Step 1: Recall physical properties of nitrogen oxides.
Different oxides of nitrogen have characteristic colours depending on their electronic structure and molecular association.
Step 2: Analyse each oxide.
\(\mathrm{N_2O_5}\) is a colourless solid.
\(\mathrm{N_2O_3}\) is deep blue in liquid state.
\(\mathrm{NO_2}\) is a brown coloured toxic gas.
\(\mathrm{N_2O_4}\) is colourless due to dimerisation of \(\mathrm{NO_2}\).
Step 3: Conclusion.
Nitrogen dioxide (\(\mathrm{NO_2}\)) is the brown coloured gas.