(Solved): 1. A piston-cylinder device (Fig. 1) contains \( 1 \mathrm{~kg} \) of air at the initial pressure ...

1. A piston-cylinder device (Fig. 1) contains \( 1 \mathrm{~kg} \) of air at the initial pressure \( P=1 \mathrm{MPa} \) and \( T=-23^{\circ} \mathrm{C} \). The air undergoes isothermal expansion until the air pressure reaches the pressure of the surroundings, \( P^{(e)}=100 \mathrm{kPa} \). The temperature of the surroundings is \( T^{(e)}=25^{\circ} \mathrm{C} \). (a) Plot the process described above on the \( P-v, T-v, P-s \), and \( T-s \) diagrams and mark the temperature and pressure of the surroundings on the graphs. (b) By integrating the relation \[ W^{\mathrm{u}}=\left(P-P^{(e)}\right) \mathrm{d} V \] determine the useful work performed in this process. (c) Briefly explain why both the gas pressure \( P \) and the external pressure \( P^{(e)} \) are present in Eq. (1.1). Figure 1: Air undergoing isothermal expansion at a fixed temperature and pressure of the surroundings.