Animated Solution for Chemistry - Electrochemistry: For the following cell with hydrogen electrodes at two different pressures $p_1$ and $p_2$ $Pt(H_2)|H^{+}(aq)|Pt(H_2)$ $p_1$ $1M$ $p_2$ emf is given by

Imagine you are looking at a fascinating electrochemical setup known as a gas concentration cell. In this specific problem, we have two identical hydrogen electrodes, but there is a twist—they are operating at different pressures, $p_1$ and $p_2$.

The cell representation is given as:

By convention, the left side of the cell representation is always the anode, where oxidation takes place. The right side is the cathode, where reduction occurs. Because both electrodes are made of the exact same chemical species (hydrogen), the standard cell potential $E_{cell}^{\circ }$ is exactly zero. The only driving force for this cell is the difference in pressure!

Let's break down the chemistry happening at each electrode.

At the anode, hydrogen gas loses electrons:

At the cathode, hydrogen ions gain electrons:

When we add these two half-reactions together, the $H^{+}$ ions and the electrons perfectly cancel out. The overall net reaction is beautifully simple:

From this balanced equation, we can extract two critical pieces of information for the Nernst equation. First, the number of electrons transferred is $n=2$. Second, the reaction quotient $Q$ is the ratio of the product pressure to the reactant pressure, which gives us $Q=\frac{p_2}{p_1}$.

Now, we bring in the heavy artillery—the Nernst equation:

Substituting our known values into the equation:

To clean up the expression and remove the negative sign, we simply take the reciprocal of the fraction inside the logarithm. This is a standard mathematical trick that makes the final answer look much more elegant:

And there we have it! The cell potential is directly proportional to the natural logarithm of the pressure ratio. For the cell to be spontaneous ($E_{cell}>0$), the initial pressure $p_1$ must be greater than the final pressure $p_2$. It perfectly aligns with our physical intuition: gas naturally wants to expand from a region of high pressure to a region of low pressure!