Final answer:
During the 5-minute interval, the air in the tank undergoes energy transfers from a paddle wheel and by heat. The change in specific internal energy of the air is 7.5 kJ, which is the sum of the energy transferred by the paddle wheel (5 kJ) and the energy transferred by heat (2.5 kJ). The initial temperature of the air is 200°C, and using the specific heat capacity of air and the energy transferred by heat, we can calculate the final temperature of the air.
Step-by-step explanation:
To determine the change in specific internal energy of the air, we need to calculate the total energy transferred to the air during the 5-minute interval. The paddle wheel transfers energy to the air at a rate of 1 kW, so the total energy transferred by the paddle wheel is 1 kW x 5 minutes = 5 kJ. The air also receives energy by heat transfer at a rate of 0.5 kW, so the total energy transferred by heat is 0.5 kW x 5 minutes = 2.5 kJ.
The change in specific internal energy of the air is the sum of the energy transferred by the paddle wheel and the energy transferred by heat. Therefore, the change in specific internal energy is the sum of 5 kJ and 2.5 kJ, which is 7.5 kJ.
To determine the final temperature of the air, we can use the first law of thermodynamics, which states that the change in internal energy of a system is equal to the energy transferred to the system as heat minus the work done by the system. In this case, there is no work done by the system (no change in kinetic or potential energy), so the change in internal energy is equal to the energy transferred as heat.
The air initially has a temperature of 200°C and receives 2.5 kJ of energy by heat transfer. To determine the final temperature, we need to use the specific heat capacity of the air. Assuming the ideal gas model, the specific heat capacity of air at constant pressure can be given by Cp = (f/2)R, where f is the degree of freedom for the gas (which is 5 for diatomic gases like air) and R is the molar gas constant.
Using the ideal gas equation PV = nRT, where P is the pressure, V is the volume, n is the number of moles, and R is the molar gas constant, we can calculate the number of moles of air in the tank. Given that the tank contains 1.6 kg of air, we can use the molar mass of air (approximately 28.97 g/mol) to calculate the number of moles. Once we have the number of moles, we can calculate the change in specific internal energy as the change in internal energy per mole of air.
Using the specific heat capacity of air at constant pressure, the number of moles of air, and the energy transferred by heat, we can calculate the final temperature of the air. The specific heat capacity of air at constant pressure is approximately 20.786 J/(mol · K). By rearranging the equation Q = mcΔT, where Q is the energy transferred by heat, m is the mass, c is the specific heat capacity, and ΔT is the change in temperature, we can solve for the change in temperature ΔT. Subtracting the change in temperature from the initial temperature of 200°C, we can find the final temperature of the air.