$2.(60$ pts$.)$ An insulated vertical piston$-$cylinder assembly contains $10$ kg of a two$-$phase steam water mixture of quality $0\mathrm{.}60.$ The piston is exposed to the atmosphere and the atmospheric pressure is found to be $100$ kPa $.$ The mass of the piston is $100$ kg $,$ its cross$-$sectional area is $\backslash (0\mathrm{.}01\backslash$mathrm$\{$~m$\}^\{2\}\backslash )$ and the acceleration of gravity is $\backslash (10\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}^\{2\}\backslash ).$ The cylinder is connected through a valve to a steam supply line that provides steam at $0\mathrm{.}5$ MPa and $\backslash (350^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ).$ The valve is opened, steam flows into the cylinder and the piston moves upwards. When the valve is closed, the cylinder is found to contain saturated steam only. Find the temperature in the cylinder and the mass of the steam that has entered. Ignore kinetic and potential energy effects. Friction between the piston and the cylinder wall can be neglected. Use the steam tables for $\backslash (\backslash$mathrm$\{$H$\}\_\{2\}\backslash$mathrm$\{$O$\}\backslash )$ data.