The equivalent conductance of NaCl at concentration C and at infinite dilution are $$\lambda_C$$ and $$\lambda_{\infty}$$, respectively. The correct relationship between $$\lambda_C$$ and $$\lambda_{\infty}$$ is given as: (where the constant B is positive)

We begin by recalling Kohlrausch’s law of independent migration of ions, which gives the equivalent conductance at infinite dilution:

$$\lambda_{\infty}= \lambda_{+}^{\infty} + \lambda_{-}^{\infty}$$

At infinite dilution the ions are so far apart that inter-ionic attractions are negligible, so $$\lambda_{\infty}$$ is a constant characteristic of the electrolyte (here, $$\text{NaCl}$$).

When the solution is at a finite molar concentration $$C$$, two main effects lower the conductance:

(i) Electrophoretic effect (the dragging of the ionic atmosphere in the opposite direction), and
(ii) Relaxation effect (distortion of the ionic atmosphere).

The combined influence of these two effects on a strong 1 : 1 electrolyte is expressed by the Debye-Hückel-Onsager equation. At 25 °C this equation is usually written in the simplified form

$$\lambda_C = \lambda_{\infty} - (A + B\,\lambda_{\infty})\sqrt{C},$$

where $$A$$ and $$B$$ are positive temperature-dependent constants. Because $$\lambda_{\infty}$$ itself is a positive quantity, we can absorb the entire factor $$(A + B\,\lambda_{\infty})$$ into a single positive constant, say $$B'$$. We then obtain the relation

$$\lambda_C = \lambda_{\infty} - B'\sqrt{C},$$

where $$B' \gt 0$$.

Thus, for any positive concentration, the second term $$B'\sqrt{C}$$ is positive, and it is subtracted from $$\lambda_{\infty}$$, showing that $$\lambda_C$$ is always less than $$\lambda_{\infty}$$ and decreases approximately with the square root of the concentration.

Comparing this derived relation with the four options given, we see that it matches exactly with option C:

$$\lambda_C = \lambda_{\infty} - (B)\sqrt{C}.$$

Hence, the correct answer is Option C.