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Question 69

Correct order of limiting molar conductivity for cations in water at 298 K is :

The limiting molar conductivity $$\Lambda_m^{\circ}$$ of an ion in water at $$298 \text{ K}$$ is directly proportional to its ionic mobility $$u^{\circ}$$:
$$\Lambda_m^{\circ} = z \, F \, u^{\circ}$$, where $$z$$ is the ionic charge and $$F$$ is Faraday’s constant.

Ionic mobility depends mainly on (i) ionic size (including hydration), (ii) charge density and (iii) special transport mechanisms, if any.

Case 1: $$H^{+}$$
$$H^{+}$$ is transported via the Grotthuss or proton-hopping mechanism. This “relay race” of protons through the hydrogen-bond network gives it an extremely high mobility, so $$\Lambda_m^{\circ}(H^{+})$$ is the largest for any ion.

Case 2: Monovalent alkali metal ions $$K^{+}$$ and $$Na^{+}$$
Both ions move through ordinary diffusion. Hydration is stronger for the smaller $$Na^{+}$$, which makes its effective (hydrated) radius larger than that of $$K^{+}$$. Therefore
$$u^{\circ}(K^{+}) \gt u^{\circ}(Na^{+})$$ and hence $$\Lambda_m^{\circ}(K^{+}) \gt \Lambda_m^{\circ}(Na^{+})$$.

Case 3: Divalent alkaline-earth ions $$Ca^{2+}$$ and $$Mg^{2+}$$
Both carry double charge, but the smaller $$Mg^{2+}$$ has a much higher charge density, so it is more strongly hydrated and moves more sluggishly than $$Ca^{2+}$$. Hence
$$u^{\circ}(Ca^{2+}) \gt u^{\circ}(Mg^{2+})$$ and $$\Lambda_m^{\circ}(Ca^{2+}) \gt \Lambda_m^{\circ}(Mg^{2+})$$.

Numerical data at $$298 \text{ K}$$ (S cm2 mol−1):
$$\Lambda_m^{\circ}(H^{+}) \approx 349.6$$
$$\Lambda_m^{\circ}(Ca^{2+}) \approx 119.0$$
$$\Lambda_m^{\circ}(Mg^{2+}) \approx 106.1$$
$$\Lambda_m^{\circ}(K^{+}) \approx 73.5$$
$$\Lambda_m^{\circ}(Na^{+}) \approx 50.1$$

Correct descending order:
$$H^{+} \gt Ca^{2+} \gt Mg^{2+} \gt K^{+} \gt Na^{+}$$

This matches Option B.

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