Questions: HII regions can exist only if there is a nearby star hot enough to ionize hydrogen. Hydrogen is ionized only by radiation with wavelengths shorter than 91.2 nm. What is the temperature in kelvins of a star that emits its maximum energy at 91.2 nm? (Use Wien's law.) Temperature = K

HII regions can exist only if there is a nearby star hot enough to ionize hydrogen. Hydrogen is ionized only by radiation with wavelengths shorter than 91.2 nm.
What is the temperature in kelvins of a star that emits its maximum energy at 91.2 nm? (Use Wien's law.)
Temperature = K

Transcript text: HII regions can exist only if there is a nearby star hot enough to ionize hydrogen. Hydrogen is ionized only by radiation with wavelengths shorter than 91.2 nm. What is the temperature in kelvins of a star that emits its maximum energy at 91.2 nm ? (Use Wien's law.) Temperature $=$ $\square$ K

Solution

Solution Steps

Step 1: Convert wavelength to meters

Wien's law relates the peak wavelength of a blackbody's radiation to its temperature. The formula is λ_max * T = b, where λ_max is the peak wavelength, T is the temperature in Kelvin, and b is Wien's displacement constant (2.898 x 10^-3 m⋅K). The given wavelength is 91.2 nm. To use Wien's law, we must convert this to meters: 91.2 nm * (1 m / 10⁹ nm) = 9.12 x 10^-8 m.

Step 2: Apply Wien's Law

Now we can use Wien's law to find the temperature: T = b / λ_max T = (2.898 x 10^-3 m⋅K) / (9.12 x 10^-8 m)

Step 3: Calculate Temperature

T ≈ 31776 K