- Set up algebraic equations for each element to satisfy the law of conservation of mass for both reactions.
- Solve the systems of linear equations by choosing a value for one coefficient and determining the others.
- The balanced equation for the first reaction is \(3Mg + 2Al(OH)_3 \rightarrow 3Mg(OH)_2 + 2Al\).
- The balanced equation for the second reaction is \(2Fe + 3NiI_2 \rightarrow 2FeI_3 + 3Ni\).
- Final coefficients for reaction 5: \(\boxed{3, 2, 3, 2}\).
- Final coefficients for reaction 6: \(\boxed{2, 3, 2, 3}\).
Explanation
- Balancing the Reaction of Magnesium and Aluminum Hydroxide
To balance the chemical equation for the reaction \(Mg + Al(OH)_3 \rightarrow Mg(OH)_2 + Al\), we denote the stoichiometric coefficients as \(a, b, c,\) and \(d\):
\[ aMg + bAl(OH)_3 \rightarrow cMg(OH)_2 + dAl \]
According to the law of conservation of mass, the number of atoms of each element must be equal on both sides:
- For Magnesium (Mg): \(a = c\)
- For Aluminum (Al): \(b = d\)
- For Oxygen (O): \(3b = 2c\)
- For Hydrogen (H): \(3b = 2c\)
We have a system of equations: \(a=c\), \(b=d\), and \(3b=2c\). To find the smallest integer coefficients, let \(b=2\).
Substituting \(b=2\) into \(3b=2c\), we get \(3(2) = 2c\), which simplifies to \(6 = 2c\), so \(c=3\).
Since \(a=c\), then \(a=3\).
Since \(b=d\), then \(d=2\).
The balanced equation is:
\[ 3Mg + 2Al(OH)_3 \rightarrow 3Mg(OH)_2 + 2Al \]
- Balancing the Reaction of Iron and Nickel(II) Iodide
To balance the chemical equation for the reaction \(Fe + NiI_2 \rightarrow FeI_3 + Ni\), we denote the stoichiometric coefficients as \(e, f, g,\) and \(h\):
\[ eFe + fNiI_2 \rightarrow gFeI_3 + hNi \]
Again, applying the law of conservation of mass:
- For Iron (Fe): \(e = g\)
- For Nickel (Ni): \(f = h\)
- For Iodine (I): \(2f = 3g\)
We have the equations: \(e=g\), \(f=h\), and \(2f=3g\). To find the smallest integer coefficients, let \(g=2\).
Substituting \(g=2\) into \(2f=3g\), we get \(2f = 3(2) = 6\), so \(f=3\).
Since \(e=g\), then \(e=2\).
Since \(f=h\), then \(h=3\).
The balanced equation is:
\[ 2Fe + 3NiI_2 \rightarrow 2FeI_3 + 3Ni \]
Examples
Balancing chemical equations is essential in laboratory and industrial settings to ensure that the correct proportions of reactants are used. For instance, if a pharmaceutical company needs to synthesize a specific compound, they must use precise stoichiometric ratios to maximize yield and avoid wasting expensive chemicals. This balancing process, based on the principle that atoms are neither created nor destroyed, forms the foundation for calculating quantities in chemistry, known as stoichiometry. It is the core tool for scaling up reactions from test-tube quantities to large-scale manufacturing.