A compound is optically active if it has at least one chiral center (a carbon atom attached to four different groups). Let us analyze the given compounds:
CH$_3$-CH(OH)-CH(OH)-CH$_3$: This compound has two hydroxyl groups (OH) on adjacent carbon atoms. Both hydroxyl groups are equivalent, and the molecule has a plane of symmetry, making it optically inactive.
CH$_3$-CH$_2$-CH$_2$-OH: This compound has no chiral centers, as all carbons are attached to at least two identical groups. Therefore, it is optically inactive.
CH$_3$-CH$_2$-CH-CH$_3$ (with a Cl on the second carbon):} The second carbon atom is a chiral center, as it is attached to four different groups: CH$_3$, H, Cl, and CH$_2$CH$_3$. Hence, this compound is optically active.
(CH$_3$)$_2$CH-CH$_2$-CH$_2$-Cl: The molecule does not have any chiral centers, as the carbon bonded to the chlorine atom is not attached to four different groups. Therefore, it is optically inactive.
Conclusion: Among the given compounds, only CH$_3$-CH$_2$-CH-CH$_3$ (with Cl on the second carbon) is optically active.
How many different stereoisomers are possible for the given molecule?
Assertion (A): All naturally occurring \(\alpha\)-amino acids except glycine are optically active. Reason (R): Most naturally occurring amino acids have L-configuration.
Let \( y = f(x) \) be the solution of the differential equation
\[ \frac{dy}{dx} + 3y \tan^2 x + 3y = \sec^2 x \]
such that \( f(0) = \frac{e^3}{3} + 1 \), then \( f\left( \frac{\pi}{4} \right) \) is equal to:
Find the IUPAC name of the compound.
If \( \lim_{x \to 0} \left( \frac{\tan x}{x} \right)^{\frac{1}{x^2}} = p \), then \( 96 \ln p \) is: 32