2 kg of air, initially at 5 bar, 350 K and 4 Kg of carbon monoxide initially at 2 bar, 450 K are confined to opposite sides of a rigid, well insulated container. The partition is free to move and allows conduction from one gas to the other without energy storage in the partition itself. The air and CO behave as ideal gases with constant specific heat ratio, k=1.395. Determine at equilibrium the temperature (K), the pressure (bar) and volume occupied by each gas (m3 )
Assistance needed.
To determine the equilibrium temperature, pressure, and volume for each gas, we can use the principles of thermodynamics and the equation of state for ideal gases.
Step 1: Calculate the initial volumes of each gas using the ideal gas law.
The ideal gas law is given by PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature in Kelvin.
For air:
Initial pressure (P1) = 5 bar
Initial temperature (T1) = 350 K
Number of moles (n1) = mass/Molar mass = 2 kg/(Molar mass of air)
The molar mass of air is the sum of the molar masses of oxygen and nitrogen, which are 32 g/mol and 28 g/mol respectively.
So, the molar mass of air = 0.21 * 32 + 0.79 * 28 = 28.97 g/mol = 0.02897 kg/mol
Number of moles of air (n1) = 2 kg / 0.02897 kg/mol ≈ 69.085 mol
Using the ideal gas law, we can calculate the initial volume of air (V1_air):
V1_air = (n1 * R * T1) / P1
Similarly, for carbon monoxide (CO):
Initial pressure (P2) = 2 bar
Initial temperature (T2) = 450 K
Number of moles (n2) = mass/Molar mass = 4 kg/(Molar mass of CO)
The molar mass of CO = 28 g/mol + 16 g/mol = 44 g/mol = 0.044 kg/mol
Number of moles of CO (n2) = 4 kg / 0.044 kg/mol ≈ 90.909 mol
Using the ideal gas law, we can calculate the initial volume of CO (V1_CO):
V1_CO = (n2 * R * T2) / P2
Step 2: Assuming the final volume is the same for both gases at equilibrium, we can set V_air = V_CO = V_f.
Step 3: Apply conservation of energy to determine the final temperature.
Since the container is well-insulated, the total energy of the system remains constant. Therefore, the sum of the internal energies of the gases is constant. We can express this as follows:
n1 * Cp_air * (T1 - Tf) + n2 * Cp_CO * (T2 - Tf) = 0
Cp_air and Cp_CO are the specific heat capacities of air and CO, respectively.
Step 4: Solve the equations from Step 1 and Step 3 simultaneously to find the equilibrium temperature (Tf).
Step 5: Calculate the equilibrium pressures (P_f) and volumes (V_f) for each gas using the ideal gas law and the equilibrium temperature (Tf).
Using the ideal gas law:
For air: P_f_air = (n1 * R * Tf) / V_f
For CO: P_f_CO = (n2 * R * Tf) / V_f
By following these steps, you should be able to determine the equilibrium temperature, pressure, and volume for each gas in the given scenario.