Verified Answer :

  • Represent the unbalanced reaction with stoichiometric coefficients: \(aC + bH_2 \rightarrow cC_2H_6\).
  • Establish the system of equations from the law of conservation of mass: \(a = 2c\) for Carbon and \(2b = 6c\) for Hydrogen.
  • Solve the system by setting the smallest integer \(c = 1\), resulting in \(a = 2\) and \(b = 3\).
  • Conclude that the balanced equation is \(\boxed{2C(s) + 3H_2(g) \rightarrow C_2H_6(g)}\).

Explanation

  1. Understanding the Problem and Data Analysis

To balance the chemical equation \(C(s) + H_2(g) \rightarrow C_2H_6(g)\), we must find the smallest positive integer stoichiometric coefficients \(a\), \(b\), and \(c\) such that the equation becomes \(aC + bH_2 \rightarrow cC_2H_6\). The principle of the conservation of mass requires that the number of atoms of each element on the left side of the reaction must equal the number of atoms of the same element on the right side.

  1. Setting Up the Balancing Equations

We set up the balance equations for each element involved in the reaction:

  • For Carbon (C): On the left side, there are \(a\) atoms. On the right side, there are \(2c\) atoms (since each molecule of \(C_2H_6\) contains two carbons). Thus, \(a = 2c\).
  • For Hydrogen (H): On the left side, there are \(2b\) atoms (since each molecule of \(H_2\) contains two hydrogens). On the right side, there are \(6c\) atoms (since each molecule of \(C_2H_6\) contains six hydrogens). Thus, \(2b = 6c\), which simplifies to \(b = 3c\).
  1. Solving for Stoichiometric Coefficients

To find the smallest whole number coefficients, we express \(a\) and \(b\) in terms of \(c\). By choosing the smallest positive integer for \(c\), we can find the corresponding integer values for \(a\) and \(b\).
Let \(c = 1\).
Then, \(a = 2(1) = 2\).
And, \(b = 3(1) = 3\).
Substituting these values back into the equation, we get \(2C + 3H_2 \rightarrow 1C_2H_6\). Since all coefficients are positive integers and their greatest common divisor is 1, these are the smallest possible coefficients.

  1. Conclusion and Verification

The balanced equation is \(2C(s) + 3H_2(g) \rightarrow C_2H_6(g)\).

  • Carbon atoms: \(2 \times 1 = 2\) on the left; \(1 \times 2 = 2\) on the right.
  • Hydrogen atoms: \(3 \times 2 = 6\) on the left; \(1 \times 6 = 6\) on the right.

The equation is now balanced. The coefficients are \(a = 2\), \(b = 3\), and \(c = 1\).

Examples

Balancing chemical equations is essential in fields like pharmacology and environmental science. For instance, in industrial chemical production, knowing the precise ratio of reactants ensures that no materials are wasted and that the desired chemical products are formed efficiently. Just like finding the balance between these elements, understanding stoichiometry helps engineers optimize fuel combustion for cleaner energy or synthesize new materials in a cost-effective way.

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