Question

Convert the following Kelvin temperatures to Celsius and Fahrenheit degrees. a. the temperature that registers the same value on both the Fahrenheit and Celsius scales, \(233 \mathrm{~K}\) b. the boiling point of helium, \(4 \mathrm{~K}\) c. the temperature at which many chemical quantities are determined, \(298 \mathrm{~K}\) d. the melting point of tungsten, \(3680 \mathrm{~K}\)

Short answer

a. $233 \mathrm{~K}$: \(T\mathrm{~C} = 233.15\mathrm{~C}\), \(T\mathrm{~F} = -40\mathrm{~F}\) b. $4 \mathrm{~K}$: \(T\mathrm{~C} = -269.15\mathrm{~C}\), \(T\mathrm{~F} = -452.47\mathrm{~F}\) c. $298 \mathrm{~K}$: \(T\mathrm{~C} = 24.85\mathrm{~C}\), \(T\mathrm{~F} = 76.73\mathrm{~F}\) d. $3680 \mathrm{~K}$: \(T\mathrm{~C} = 3406.85\mathrm{~C}\), \(T\mathrm{~F} = 6154.33\mathrm{~F}\)

Step-by-step solution

1. Recall the conversion formulas

For converting Kelvin to Celsius and Fahrenheit, we use the following formulas: \(K - 273.15 = C\) (Conversion to Celsius) \(\left( \frac{9}{5} \right)(K - 273.15) + 32 = F\) (Conversion to Fahrenheit)

a. Find the temperature that registers the same value on both Fahrenheit and Celsius scales

Let \(T\) be the temperature that registers the same value on both the Fahrenheit and Celsius scales. From given information, \(T\mathrm{~C} = T\mathrm{~F}\). We can convert \(T\mathrm{~C}\) to Fahrenheit using the conversion formula: \(T\mathrm{~F} = \left( \frac{9}{5} \right) T\mathrm{~C} + 32\) Now, set the equations equal to each other: \(T\mathrm{~C} = \left( \frac{9}{5} \right) T\mathrm{~C} + 32\) Solving for \(T\mathrm{~C}\), we obtain \(T\mathrm{~C} = - 40^\circ\). This means \(T\mathrm{~F} = - 40^\circ\). Now we can convert this value to Kelvin using the Kelvin to Celsius conversion formula mentioned above.

2. Convert given temperatures to Celsius and Fahrenheit

a. Temperature in Celsius: \(T\mathrm{~C} = -40 + 273.15 = 233.15\mathrm{~K}\) Temperature in Fahrenheit: \(T\mathrm{~F} = \left( \frac{9}{5} \right)(233 - 273.15) + 32 = -40\mathrm{~F}\) b. Boiling point of helium in Celsius: \(T\mathrm{~C} = 4 - 273.15 = -269.15\mathrm{~C}\) Boiling point of helium in Fahrenheit: \(T\mathrm{~F} = \left( \frac{9}{5} \right)(4 - 273.15) + 32 = -452.47\mathrm{~F}\) c. Temperature in Celsius: \(T\mathrm{~C} = 298 - 273.15 = 24.85\mathrm{~C}\) Temperature in Fahrenheit: \(T\mathrm{~F} = \left( \frac{9}{5} \right)(298 - 273.15) + 32 = 76.73\mathrm{~F}\) d. Melting point of tungsten in Celsius: \(T\mathrm{~C} = 3680 - 273.15 = 3406.85\mathrm{~C}\) Melting point of tungsten in Fahrenheit: \(T\mathrm{~F} = \left( \frac{9}{5} \right)(3680 - 273.15) + 32 = 6154.33\mathrm{~F}\)

Key concepts

Kelvin to Celsius conversion

Temperature conversion can seem tricky at first, especially when juggling between different scales like Kelvin and Celsius. But it's simpler than it seems!
One of the foundational formulas for temperature conversion involves switching from Kelvin to Celsius. The beauty of this conversion is that it’s a straightforward subtraction. To convert a temperature given in Kelvin to Celsius, all you need to do is subtract 273.15.

• For instance, to convert 4 Kelvin to Celsius, you simply do: \( 4 - 273.15 = -269.15 \)°C. This indicates how cold helium's boiling point is!
• Similarly, for the melting point of tungsten at 3680 Kelvin: \( 3680 - 273.15 = 3406.85 \)°C.

This formula reflects the relationship between absolute temperature (Kelvin) and the Celsius scale, anchored at the freezing point of water in Celsius, which is 0°C or 273.15 Kelvin.

Kelvin to Fahrenheit conversion

Converting temperatures from Kelvin to Fahrenheit requires a few more steps but it’s still manageable! This formula combines the earlier Kelvin to Celsius conversion and shifts it to Fahrenheit.
Start with the Kelvin to Celsius conversion, and then apply the formula for Celsius to Fahrenheit: multiply the Celsius temperature by \( \frac{9}{5} \) and add 32.

• For example, 4 Kelvin translates to \(-269.15\)°C. Convert this to Fahrenheit by using the formula: \( \left( \frac{9}{5} \right)(-269.15) + 32 = -452.47 \)°F. That’s the boiling point of helium in Fahrenheit!
• For 3680 Kelvin, first convert to Celsius to get 3406.85°C, then to Fahrenheit using:\( \left( \frac{9}{5} \right)(3406.85) + 32 = 6154.33 \)°F.

Though this involves two steps, each part is straightforward once you remember the formulas.

Boiling point of helium

Helium, one of the noble gases, boasts some interesting thermal properties, especially its boiling point. Helium boils at an incredibly low temperature : just 4 Kelvin.
Understanding its boiling point requires conversion:

• In Celsius, this becomes \(-269.15\)°C.
• Meanwhile, in Fahrenheit, the boiling point is a chilly \(-452.47\)°F.

This extremely low boiling point implies that helium remains gaseous under most earthly conditions. Helium is commonly used in cryogenics because its very low boiling point can help researchers achieve ultracold temperatures.

Melting point of tungsten

Tungsten is renowned for having the highest melting point of all metals, anchoring its use in applications requiring extreme heat resistance. Its melting point is a staggering 3680 Kelvin.
Understanding the melting point of tungsten in different temperature scales is crucial:

• In Celsius, tungsten melts at \( 3406.85\)°C.
• In Fahrenheit, the temperature is an enormous \(6154.33\)°F.

These values highlight tungsten's unique capacity to withstand high temperatures, making it ideal for use in light bulb filaments and cutting tools.