- Apply the principle of conservation of mass to set up a system of equations based on the number of atoms for carbon, hydrogen, and oxygen.
- Set the coefficient of glucose to \(a = 1\) to determine the values of the other coefficients through substitution.
- Solve the resulting linear equations to find \(b=6\), \(c=6\), and \(d=6\).
- Verify the balanced equation to confirm the correct sequence of coefficients is \(\boxed{1, 6, 6, 6}\).
Explanation
- Analyzing the Chemical Reaction Balance Requirements
To balance the chemical equation for the combustion of sugar (glucose), represented as \(a C_6H_{12}O_6(s) + b O_2(g) \rightleftarrows c CO_2(g) + d H_2O(l)\), we must ensure that the number of atoms for each element is conserved on both sides of the reaction. This means the number of carbon (C), hydrogen (H), and oxygen (O) atoms in the reactants must equal the number of these atoms in the products.
We can establish the following conservation equations:
- Carbon balance: \(6a = c\)
- Hydrogen balance: \(12a = 2d\), which simplifies to \(6a = d\)
- Oxygen balance: \(6a + 2b = 2c + d\)
- Calculating the Coefficients
Assuming the simplest integer coefficient for glucose, we set \(a = 1\). Using this value, we can determine the other coefficients:
- From the carbon balance equation, \(6a = c\), we get \(c = 6(1) = 6\).
- From the hydrogen balance equation, \(6a = d\), we get \(d = 6(1) = 6\).
- Now, substitute \(a = 1\), \(c = 6\), and \(d = 6\) into the oxygen balance equation: \(6a + 2b = 2c + d\).
This gives:
\(6(1) + 2b = 2(6) + 6\)
\(6 + 2b = 12 + 6\)
\(6 + 2b = 18\)
\(2b = 12\)
\(b = 6\)
Thus, the balanced coefficients are \(a=1, b=6, c=6, d=6\).
- Verification and Conclusion
We have identified the sequence of coefficients to be \(1, 6, 6, 6\). We can verify this set of coefficients by substituting them back into the chemical equation:
\(1 C_6H_{12}O_6(s) + 6 O_2(g) \rightleftarrows 6 CO_2(g) + 6 H_2O(l)\)
Checking the atoms:
- Carbon: 6 on the left, 6 on the right.
- Hydrogen: 12 on the left, 12 on the right.
- Oxygen: \(6 + 2(6) = 18\) on the left, \(2(6) + 6 = 18\) on the right.
The equation is balanced. The sequence of coefficients is \(\boxed{1, 6, 6, 6}\).
Examples
Balancing chemical equations is essential in chemistry for understanding the stoichiometry of reactions, such as in industrial processes like synthesizing ammonia or determining the oxygen required for metabolic processes in the human body. By applying principles of conservation of mass, we can predict how much reactant is needed to produce a specific amount of product, which is fundamental in pharmacology, engineering, and environmental science. Understanding these balanced equations allows for efficient resource management and safe handling of chemical reactions.