The vibrational energy of the n^th state of HCI is approximately given as

G(n) = 3000 \(\left(n+\frac{1}{2}\right)\) - 50 \(\left(n+\frac{1}{2}\right)^2\) (in cm^-1)

The vibrational quantum number, n_max, beyond which HCI undergoes dissociation is

Concept:

The vibrational quantum number, often denoted as "ν", is a quantum mechanical parameter used to describe the vibrational energy levels of molecules. It is a fundamental concept in the field of molecular spectroscopy and quantum mechanics and is particularly important in the study of molecular vibrations using techniques like infrared (IR) spectroscopy and Raman spectroscopy.

Explanation:

We know that, 

\(E=({n+{1 \over 2})ω_e - (n+{1 \over 2})^2 ω_e X_e}\)

Comparing, G(n) with E

\({[3000(n+{1 \over 2})ω_e - 50(n+{1 \over 2})^2 ω_e X_e]} \) \(={[(n+{1 \over 2})ω_e - (n+{1 \over 2})^2 ω_e X_e]} \)

∴ ω_e =3000

ω_eX_e = 50

X_e = 1/60

Now, 

Putting the value of X_e in n_max i.e.

\(n_{max}= {1\over 2 X_e}-1\)

⇒n_max = 29

Conclusion:

Therefore, the value of n_max is 29