For a galvanic cell operating at 298 K , the overall cell reaction...

The given cell reaction is:

$2\text{Al(s)} + 3\text{Cu}^{2+}(0.001\text{ M}) \rightarrow 2\text{Al}^{3+}(0.001\text{ M}) + 3\text{Cu(s)}$

The number of electrons transferred in the balanced reaction is $n = 6$.

The reaction quotient $Q$ is given by:

$Q = \dfrac{[\text{Al}^{3+}]^2}{[\text{Cu}^{2+}]^3}$

Substituting the given concentrations:

$Q = \dfrac{(10^{-3})^2}{(10^{-3})^3} = \dfrac{10^{-6}}{10^{-9}} = 10^3$

Using the Nernst equation:

$E_{\text{cell}} = E^\circ_{\text{cell}} - \dfrac{0.059}{n} \log Q$

Substituting the values:

$E_{\text{cell}} = 2.00 - \dfrac{0.059}{6} \log(10^3)$

$E_{\text{cell}} = 2.00 - \dfrac{0.059}{6} \times 3$

$E_{\text{cell}} = 2.00 - \dfrac{0.059}{2}$

$E_{\text{cell}} = 2.00 - 0.0295 = 1.9705\text{ V}$

Answer: $1.9705\text{ V}$