Step 1: Recall Dobereiner's law of triads.
Dobereiner's law of triads states that when elements are arranged in increasing order of atomic weight, the atomic weight of the middle element in a triad is approximately the average of the atomic weights of the first and third elements.
Step 2: Apply the law to the given data.
The atomic weights of lithium (\( \text{Li} \)) and potassium (\( \text{K} \)) are 7 and 39, respectively. Sodium (\( \text{Na} \)) lies between them in the triad. Using the formula:
\[ \text{Atomic weight of Na} = \frac{\text{Atomic weight of Li} + \text{Atomic weight of K}}{2}. \]
Substitute the values:
\[ \text{Atomic weight of Na} = \frac{7 + 39}{2} = \frac{46}{2} = 23. \]
Final Answer: The atomic weight of sodium is \( \mathbf{23} \), which corresponds to option \( \mathbf{(4)} \).