Final answer:
The change in internal energy of a gas when compressed from 75.9 L to 3.5 L at 0.50 atmospheres while receiving 4200 J of heat is 7.86797 kJ.
Step-by-step explanation:
The question is asking about the change in internal energy of a gas during compression and heat transfer. To find the change in internal energy (ΔU), we need to apply the first law of thermodynamics, which is ΔU = Q - W. Here, Q is the heat added to the system, and W is the work done by the system.
In this scenario, the gas is compressed, so work is done on the gas, and we take W as negative. Given that the gas is compressed from 75.9 L to 3.5 L at a constant pressure of 0.50 atmospheres and gains 4200 J of heat from the surroundings, we can calculate W using W = PΔV, where P is the pressure and ΔV is the change in volume.
However, we need to convert the volume from liters to cubic meters (m³) and pressure from atmospheres to Pascals (Pa) to calculate work in joules since 1 L = 1 x 10-3 m³ and 1 atm = 101325 Pa.
First, calculate the work done: W = PΔV = (0.50 atm)(75.9 L - 3.5 L) = (0.50 atm × 101325 Pa/atm) × (72.4 L × 1 x 10-3 m³/L) = (50662.5 Pa) × (0.0724 m³) = 3667.97 J. The work done on the gas is positive, so we use W = -3667.97 J to represent work done on the system.
Next, calculate the change in internal energy: ΔU = Q - W = 4200 J - (-3667.97 J) = 7867.97 J. Converting this to kilojoules (since 1 kJ = 1000 J), we get ΔU = 7.86797 kJ.
Therefore, the change in internal energy of the gas is 7.86797 kJ.