This question requires an analysis of the boiling points of ammonia (NH₃) and phosphine (PH₃) in the context of their intermolecular forces, as described in an assertion and a corresponding reason.
The boiling point of a substance is the temperature at which its vapor pressure equals the pressure surrounding the liquid, and the liquid changes into a vapor. The boiling point is directly related to the strength of the intermolecular forces (IMFs) between the molecules. Stronger IMFs require more energy (and thus a higher temperature) to overcome, leading to a higher boiling point.
The primary types of intermolecular forces relevant here are:
Step 1: Analyze the Assertion (A).
The assertion states: "PH₃ has lower boiling point than NH₃."
Let's compare the experimentally determined boiling points of these two compounds:
Since -87.7 °C is a lower temperature than -33.34 °C, the boiling point of PH₃ is indeed lower than that of NH₃. Therefore, the statement made in Assertion (A) is true.
Step 2: Analyze the Reason (R).
The reason states: "In liquid state NH₃ molecules are associated through vander waal’s forces, but PH₃ molecules are associated through hydrogen bonding."
Let's examine the intermolecular forces present in each liquid:
The statement in the Reason claims the opposite: it incorrectly assigns van der Waals forces to NH₃ and hydrogen bonding to PH₃. Therefore, the statement made in Reason (R) is false.
Step 3: Conclude the relationship between Assertion and Reason.
We have established that Assertion (A) is a true statement, but Reason (R) is a false statement. The actual reason for Assertion (A) being true is that NH₃ has a higher boiling point due to the presence of strong intermolecular hydrogen bonds, which are absent in PH₃.
Based on the analysis, Assertion (A) is a correct statement, but Reason (R) is an incorrect statement.
Therefore, the most appropriate answer is: (A) is true but (R) is false.


Let \( f : \mathbb{R} \to \mathbb{R} \) be a twice differentiable function such that \[ (\sin x \cos y)(f(2x + 2y) - f(2x - 2y)) = (\cos x \sin y)(f(2x + 2y) + f(2x - 2y)), \] for all \( x, y \in \mathbb{R}. \)
If \( f'(0) = \frac{1}{2} \), then the value of \( 24f''\left( \frac{5\pi}{3} \right) \) is: