1. We have a closed system containing nitrogen initially at State 1: $p_1 = 1.0$ bar, $T_1 = 300$ K. A three-leg process involving this system occurs as follows:
• Leg A (1$\to$2): Isochoric heating until the pressure doubles.
• Leg B (2$\to$3): Isothermal expansion at the new temperature until the pressure returns to 1.0 bar.
• Leg C (3$\to$4): Isobaric cooling at $p = 1.0$ bar down to $T_4 = 400$ K.
You may consider the following:
– The gas behaves as thermally perfect throughout
– $R$ for nitrogen = 0.2968 kJ/(kg.K), and the specific heat at constant volume is modeled as
$c_v(T) = 0.743 + 2 \times 10^{-4} (T - 300)$ kJ/(kg.K) (1)
(a) Carefully sketch qualitative $p-v$ and $T-v$ diagrams for this process, labeling legs A, B, and C, and indicating whether $v$ increases or decreases on each leg.
(b) Determine $T_2$, $T_3$, and the following specific volume ratios: $v_2/v_1$, $v_3/v_2$, $v_4/v_3$, and $v_4/v_1$.
(c) We previously encountered an expression for the specific work $w$ as
$w = \int_{v_1}^{v_2} pdv$ (2)
Show that the isothermal work can be written
$w = RT\ln \left(\frac{v_3}{v_2}\right)$ (3)
(d) Similarly, show that the isobaric work can be written
$w = R(T_4 - T_3)$ (4)
When expansion occurs under these conditions, should the work be positive or negative? Briefly justify your answer using this expression for the work.
(e) For each leg, compute $\Delta u$, $\Delta h$, and the work $w$ (paying attention to the correct sign convention for $w$). You may tabulate the values you calculate for legibility. Determine the total $\Delta u$, $\Delta h$, and the work $w$ for 1$\to$4.
