Answer
$5d$
Work Step by Step
The $5d$ state has higher energy than $4f$ because the energy level $ E_n $ is given by
$$
E_n = -\frac{13.6 \, \text{eV}}{n^2}
$$
Where $ n $ is the principal quantum number. For the $5d$ state, $ n = 5 $, and for the $4f$ state, $ n = 4 $. The energy comparison is based solely on the value of $ n $, so we can substitute the values:
For the $5d$ state:
$$
E_5 = -\frac{13.6 \, \text{eV}}{5^2} = -\frac{13.6 \, \text{eV}}{25} = -0.544 \, \text{eV}
$$
For the $4f$ state:
$$
E_4 = -\frac{13.6 \, \text{eV}}{4^2} = -\frac{13.6 \, \text{eV}}{16} = -0.85 \, \text{eV}
$$
Since $ E_5 > E_4 $, the $5d$ state has higher energy than the $4f$ state.
