The reaction involved is:
\[2\text{NH}_4\text{Cl} + \text{MnO}_2 + 2\text{H}_2\text{SO}_4 \xrightarrow{\Delta} \text{MnSO}_4 + (\text{NH}_4)_2\text{SO}_4 + 2\text{H}_2\text{O} + \text{Cl}_2 \uparrow\]
In this reaction, chlorine gas (\( \text{Cl}_2 \)) is liberated, which has a greenish-yellow color.
Let $ P_n = \alpha^n + \beta^n $, $ n \in \mathbb{N} $. If $ P_{10} = 123,\ P_9 = 76,\ P_8 = 47 $ and $ P_1 = 1 $, then the quadratic equation having roots $ \alpha $ and $ \frac{1}{\beta} $ is: