Q. \[ 2\mathrm{H}_2\mathrm{O}_2 \rightarrow 2\mathrm{H}_2\mathrm{O} + \mathrm{O}_2 \]

Goal. Balance the chemical equation for the decomposition of hydrogen peroxide given as \( \mathrm{H}_2\mathrm{O}_2 \) forming water and oxygen. We will find integer coefficients \(a,b,c\) for the general form \( a\,\mathrm{H}_2\mathrm{O}_2 = b\,\mathrm{H}_2\mathrm{O} + c\,\mathrm{O}_2 \).

Step 1. Write atom-balance equations. Count hydrogen and oxygen atoms on each side and write algebraic equations for those counts.

Hydrogen balance. Each molecule of \( \mathrm{H}_2\mathrm{O}_2 \) contains 2 hydrogen atoms, and each molecule of \( \mathrm{H}_2\mathrm{O} \) contains 2 hydrogen atoms. Therefore the hydrogen equation is \( 2a = 2b \).

Oxygen balance. Each molecule of \( \mathrm{H}_2\mathrm{O}_2 \) contains 2 oxygen atoms. Each molecule of \( \mathrm{H}_2\mathrm{O} \) contains 1 oxygen atom, and each molecule of \( \mathrm{O}_2 \) contains 2 oxygen atoms. Therefore the oxygen equation is \( 2a = b + 2c \).

Step 2. Solve the system. From the hydrogen equation \( 2a = 2b \) we get \( a = b \). Substitute \( b = a \) into the oxygen equation \( 2a = b + 2c \) to obtain \( 2a = a + 2c \). Rearranging gives \( a = 2c \). To get the smallest positive integer coefficients, choose \( c = 1 \). Then \( a = 2 \) and \( b = a = 2 \).

Step 3. Write the balanced equation. With coefficients \( a=2 \), \( b=2 \), \( c=1 \) the balanced equation is

\[ 2\,\mathrm{H}_2\mathrm{O}_2 = 2\,\mathrm{H}_2\mathrm{O} + \mathrm{O}_2 \]

Step 4. Verify by counting atoms. Left side: hydrogen \(2\times 2 = 4\) atoms, oxygen \(2\times 2 = 4\) atoms. Right side: hydrogen \(2\times 2 = 4\) atoms from \(2\,\mathrm{H}_2\mathrm{O}\), oxygen \(1\times 2 = 2\) atoms from \(2\,\mathrm{H}_2\mathrm{O}\) plus \(2\) atoms from \(\mathrm{O}_2\), total oxygen \(2 + 2 = 4\). Both elements balance, so the equation is correctly balanced.