$3$ Closed System with Weight $/$ Internal Energy U

Gas at a pressure of $p_1=0\mathrm{.}4$MPa and a temperature of $t_1=376\mathrm{.}85C$ is enclosed in a cylinder with a frictionless moving piston of an area $A=0\mathrm{.}6m^2$ at an initial height of $2*z.$ On top of the piston, additional mass $m$ is placed. By heat transfer $Q_{12}$ to the environment, the enclosed gas is cooled until the piston rests on the protrusions at a height $z=1m$ within the cylinder. Subsequently, the gas is cooled by heat transfer $Q_{23}$ until the temperature within the cylinder is in thermal equilibrium with the environment $)=\frac{3\mathrm{.}9}{b}ar,t_a=(21\mathrm{.}85C.$ The gas can be modelled as ideal.

Figure $0\mathrm{.}2$: Sketch of piston at state $1$

a$)$ Sketch the process of the system. Name the states and mark the transferred heat and work for each process step.

b$)$ Draw the process in a $p,v-$diagram.

c$)$ Determine the mass $m$ on the piston.

d$)$ Calculate the temperature $t_2$ in state $2$ and the pressure $p_3$ in state $3.$

e$)$ Determine the transferred heat in every process step

Given: $R_{\text{g}\text{a}\text{s}}=287\mathrm{.}09Jk{g}^{-1}{K}^{-1},c_{v,\text{g}\text{a}\text{s}}^{ig}=0\mathrm{.}718kJk{g}^{-1}{K}^{-1},c_{p,\text{g}\text{a}\text{s}}^{ig}=1\mathrm{.}005kJk{g}^{-1}{K}^{-1}$