To find the molar ratio of $$H$_2$ and $$O$_2$ in the mixture, we need to follow these steps:$\\$1. Determine the mass of each gas in the mixture.$\\$• Let the total mass of the mixture be $m. \\$• The mass of $$H$_2$ is $\frac{m}{5}$ (since the ratio is 1:4).$\\$• The mass of $$O$_2$ is $\frac{4m}{5}. \\$2. Calculate the number of moles of each gas.$\\$• Molar mass of $$H$_2$ is $2$ g/mol.$\\$• Moles of $$H$_2 = \frac{\frac{m}{5}}{2} = \frac{m}{10}. \\$• Molar mass of $$O$_2$ is $32$ g/mol.$\\$• Moles of $$O$_2 = \frac{\frac{4m}{5}}{32} = \frac{m}{40}. \\$3. Determine the molar ratio of $$H$_2$ to $$O$_2. \\$Molar ratio = \frac{Moles of $$H_2}{Moles of $$O_2} = \frac{\frac{m}{10}}{\frac{m}{40}} = \frac{m}{10} \times \frac{40}{m} = 4. \\ Thus, the molar ratio of $$H$_2$ to $$O$_2$ is $4:1. \\$Therefore, the correct option is Option 1.