A metal M has a standard reduction potential E^ _ M ²⁺/ M = +0.06 V . A cell is constructed where metal M is placed in a solution containing M ²⁺ ions at a concentration of 10⁻⁴ M and is connected to a hydrogen electrode operating at 1 atm pressure of H ₂ gas. What is the maximum pH of the solution at which metal M can spontaneously liberate H ₂ gas at 298 K ? (Given: 2.303 RT F = 0.06 V )

Step-by-step solution

The reaction for the liberation of hydrogen gas by metal M is: M (s) + 2 H ^+(aq) M ²⁺(aq) + H ₂(g) The standard cell potential is: E^ _ cell = E^ _ H ^+/ H ₂ - E^ _ M ²⁺/ M = 0 - 0.06 = -0.06 V Using the Nernst equation for the cell at 298 K : E_ cell = E^ _ cell - 0.06 n [ M ²⁺][P_ H ₂ ] [ H ^+]^2 Here, n = 2 , [ M ²⁺] = 10⁻⁴ M , and P_ H ₂ = 1 atm . E_ cell = -0.06 - 0.06 2 10⁻⁴ [ H ^+]^2 For the reaction to be spontaneous, E_ cell 0 . -0.06 - 0.03 ( 10⁻⁴ - 2 [ H ^+] ) 0 Substitute pH = - [ H ^+] : -0.06 - 0.03