15:I[8216,["972","static/chunks/972-b28da0de92681928.js","47","static/chunks/47-a7f06e9eda1ac756.js","115","static/chunks/app/%5Bexam%5D/questions/%5Byear%5D/%5Bsubject%5D/%5Bchapter%5D/%5BquestionId%5D/page-7b6f0a12b9cc0206.js"],"BackButton"] 16:I[1553,["972","static/chunks/972-b28da0de92681928.js","47","static/chunks/47-a7f06e9eda1ac756.js","115","static/chunks/app/%5Bexam%5D/questions/%5Byear%5D/%5Bsubject%5D/%5Bchapter%5D/%5BquestionId%5D/page-7b6f0a12b9cc0206.js"],"QuestionViewer"] 17:T7db,

To determine which hydrogen-like species will have the same radius as the 1st Bohr orbit of the hydrogen atom, we first need to recall the formula for the radius of the nth orbit in a hydrogen-like atom. The radius for the nth orbit in a hydrogen-like atom is given by:

$$r_n = \frac{n^2 h^2}{4 \pi^2 m e^2 Z} \cdot \frac{1}{1}$$

where:

$$r_n$$ is the radius of the nth orbit.
$$n$$ is the principal quantum number.
$$h$$ is Planck's constant.
$$m$$ is the mass of the electron.
$$e$$ is the charge of the electron.
$$Z$$ is the atomic number of the nucleus.

For a hydrogen atom in the 1st orbit ($$n=1$$, $$Z=1$$), the radius is:

$$r_1 = \frac{1^2 \cdot h^2}{4 \pi^2 \cdot m \cdot e^2 \cdot 1}$$

Now, for a hydrogen-like species with atomic number $$Z$$ and in the nth orbit, the radius is:

$$r_n = \frac{n^2 \cdot h^2}{4 \pi^2 \cdot m \cdot e^2 \cdot Z}$$

We need to find the species and quantum number $$n$$ such that its radius equals the radius of the 1st Bohr orbit of hydrogen, i.e., $$r_1$$ of the hydrogen atom. This can be written as:

$$\frac{n^2 \cdot h^2}{4 \pi^2 \cdot m \cdot e^2 \cdot Z} = \frac{1^2 \cdot h^2}{4 \pi^2 \cdot m \cdot e^2 \cdot 1}$$

Simplifying, we get:

$$\frac{n^2}{Z} = 1$$

This gives us:

$$n^2 = Z$$

So, the species must satisfy this condition:

For $$n=2$$, $$n^2 = 2^2 = 4.$$ So, $$Z = 4$$.
For $$n=3$$, $$n^2

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