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, \(3680 \mathrm{~K}\)

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)

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.

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: \(T\mathrm{~F} = \left( \frac{9}{5} \right)(3680 - 273.15) + 32 = 6154.33\mathrm{~F}\)