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(A) Using the Rydberg formula for hydrogen-like ions, the wavelength $\lambda$ of the emission line is given by:
$\frac{1}{\lambda} = R Z^{2} \left(

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$2$

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(A) Using the Rydberg formula for hydrogen-like ions, the wavelength $\lambda$ of the emission line is given by:
$\frac{1}{\lambda} = R Z^{2} \left( \frac{1}{n_{f}^{2}} - \frac{1}{n_{i}^{2}} ight)$
Given:
Rydberg constant $R \approx 1.097 	imes 10^{7} \, m^{-1} \approx 1.1 	imes 10^{7} \, m^{-1}$
Atomic number of iron $Z = 26$
Initial state $n_{i} = 2$
Final state $n_{f} = 1$
Substituting the values:
$\frac{1}{\lambda} = (1.1 	imes 10^{7}) 	imes (26)^{2} 	imes \left( \frac{1}{1^{2}} - \frac{1}{2^{2}} ight)$
$\frac{1}{\lambda} = 1.1 	imes 10^{7} 	imes 676 	imes ( 1 - 0.25 )$
$\frac{1}{\lambda} = 1.1 	imes 10^{7} 	imes 676 	imes 0.75$
$\frac{1}{\lambda} = 5.577 	imes 10^{9} \, m^{-1}$
Calculating $\lambda$:
$\lambda = \frac{1}{5.577 	imes 10^{9}} \approx 1.79 	imes 10^{-10} \, m$
Converting to $\mathring{A}$ $(1 \, \mathring{A} = 10^{-10} \, m)$:
$\lambda \approx 1.79 \, \mathring{A}$
Thus, the wavelength is closest to $2 \, \mathring{A}$.

The Hitomi satellite recently observed the Lyman alpha emission line ($n=2$ to $n=1$) of hydrogen-like iron ion (atomic number of iron is $26$) from the Perseus galaxy cluster. The wavelength of the line is closest to ............... $\mathring{A}$.

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