(A) both total temperature and total pressure decrease
(B) both total temperature and total pressure remain constant
(C) total pressure remains constant but total temperature decreases
(D) total temperature remains constant but total pressure decreases
Official Answer Key:
D
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2007 Q.13)**
**1. Normal Shock Governing Relations**:
Across a normal shock wave in a calorifically perfect gas:
- **Total Temperature ($T_0$)**: Because the shock is an adiabatic process with no work or heat transfer ($\dot{q} = 0, \dot{w} = 0$), the total enthalpy and therefore total temperature remain strictly constant:
$$T_{02} = T_{01}$$
- **Total Pressure ($P_0$)**: The normal shock involves severe internal dissipation and entropy generation ($\Delta s > 0$). By the thermodynamic $T\,ds$ relation:
$$\frac{P_{02}}{P_{01}} = e^{-\Delta s / R} < 1$$
Thus, total pressure decreases across the shock wave.
**2. Official Master Key**:
- **Official Answer Key**: **D** (total temperature remains constant but total pressure decreases)
- **Exam Takeaway**: Shock waves are adiabatic ($T_0 = \text{const}$) but irreversible (entropy increases, causing a loss in total pressure $P_0$).