04 Closed Systems

\[ \begin{aligned} \delta Q - \delta W &= dU \\ Q_\text{net} - W_\text{net} &= \Delta U \\ \dot Q_\text{net} - \dot W_\text{net} &= \frac{\mathrm{d} E_\text{sys}}{dt} & \left(\ne \frac{\Delta U}{\Delta t} \right) \\ \text{For a cycle, } Q_\text{net} &= W_\text{net} & (\Delta U = 0) \end{aligned} \]

\[ \begin{aligned} \Delta u &= \int \limits_{T_1}^{T_2} C_V \cdot \mathrm{d} T \\ &= u[T_2] - u[T_1] & \text{(A.7)}\\ \text{For Solids and Liquids, } \Delta u &= C_V \Delta T \\ \text{For Insulated Rigid Tank, } \Delta u &= \Delta U = 0 & (Q_\text{net} = W_\text{net} = 0) \\ \Delta h &= \int \limits_{T_1}^{T_2} C_P \cdot \mathrm{d} T \\ &= h[T_2] - h[T_1] & \text{(A.8)} \\ C_V &= C_P - R \\ C_P &= \sum_0^3 C_n \theta^n & \left( \theta = \frac{T[K]}{1000} \right) \\ &= C_0 + C_1 \theta + C_2 \theta^2 + C_3 \theta^3 \\ \end{aligned} \]