GATE 2026 CY – Question 35
The ionic mobility of Ag+ in very dilute aqueous AgNO3 at 298 K is $y\times10^{-8}$ m² V−1 s−1. The value of y (two decimal places) is _____. Given water viscosity $8.94\times10^{-4}$ kg m−1 s−1, $k=1.38\times10^{-23}$ J/K, F = 96500 C/mol, R = 8.314 J mol−1 K−1, and Stokes radius 0.15 nm.
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Correct answer: 6.31 to 6.37
Explanation
**Step 1: Stokes-Einstein relation** for the diffusion coefficient of an ion of Stokes radius $r$ in a solvent of viscosity $\eta$:
$$D=\frac{k_BT}{6\pi\eta r} .$$
**Step 2: Nernst-Einstein relation** between the mobility and the diffusion coefficient of a singly charged ion:
$$u=\frac{zFD}{RT}=\frac{F\,k_B}{6\pi\eta\,r\,R}\quad(z=1),$$
where the temperature cancels.
**Numbers.**
$$u=\frac{96\,500\times1.38\times10^{-23}}{6\pi\times8.94\times10^{-4}\times0.15\times10^{-9}\times8.314}=\frac{1.3317\times10^{-18}}{2.1015\times10^{-11}}=6.337\times10^{-8}\ \text{m}^2\text{V}^{-1}\text{s}^{-1}.$$
So $y=\mathbf{6.34}$.