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Identify the correct coefficients to balance the redox reaction with the lowest possible integer coefficients.

Monica [59]

Answer:

$\rm 3\; Ag^{1+} + 1\; Al \to 1\; Al^{3+} + 3\; Ag$ .

Explanation:

Electrons are conserved in a chemical equation.

The superscript of $\rm Ag^{1+}$ indicates that each of these ions carries a charge of $+1$ . That corresponds to the shortage of one electron for each $\rm Ag^{+}$ ion.

Similarly, the superscript $+3$ on each $\rm Al^{3+}$ ion indicates a shortage of three electrons per such ion.

Assume that the coefficient of $\rm Ag^{+}$ (among the reactants) is $x$ , and that the coefficient of $\rm Al^{3+}$ (among the reactants) is $y$ .

$\rm \mathnormal{x}\; Ag^{1+} + ?\; Al \to \mathnormal{y}\; Al^{3+} + ?\; Ag$ .

There would thus be $x$ silver ( $\rm Ag$ ) atoms and $y$ aluminum ( $\rm Al$ ) atoms on either side of the equation. Hence, the coefficient for $\rm Al\!$ and $\rm Ag\!$ would be $y\!$ and $x\!$ , respectively.

$\rm \mathnormal{x}\; Ag^{1+} + \mathnormal{y}\; Al \to \mathnormal{y}\; Al^{3+} + \mathnormal{x}\; Ag$ .

The $x$ $\rm Ag^{1+}$ ions on the left-hand side of the equation would correspond to the shortage of $x$ electrons. On the other hand, the $y$ $Al^{3+}$ ions on the right-hand side of this equation would correspond to the shortage of $3\, y$ electrons.

Just like atoms, electrons are also conserved in a chemical reaction. Therefore, if the left-hand side has a shortage of $x$ electrons, the right-hand side should also be $x\!$ electrons short of being neutral. On the other hand, it is already shown that the right-hand side would have a shortage of $3\, y$ electrons. These two expressions should have the same value. Therefore, $x = 3\, y$ .