Find the total change in the internal energy of a gas that is subjected to the following two-step process. In the first step the gas is made to go through isochoric heating until it gains 5963 J and its pressure is 3.12 105 Pa. In the second step it is subjected to isobaric adiabatic compression until its volume decreases by 7.90 10-3 m3. What is the total change in internal energy of this gas?
To find the total change in the internal energy of the gas, we need to calculate the change in internal energy for each step and then find the sum.
Step 1: Isochoric heating
In an isochoric process, the volume remains constant, so the work done on the gas is zero (W = 0). Therefore, according to the first law of thermodynamics:
ΔU = Q - W
Since W = 0, the change in internal energy (ΔU) is equal to the heat added (Q):
ΔU = Q = 5963 J
Step 2: Isobaric adiabatic compression
In an isobaric process, the pressure remains constant. Since the process is adiabatic, no heat is exchanged with the surroundings (Q = 0). Therefore, the change in internal energy can be calculated using:
ΔU = Q - W
Since Q = 0, the work done on the gas (W) is equal to the negative of the work done on the surroundings. The work done in an isobaric process is given by:
W = -PΔV
Where P is the pressure and ΔV is the change in volume. The work done can be calculated as follows:
W = -(3.12 x 10^5 Pa)(-7.90 x 10^-3 m^3) = 2.46 x 10^3 J
Therefore, the change in internal energy (ΔU) for step 2 is:
ΔU = Q - W = 0 - 2.46 x 10^3 J = -2.46 x 10^3 J
Now, we can find the total change in internal energy by summing the changes from each step:
Total change in internal energy = ΔU_step1 + ΔU_step2 = 5963 J + (-2.46 x 10^3 J)
Therefore, the total change in internal energy of the gas is 3493 J.
To find the total change in internal energy of the gas, we need to consider the change in internal energy in each step and then add them up.
Step 1: Isochoric Heating
In this step, the gas undergoes isochoric (constant volume) heating. The change in internal energy is given by the formula:
ΔU = Q
Where ΔU is the change in internal energy and Q is the heat added to the system.
Given that the gas gains 5963 J of heat during this step, ΔU = 5963 J.
Step 2: Isobaric Adiabatic Compression
In this step, the gas undergoes isobaric (constant pressure) adiabatic (no heat transfer) compression. The change in internal energy is given by the formula:
ΔU = W
Where ΔU is the change in internal energy and W is the work done on the system.
To calculate the work done, we need to use the formula:
W = PΔV
Where P is the pressure and ΔV is the change in volume.
Given that the pressure is 3.12 × 10^5 Pa and the volume decreases by 7.90 × 10^-3 m^3, we can calculate the work done:
W = (3.12 × 10^5 Pa) × (7.90 × 10^-3 m^3)
W ≈ 2462 J
Therefore, the change in internal energy in the second step is ΔU = 2462 J.
Total Change in Internal Energy
To find the total change in internal energy, we need to add up the changes in internal energy from each step:
Total ΔU = ΔU1 + ΔU2
Total ΔU = 5963 J + 2462 J
Total ΔU ≈ 8425 J
Therefore, the total change in internal energy of the gas is approximately 8425 J.