Gaseous State

Concept:For ideal gas mixtures at same temperature and volume, pressure is directly proportional to total moles ($n$).Explanation:Let the molar mass of He be $M$. Then Ar has molar mass $10M$ (given).First cylinder: He mass $m_1$, Ar mass $m_2$.Moles: He $= \frac{m_1}{M}$, Ar $= \frac{m_2}{10M}$.Total moles: $n_1 = \frac{m_1}{M} + \frac{m_2}{10M}$.Second cylinder: He mass $m_2$, Ar mass $m_1$.Moles: He $= \frac{m_2}{M}$, Ar $= \frac{m_1}{10M}$.Total moles: $n_2 = \frac{m_2}{M} + \frac{m_1}{10M}$.Given $P_1 = 5P_2$ at equal $V$ and $T$, so from $PV=nRT$, $n_1 = 5n_2$.Hence: $\frac{m_1}{M} + \frac{m_2}{10M} = 5\left(\frac{m_2}{M} + \frac{m_1}{10M}\right)$.Multiply by $10M$: $10m_1 + m_2 = 50m_2 + 5m_1$.Simplify: $10m_1 - 5m_1 = 50m_2 - m_2 \implies 5m_1 = 49m_2$.Thus $\frac{m_1}{m_2} = \frac{49}{5}$.Answer:$\frac{49}{5}$ (or $9.80$).