Initially 1.200 mol of an ideal gas in a container occupies a volume of 3.50 l at a pressure of 3.30 atm with an internal energy U1 = 547.2 J. The gas is cooled at a constant volume until its pressure is 1.80 atm. Then it is allowed to expand at constant pressure until its volume is 7.90 l. The final internal energy is U2 = 673.7 J. All processes are quasi static. What is the work done by the gas?
To find the work done by the gas, we can use the first law of thermodynamics, which states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system.
The equation for the first law of thermodynamics is:
ΔU = Q - W
Where:
ΔU = change in internal energy
Q = heat added to the system
W = work done by the system
In this case, we are given the initial and final internal energy values (U1 and U2, respectively). We need to find the work done by the gas.
Let's break down the problem and solve it step by step:
Step 1: Finding the change in internal energy (ΔU)
ΔU = U2 - U1
ΔU = 673.7 J - 547.2 J
ΔU = 126.5 J
Step 2: Finding the heat added to the system (Q)
Since the process is at constant volume (isochoric), no work is done by the system. Therefore, all the heat added to the system goes into changing its internal energy.
Q = ΔU (for isochoric process)
Q = 126.5 J
Step 3: Finding the work done by the system (W)
Since the process changes from constant volume to constant pressure, we can calculate the work done by the gas using the equation:
W = PΔV
Where:
W = work done by the system
P = pressure
ΔV = change in volume
Step 4: Calculating the change in volume (ΔV)
The initial volume is 3.50 L, and the final volume is 7.90 L.
ΔV = V2 - V1
ΔV = 7.90 L - 3.50 L
ΔV = 4.40 L
Step 5: Calculating the work done by the system (W)
W = PΔV
W = 1.80 atm * (4.40 L)
W = 7.92 atm * L
Step 6: Converting the units to Joules
To convert atm * L to Joules, we use the conversion factor:
1 atm * L = 101.3 J
W = 7.92 atm * L * 101.3 J / (atm * L)
W = 802.296 J
Therefore, the work done by the gas is approximately 802.296 J.
To find the work done by the gas, you can use the first law of thermodynamics, which states that the change in internal energy (ΔU) of a system is equal to the heat transferred (Q) to the system minus the work done (W) by the system.
Mathematically, it can be written as:
ΔU = Q - W
In this case, you are given the initial and final internal energy values (U1 = 547.2 J and U2 = 673.7 J), so you can calculate the change in internal energy (ΔU) as:
ΔU = U2 - U1
= 673.7 J - 547.2 J
= 126.5 J
Now, in the first step, the gas is cooled at a constant volume, which means no work is done (W1 = 0). So, the change in internal energy (ΔU) is equal to the heat transferred (Q1) during this step.
ΔU = Q1
In the second step, the gas expands at constant pressure. In this case, the work done (W2) can be calculated using the formula:
W = PΔV
Where P is the pressure and ΔV is the change in volume.
The initial pressure and volume are given as P1 = 3.30 atm and V1 = 3.50 L. The final pressure and volume are given as P2 = 1.80 atm and V2 = 7.90 L. So, the change in volume (ΔV) can be calculated as:
ΔV = V2 - V1
= 7.90 L - 3.50 L
= 4.40 L
Now, substitute the values into the formula to calculate the work done (W2):
W2 = P2ΔV
= 1.80 atm * 4.40 L
= 7.92 atm⋅L
Note that atm⋅L is a unit of energy, so you can convert it to Joules by using the conversion factor 1 atm⋅L = 101.325 J:
W2 = 7.92 atm⋅L * 101.325 J/atm⋅L
= 802.284 J
Therefore, the total work done by the gas (W) is the sum of the work done in both steps:
W = W1 + W2
= 0 + 802.284 J
= 802.284 J
So, the work done by the gas is 802.284 J.