Final answer:
The initial volume of the gas system was approximately 12.262 liters, calculated using the first law of thermodynamics and the given data.
Step-by-step explanation:
We are asked to calculate the initial volume of a gas system given that the change in internal energy of the gas is -408.119 J, and the gas gains 0.929 kJ of heat (equivalent to 929 J) when it expands to a final volume of 16.661 L against a constant external pressure of 3 atm.
To solve this problem, we'll use the first law of thermodynamics, which states that the change in the internal energy of a system (ΔU) is equal to the heat added to the system (Q) minus the work done by the system (W).
ΔU = Q - W
In this case, the work done (W) can be calculated because we know the pressure and the change in volume (ΔV). Work done by a gas at constant pressure is given by W = PΔV, where P is the external pressure and ΔV is the change in volume.
Given that P is constant and measured in atmospheres, we must convert it to joules by using the conversion factor: 1 atm = 101.325 J/L.
W = PΔV = 3 atm * 101.325 J/L * (ΔV in L)
= 303.975 J/L * (ΔV in L)
Substituting the value of W and Q in the first law, we get:
-408.119 J = 929 J - (303.975 J/L * ΔV)
ΔV can be calculated by rearranging the equation:
ΔV = (929 J + 408.119 J) / (303.975 J/L)
ΔV = 4.399 L
The initial volume (V_initial) can now be found by subtracting the change in volume from the final volume:
V_initial = V_final - ΔV
V_initial = 16.661 L - 4.399 L
V_initial = 12.262 L
Therefore, the initial volume of the gas system was approximately 12.262 liters.