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Step-by-Step Solution
Step 1: Understand the Concept (Avogadro’s Hypothesis)
Avogadro’s law states that, under the same conditions of temperature and pressure, equal volumes of different gases contain the same number of molecules (or moles). Hence, for gases at identical conditions, the ratio of their volumes is the same as the ratio of the moles of each gas.
Step 2: Express the Number of Moles
We are given equal masses of hydrogen ($H_{2}$), oxygen ($O_{2}$), and methane ($CH_{4}$). Let the common mass of each gas be $w$.
Number of moles of $H_{2}$:
$$
n_{H_{2}} = \frac{w}{2}
$$
(because the molar mass of $H_{2}$ is 2 g/mol)
Number of moles of $O_{2}$:
$$
n_{O_{2}} = \frac{w}{32}
$$
(because the molar mass of $O_{2}$ is 32 g/mol)
Number of moles of $CH_{4}$:
$$
n_{CH_{4}} = \frac{w}{16}
$$
(because the molar mass of $CH_{4}$ is 16 g/mol)
Step 3: Calculate the Ratio of Moles
To find the ratio of moles (which, by Avogadro’s law, will also be the ratio of volumes), we write:
$$
n_{H_{2}} : n_{O_{2}} : n_{CH_{4}}
= \frac{w}{2} : \frac{w}{32} : \frac{w}{16}
$$
Canceling $w$ from each term gives:
$$
\frac{1}{2} : \frac{1}{32} : \frac{1}{16}
$$
Step 4: Simplify the Ratio
To remove the fractions, multiply each term by 32:
$$
\left(\frac{1}{2} \times 32\right) : \left(\frac{1}{32} \times 32\right) : \left(\frac{1}{16} \times 32\right)
= 16 : 1 : 2
$$
Step 5: State the Final Answer
Therefore, the ratio of the volumes of the gases $H_{2}$ : $O_{2}$ : $CH_{4}$ when equal masses are taken is:
$$
16 : 1 : 2
$$
