Question #44102

An aqueous solution is prepared at $273\text{ K}$ by mixing $50\text{ mL}$ of $2.0 \times 10^{-3}\text{ M } \ce{BaCl2}$ and $50\text{ mL}$ of $2.0 \times 10^{-3}\text{ M } \ce{Na2SO4}$ in a perfectly clean, dust-free container. For $\ce{BaSO4}$, $K_{sp} = 1.0 \times 10^{-10}$ at $273\text{ K}$. The homogeneous nucleation rate $J$ (nuclei $\text{m}^{-3}\text{ s}^{-1}$) is given by $J = A e^{-\frac{\Delta G^*}{RT}}$, where the pre-exponential factor $A = 10^{30}\text{ m}^{-3}\text{ s}^{-1}$ and the critical free energy of nucleation $\Delta G^* = 156.6\text{ kJ mol}^{-1}$.

Given $R = 8.314\text{ J K}^{-1}\text{ mol}^{-1}$ and $e^{-69} \approx 10^{-30}$, what is the concentration of $\ce{Ba^{2+}_{(aq)}}$ remaining in the solution after $1\text{ minute}$?