Question
Two cylinders, both fitted with frictionless pistons, are filled with mixtures of He and Ar gases. In the first cylinder, the masses of He and Ar are m_1 and m_2, respectively. In the second cylinder, the masses of He and Ar are m_2 and m_1, respectively. The molar mass of Ar is 10 times the molar mass of He. The external pressure applied by the piston on the first cylinder needs to be 5 times that on the second cylinder so that the volume of the gas mixtures in both the cylinders are equal at the same temperature. Assuming He and Ar behave like ideal gases, the value of (m_1/m_2) is ____.
Step-by-step solution
Let the molar mass of He be M. Then the molar mass of Ar is 10M. In the first cylinder, the total number of moles is given by: n_1 = m_1 M + m_2 10M = 10m_1 + m_2 10M In the second cylinder, the total number of moles is given by: n_2 = m_2 M + m_1 10M = 10m_2 + m_1 10M Using the ideal gas equation PV = nRT, since the volume and temperature are the same for both cylinders, the pressure is directly proportional to the number of moles. Thus, P_1 P_2 = n_1 n_2 . Given that P_1 = 5P_2, we have: n_1 n_2 = 5 Substituting the expressions for n_1 and n_2: 10m_1 + m_2 10m_2 + m_1 = 5 10m_1 + m_2 = 50m_2 + 5m_1 5m_1 = 49m_2 m_1 m_2 = 49 5 = 9.8 Answer: 9.8