a 100 W electric heater operates for 10 minutes to hear the gas in a cylinder. At the same time, the gas expands from 1 L to 14 L against a constant atmospheric pressure of 4.246 atm. What is the change in internal energy in Joules of the gas?
dE = q+w
q = 100 watts for 10 minutes converted to joules.
work is pdV. p is 1 atm and dV is 14L so work is 14 L*atm. Convert that to joules by 14 L*atm x 101.325 = ?J. Since this is work done by the system make it a negative number so it becomes
dE = q + (-)1*14*101.325 = xJ.
Watch for the typo. I typed in 1 atm for pressure but the problem states 4.246 atm Sorry 'bout that.
To find the change in internal energy of the gas, we can use the formula:
ΔU = q + w
where ΔU is the change in internal energy, q is the heat transferred to the system, and w is the work done by the system.
First, let's calculate the heat transferred to the system:
q = PΔV
where P is the atmospheric pressure and ΔV is the change in volume.
Given:
P = 4.246 atm
ΔV = (14 L - 1 L)
q = (4.246 atm) * (14 L - 1 L)
Now, let's calculate the work done by the system. Since the gas is expanding against a constant pressure, the work done can be calculated as:
w = -PΔV
w = -(4.246 atm) * (14 L - 1 L)
Now, let's calculate ΔU:
ΔU = q + w
Plug in the values we calculated:
ΔU = (4.246 atm) * (14 L - 1 L) + -(4.246 atm) * (14 L - 1 L)
Finally, to convert the units from atm*L to Joules, we'll use the conversion factor:
1 atm*L = 101.325 J
Thus, the change in internal energy of the gas can be calculated by substituting the values:
ΔU = (4.246 atm) * (14 L - 1 L) * 101.325 J/atm*L + -(4.246 atm) * (14 L - 1 L) * 101.325 J/atm*L
Now you can calculate the change in internal energy.
To find the change in internal energy of the gas, we need to consider 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.
In this scenario, the heat added to the gas by the electric heater is given by the equation:
Heat = Power x Time
Given that the electric heater operates at 100 W for 10 minutes, we can convert the time to seconds:
Time = 10 minutes x 60 seconds/minute = 600 seconds
Now we can calculate the heat added:
Heat = 100 W x 600 s = 60,000 J
Next, we need to calculate the work done by the gas. Since the gas expands against a constant atmospheric pressure, we can use the equation:
Work = Pressure x Change in Volume
The change in volume is given as the final volume minus the initial volume:
Change in Volume = 14 L - 1 L = 13 L
Now we can calculate the work done by the gas:
Work = 4.246 atm x 13 L = 55.198 J
Finally, we can find the change in internal energy:
Change in Internal Energy = Heat - Work
Change in Internal Energy = 60,000 J - 55.198 J = 59,944.802 J
Therefore, the change in internal energy of the gas is approximately 59,944.802 Joules.