If the calculated voltage of the cell is positive, the cell reaction as written will take place

spontaneously

The correct answer is option 2. Spontaneously.

Let us delve into why a positive cell voltage (\( E^\circ_{\text{cell}} > 0 \)) indicates that the cell reaction will proceed spontaneously.

Understanding Cell Potential (\( E^\circ_{\text{cell}} \)):

The cell potential (\( E^\circ_{\text{cell}} \)) is a measure of the tendency of an electrochemical cell to undergo a redox reaction. It is determined by the difference in the standard electrode potentials (\( E^\circ \)) of the two half-cells involved in the cell reaction.

Standard Gibbs Free Energy Change (\( \Delta G^\circ \)):

The relationship between \( E^\circ_{\text{cell}} \) and \( \Delta G^\circ \) (standard Gibbs free energy change) is given by the equation:

\(\Delta G^\circ = -nF E^\circ_{\text{cell}} \)

where:

\( n \) is the number of moles of electrons transferred in the cell reaction,

\( F \) is Faraday's constant (\( \approx 96485 \) C/mol)

Thermodynamic Interpretation:

Positive \( E^\circ_{\text{cell}} \):

If \( E^\circ_{\text{cell}} > 0 \), then \( \Delta G^\circ = -nF E^\circ_{\text{cell}} < 0 \). This indicates that \( \Delta G^\circ \) is negative, meaning the reaction is thermodynamically favorable.

Spontaneity:

A negative \( \Delta G^\circ \) (\( \Delta G^\circ < 0 \)) indicates that the reaction will proceed spontaneously in the forward direction under standard conditions. Spontaneous means that the reaction will occur on its own accord, without requiring external energy input.

Practical Implication:

Forward Reaction: When \( E^\circ_{\text{cell}} > 0 \), the cell reaction as written will take place spontaneously. This means that in an electrochemical cell with positive \( E^\circ_{\text{cell}} \), the reactants will convert into products, releasing energy in the process.

Conclusion:

Thus, the statement "spontaneously" correctly explains why a positive cell voltage (\( E^\circ_{\text{cell}} > 0 \)) indicates that the cell reaction will occur without external assistance, proceeding in the direction as written. This spontaneous nature is a fundamental principle in understanding the behavior of electrochemical cells.