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Does the secondary quantum number tell how many subshells a specific principal quantum number shell has?

E.g., if the principal quantum number is $n$, there are ($n-1$) subshells.

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Does the secondary quantum number tell how many subshells a specific principal quantum number shell has?

No, it does not. It is the principal quantum number $n$ which does it (as you actually mentioned yourself later in the question). The second quantum number $l$ describes the shape of the orbitals in a particular subshell. Say, $l=0$ for $\mathrm{s}$-subshell of any shell and all $\mathrm{s}$-orbitals have a spherical shape.

If the principal quantum number is $n$, there are $(n−1)$ subshells.

Why $(n−1)$? Think about the most trivial example: how many subshells the first shell ($n=1$) have? One, not zero, so $(n−1)$ is obviously wrong formula. The situation is in fact simpler: $n$-th shell has $n$ subshells.

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  • $\begingroup$ A small detail to add to Wildcat's excellent answer: the $\ell$ quantum number does not tell you the total number of subshells, but you may infer the minimum value of the principal quantum number $n$ and therefore the minimum number of subshells. For example, if $\ell=3$ then $n\ge 4$. $\endgroup$ – Paul Nov 5 at 18:58

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