Question #44229

An electrochemical cell is constructed as follows:

$\ce{M(s) | M^{2+}(aq, 1 M) || X^{-}(aq, 1 M) | X_2(g, V, T) | Pt(s)}$

The cathodic compartment initially contains $10 \text{ moles}$ of a real gas $\ce{X_2}$ in a volume of $2.0 \text{ L}$ at $400 \text{ K}$. The gas undergoes adiabatic free expansion against a vacuum until its volume becomes $10.0 \text{ L}$.

Given for $\ce{X_2(g)}$: van der Waals constants: $a = 6.0 \text{ L}^2 \text{ atm mol}^{-2}$, $b = 0$. Molar heat capacity at constant volume, $C_{v,m} = 3 R$.

Standard electrode potentials (assumed independent of temperature): $E^\circ_{\ce{M^{2+}/M}} = -0.76 \text{ V}$ $E^\circ_{\ce{X_2/X^-}} = +1.36 \text{ V}$

Assume the activity of $\ce{X_2(g)}$ in the Nernst equation is numerically equal to its pressure in atm. Use $R = 0.08 \text{ L atm K}^{-1} \text{ mol}^{-1}$, $\frac{2.303 R \times 300}{F} = 0.06 \text{ V}$. $\log_{10} 2 = 0.30$, $\log_{10} 3 = 0.48$.

Calculate the EMF of the cell after the expansion is complete.