The reaction sequence given below is carried out with 16 moles of X. The yield of the major product in each step is given below the product in parentheses. The amount (in grams) of S produced is ____. 
Use: Atomic mass (in amu): H = 1, C = 12, O = 16, Br = 80
Step 1: Track yields through the reaction sequence
Initial moles of X = 16 mol

Step 2: Calculate molar mass of S
S is formed from R (via NaH in ether). Given molar mass of P is 210 and the reaction doesn't drastically change the skeleton, so we consider R \( \approx \) P for molar mass estimate. Thus:
\[ \text{Approx. molar mass of S} = 210 \, \text{g/mol} \]
Step 3: Calculate mass of S
\[ n = 2 \, \text{mol}, \quad M = 210 \, \text{g/mol} \Rightarrow \text{Mass of S} = n \cdot M = 2 \times 42 = \boxed{84 \, \text{g}} \]
Final Answer: \( \boxed{84} \)


The center of a disk of radius $ r $ and mass $ m $ is attached to a spring of spring constant $ k $, inside a ring of radius $ R>r $ as shown in the figure. The other end of the spring is attached on the periphery of the ring. Both the ring and the disk are in the same vertical plane. The disk can only roll along the inside periphery of the ring, without slipping. The spring can only be stretched or compressed along the periphery of the ring, following Hooke’s law. In equilibrium, the disk is at the bottom of the ring. Assuming small displacement of the disc, the time period of oscillation of center of mass of the disk is written as $ T = \frac{2\pi}{\omega} $. The correct expression for $ \omega $ is ( $ g $ is the acceleration due to gravity): 
Let $ a_0, a_1, ..., a_{23} $ be real numbers such that $$ \left(1 + \frac{2}{5}x \right)^{23} = \sum_{i=0}^{23} a_i x^i $$ for every real number $ x $. Let $ a_r $ be the largest among the numbers $ a_j $ for $ 0 \leq j \leq 23 $. Then the value of $ r $ is ________.