm{F}\) (c) \(506.15 \mathrm{K}\) (d) \(107.33^{\circ}\mathrm{F}\) (e) \(1649.82 \mathrm{K}\) (f) \(-459.67^{\circ}\mathrm{F}\)

Step-by-step solution

Apply the Fahrenheit to Celsius formula

We'll use the Fahrenheit to Celsius formula here: \(°C = \frac{5}{9}(°F - 32)\)

Compute the Celsius temperature

Plug in Fahrenheit temperature and solve for Celsius temperature: \(°C = \frac{5}{9}(72°F - 32) = 22.2 °C\) (b) Convert \(216.7^{\circ}\mathrm{C}\) to \(^{\circ}\mathrm{F}\):

Apply the Celsius to Fahrenheit formula

We'll use the Celsius to Fahrenheit formula here: \(°F = \frac{9}{5}(°C) +32\)

Compute the Fahrenheit temperature

Plug in Celsius temperature and solve for Fahrenheit temperature: \(°F= \frac{9}{5}(216.7°C) + 32 = 422.06 °F\) (c) Convert \(233^{\circ}\mathrm{C}\) to \(K\):

Apply the Celsius to Kelvin formula

We'll use the Celsius to Kelvin formula here: \(K = °C + 273.15\)

Compute the Kelvin temperature

Plug in Celsius temperature and solve for Kelvin temperature: \(K = 233°C + 273.15 = 506.15 K\) (d) Convert \(315 K\) to \(^{\circ}\mathrm{F}\):

Apply Kelvin to Celsius and Celsius to Fahrenheit formulas

First, convert Kelvin to Celsius using \(°C = K - 273.15\), then convert Celsius to Fahrenheit using \(°F = \frac{9}{5}(°C) +32\)

Compute Fahrenheit temperature

First, find Celsius temperature,\(°C = 315 K -273.15 = 41.85°C\). Then, find Fahrenheit temperature: \(°F = \frac{9}{5}(41.85°C) + 32 = 107.33 °F\) (e) Convert \(2500^{\circ}\mathrm{F}\) to \(K\):

Apply Fahrenheit to Celsius and Celsius to Kelvin formulas

First, convert Fahrenheit to Celsius using \(°C = \frac{5}{9}(°F - 32)\), then convert Celsius to Kelvin using \(K = °C + 273.15\)

Compute Kelvin temperature

First, find Celsius temperature, \(°C = \frac{5}{9}(2500°F-32) = 1376.67°C\). Then, find Kelvin temperature: \(K = 1376.67°C + 273.15 = 1649.82 K\) (f) Convert \(0 K\) to \(^{\circ}\mathrm{F}\):

Apply Kelvin to Celsius and Celsius to Fahrenheit formulas

First, convert Kelvin to Celsius using \(°C = K - 273.15\), then convert Celsius to Fahrenheit using \(°F = \frac{9}{5}(°C) +32\)

Compute Fahrenheit temperature

First, find Celsius temperature, \(°C = 0 K -273.15 = -273.15°C\). Then, find Fahrenheit temperature: \(°F = \frac{9}{5}(-273.15°C) + 32 = -459.67 °F\)

Key concepts

Fahrenheit to Celsius conversion

To convert temperatures from Fahrenheit to Celsius, you need to use a specific formula: \[°C = \frac{5}{9}(°F - 32)\]This equation subtracts 32 from the Fahrenheit temperature and then multiplies by \(\frac{5}{9}\) to adjust for the different scales of measurement. Here’s why it works:

• The number 32 is subtracted because the Fahrenheit scale starts at that point, i.e., freezing point of water in °F is 32°, while in °C it's 0°.
• The fraction \(\frac{5}{9}\) is used because the Celsius scale is 5/9 of the Fahrenheit scale in terms of temperature difference per degree.

Let's try this conversion with a practical example. Assume we have a temperature of 72°F. Plug it into our formula:\[°C = \frac{5}{9}(72 - 32) = 22.2°C\]This shows that 72°F is equivalent to 22.2°C.

Celsius to Fahrenheit conversion

Converting from Celsius to Fahrenheit is also straightforward. The formula used is: \[°F = \frac{9}{5}(°C) + 32\]This equation adjusts for the scale difference and adds 32 because the Fahrenheit scale starts higher. Here’s a breakdown:

• Multiply by \(\frac{9}{5}\) to scale Celsius into Fahrenheit intervals.
• Add 32 to shift the freezing point from 0°C to 32°F.

Let’s convert 216.7°C as an example. Just plug into the formula:\[°F= \frac{9}{5}(216.7) + 32 = 422.06°F\]This shows that 216.7°C converts to 422.06°F with our formula.

Celsius to Kelvin conversion

Converting Celsius to Kelvin is perhaps the easiest of all temperature conversions. The formula used is simple:\[K = °C + 273.15\]Kelvin and Celsius are directly related through this addition:

• Both scales are based on a similar degree size, but Kelvin starts at absolute zero, about -273.15°C.

For example, converting 233°C to Kelvin:\[K = 233 + 273.15 = 506.15K\]This conversion shows that 233°C equals 506.15K, underlining the straightforward nature of the relationship between Kelvin and Celsius.

Kelvin to Fahrenheit conversion

To convert Kelvin to Fahrenheit, you use a two-step process involving both Celsius and Fahrenheit conversions, because Kelvin is not directly convertible to Fahrenheit. Here’s how:

• First, convert Kelvin to Celsius: \(°C = K - 273.15\)
• Next, convert Celsius to Fahrenheit: \(°F = \frac{9}{5}(°C) + 32\)

For instance, to convert 315K into Fahrenheit:Start by finding Celsius:\[°C = 315 - 273.15 = 41.85°C\]Then, convert Celsius to Fahrenheit:\[°F = \frac{9}{5}(41.85) + 32 = 107.33°F\]Thus, a temperature of 315K converts to 107.33°F, illustrating how these conversions smoothly transition between the scales.