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In the nernst equation, when a gas - ion electrode is used eg when we write the equation for standard reduction potential for a particular electrode : we get a term of activity of h2 gas and h+ ion. We respectively put 1 atm and 1 molar. However how can we use partial pressure and molarity in the same expression for Q? Shouldn't it be one or the other?

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You can mix activity with fugacity in the expression of equilibrium coefficient or reaction quotient. In fact that is quite common when the reaction involves both solution and gas phase reactants/products.

The definition of the equilibrium constant actually involves dividing the standard pressure $p^\unicode{x29b5}=1\,\rm atm$ or the standard concentration $c^\unicode{x29b5}=1\,\rm mol/L$. In your case, the equilibrium constant will be

$$K^\unicode{x29b5}=\frac{[a(\ce{H+})/c^\unicode{x29b5}]^2}{[f(\ce{H2})/p^\unicode{x29b5}]}$$

which is dimensionless, and is related to the change of Gibbs free energy by $\Delta G=\Delta G^\unicode{x29b5}-RT\ln K^\unicode{x29b5}$.

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