Column efficiency metric: The number of theoretical plates (N) relates to peak width in what way for a given retention time measurement?

Introduction / Context:
Theoretical plates offer a convenient way to quantify column efficiency. Narrower peaks indicate higher efficiency. In HPLC, N is commonly calculated using peak width at base or at half-height.

Given Data / Assumptions:

• Formulas: N = 16*(tR/Wb)^2 or N = 5.54*(tR/W1/2)^2.
• tR is retention time; Wb is peak width at the base; W1/2 is width at half-height.
• Peak shape is assumed approximately Gaussian for these relationships.

Concept / Approach:
Because width appears in the denominator and is squared, N increases as peak width decreases. This captures the intuitive idea that sharper peaks reflect better separation efficiency.

Step-by-Step Solution:
Start from N = k*(tR/W)^2 where k is 16 or 5.54 depending on definition.Recognize the inverse-square dependence on W.Conclude that N is inversely proportional to the square of peak width.

Verification / Alternative check:
Software in chromatographic data systems reports higher N values for the same tR when the measured W is smaller, confirming the mathematical relationship.

Why Other Options Are Wrong:

• Direct proportionality to W^2 contradicts the formula.
• Cube-root dependence is not part of plate number equations.
• Options (b) and (d) repeat the same incorrect trend.

Common Pitfalls:
Mixing up base width and half-height width; use the correct constant and width definition consistently.

Final Answer:
inversely related to the square of the peak width