Understanding the Question: We need to determine the mathematical relationship between the physical size (diameter) of a semiconducting CNT and its electronic band gap.
Key Formulas and approach: The band gap ($E_g$) of a semiconducting CNT is inversely proportional to its diameter ($d$). The general relation is:
\[ E_g \propto \frac{1}{d} \]
Detailed Solution:
Step 1: Applying the inverse relationship. Based on the formula $E_g \approx 1/d$, we can see that the energy gap is dependent on the curvature of the tube.
Step 2: Analyzing the effect of increasing diameter. When the diameter $d$ increases, the value of the fraction $1/d$ becomes smaller. Consequently, the band gap $E_g$ must decrease.
Step 3: Physical Interpretation. As a nanotube gets wider, it becomes less curved and starts to resemble a flat graphene sheet. Since graphene has no band gap, the wider the tube, the smaller the gap becomes.
Conclusion: Therefore, an increase in diameter leads to a decrease in the band gap.