Molecules in the combustion chamber of a rocket engine are in a high state of random motion. When the molecules are expelled through a nozzle in a more ordered state, will their temperature be more, less, or the same as their initial temperature in the chamber before being exhausted?
Solution 35E The molecules inside the combustion chamber of the rocket were in a high state of randomness. When they will be more ordered while coming out through a nozzle, the randomness decreases. STEP 1: Here we should consider two cases. One is from statistical mechanics, which is, the average kinetic energy of a system determines the temperature of the system. So, if the average kinetic energy is high the temperature will be high. So, statistical mechanics tells that, when the molecules will come out through the nozzle, their speed will increase and it will increase the temperature too. STEP 2: The second is from the laws of thermodynamics, which is always true. From the 2nd law we know that the entropy of the universe always increases. The way we have understood temperature, independent of the material and gas chosen is, dS 1 dQ = T ----------------------(1) dQ = TdS This implies that, when we add heat to a system, how fast the entropy of the system increases defines the inverse of temperature. From the 1st law, dE = dQ + dW = TdS PdV = TdS [when the volume is constant] dE = TdS dS 1 dE = T ---------------------(2) So, this says that, the increase in entropy with respect to the increase in internal energy defines the inverse of temperature. STEP 3: The above definition of temperature tells us everything about temperature. When the molecules will come out through the nozzle from the combustion chamber, they are more ordered which shows the decrease in randomness and more precisely the decrease in entropy. But we already know from 2nd law, that the entropy always increases. How these both can be true. The only way by which the entropy can be increased in this case is, by increasing the temperature. CONCLUSION: So, the temperature will increase.
