To calculate the change in internal energy and the heat gained by the gas during an isobaric expansion, you can use the first law of thermodynamics, also known as the energy conservation principle:
ΔU = Q - W
where: ΔU is the change in internal energy of the gas Q is the heat gained by the gas W is the work done by the system (in this case, the gas)
Given that the volume of the gas has increased by a factor of two and the work done by the system is 400 J, we can proceed with the calculations.
- Determine the work done: In an isobaric process, the work done is given by the equation: W = PΔV
Since the volume has increased by a factor of two, ΔV = V_final - V_initial = 2V_initial - V_initial = V_initial.
Therefore, W = PΔV = P * V_initial.
Given that the work done is 400 J, we have: 400 J = P * V_initial.
- Calculate the change in internal energy: We can rearrange the equation to solve for P: P = 400 J / V_initial.
Now, substituting this expression for P into the equation for work done: W = P * V_initial = (400 J / V_initial) * V_initial = 400 J.
So, the work done by the system is 400 J.
- Calculate the change in internal energy: Using the equation ΔU = Q - W, we can rearrange it to solve for ΔU: ΔU = Q - W = Q - 400 J.
Since the process is isobaric, the heat gained by the gas is given by: Q = ΔU + W = ΔU + 400 J.
Substituting this expression for Q into the equation for ΔU: ΔU = (ΔU + 400 J) - 400 J = ΔU.
Therefore, the change in internal energy (ΔU) is equal to the work done by the system, which is 400 J.
In summary:
- The change in internal energy (ΔU) is 400 J.
- The heat gained by the gas (Q) is also 400 J.
Note that this assumes an idealized situation where all other forms of energy transfer, such as heat loss to the surroundings or other forms of work, are negligible.