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 5363 J and its pressure is 2.92 105 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 internal energy of the gas, we need to consider the individual changes in internal energy during each step of the process.
Step 1: Isochoric Heating
During isochoric heating, the volume of the gas remains constant, so the work done on or by the gas is zero (W = 0). In this case, the heat added to the gas (Q) is equal to the change in internal energy (ΔU).
Given:
Heat added (Q1) = 5363 J
Pressure (P1) = 2.92 x 10^5 Pa
Using the first law of thermodynamics: ΔU = Q - W, we can calculate the change in internal energy during this step.
ΔU1 = Q1 - W1
= Q1 - 0
= 5363 J
Therefore, the change in internal energy during the first step is 5363 J.
Step 2: Isobaric Adiabatic Compression
During the second step, the process is both isobaric (constant pressure) and adiabatic (no heat exchange). In this case, the work done on the gas (W) is equal to the negative of the change in heat (Q), as the internal energy decreases.
Given:
Change in volume (ΔV) = -7.90 x 10^-3 m^3
We can calculate the work done (W2) using the formula: W = -PΔV, where P is the constant pressure.
W2 = -PΔV
= -(2.92 x 10^5 Pa)(-7.90 x 10^-3 m^3)
= 2306.8 J
Since the process is adiabatic (Q = 0), the change in internal energy (ΔU) is equal to the work done (ΔU = W2).
ΔU2 = W2
= 2306.8 J
Therefore, the change in internal energy during the second step is 2306.8 J.
Total Change in Internal Energy:
To find the total change in internal energy, we sum up the changes in internal energy during each step.
Total Change in Internal Energy (ΔU) = ΔU1 + ΔU2
= 5363 J + 2306.8 J
= 7670.8 J
Therefore, the total change in internal energy of the gas is 7670.8 J.
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 add them together.
Step 1: Isochoric (Constant Volume) Heating
In an isochoric process, the volume remains constant. The change in internal energy can be calculated using the equation:
ΔU = Q
where ΔU is the change in internal energy and Q is the heat added to the gas.
Given that the gas gains 5363 J of heat, the change in internal energy for this step is:
ΔU1 = Q1 = 5363 J
Step 2: Isobaric (Constant Pressure) Adiabatic Compression
In an isobaric adiabatic process, the pressure remains constant and no heat is transferred to or from the gas. The change in internal energy can be calculated using the equation:
ΔU = W
where ΔU is the change in internal energy and W is the work done on the gas.
The work done on the gas can be calculated using the equation:
W = -PΔV
where P is the pressure and ΔV is the change in volume.
Given that the volume decreases by 7.90 * 10^-3 m3 and the pressure is 2.92 * 10^5 Pa, the work done on the gas is:
W2 = -PΔV = -(2.92 * 10^5 Pa)(7.90 * 10^-3 m^3)
Calculating the value of W2 gives:
W2 ≈ -2303.08 J
Total Change in Internal Energy
To find the total change in internal energy, we add the change in internal energy for each step:
ΔUtotal = ΔU1 + ΔU2
ΔUtotal = 5363 J + (-2303.08 J)
Calculating the value of ΔUtotal gives:
ΔUtotal ≈ 3060.92 J
Therefore, the total change in internal energy of the gas is approximately 3060.92 J.