Van der Waals constant a is proportional to the strength of intermolecular forces. Hy drogen bonding and larger molecular size significantly increase the value of a.
The van der Waals constant \( a \) represents the magnitude of intermolecular attractive forces. It increases with molecular size, molecular mass, and the presence of hydrogen bonding. Let us analyze the given gases:
Thus, the increasing order of \( a \) is:
\(Ar (A) < CH_4 (B) < H_2O (C) < C_6H_6 (D) .\)
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: