Final answer:
To find the equilibrium temperature, use the principle of energy conservation. The kinetic energy of the stirring bar is converted into thermal energy as it slows down due to air resistance. The change in internal energy of the air can be used to calculate the temperature change of the air. The equilibrium temperature is approximately 20.87°C.
Step-by-step explanation:
To find the equilibrium temperature, we need to use the principle of energy conservation. The kinetic energy of the iron stirring bar is converted into thermal energy as it slows down due to air resistance. This thermal energy is then transferred to the air in the container, raising its temperature.
The equation for energy conservation is:
K(initial) + Q = K(final)
where K(initial) is the initial kinetic energy, Q is the heat transferred to the air, and K(final) is the final kinetic energy (which is zero since the bar has come to rest).
In this case, the initial kinetic energy is given as 50.0 J. We can assume that the heat transferred to the air is equal to the change in internal energy of the air, since the container is perfectly insulated. The change in internal energy is given by the equation:
ΔU = nCΔT
where n is the number of moles of air, C is the specific heat capacity of air (which is approximately 29 J/mol·K), and ΔT is the change in temperature.
Substituting the values into the equations, we have:
50.0 J = (0.630 mol) * (29 J/mol·K) * ΔT
Solving for ΔT, we get:
ΔT = 50.0 J / ((0.630 mol) * (29 J/mol·K))
ΔT ≈ 0.87 K
Finally, to find the equilibrium temperature, we add ΔT to the initial temperature:
Equilibrium Temperature = 20.0°C + 0.87 K ≈ 20.87°C