A sample of an ideal gas at 15.0 atm and 10.0 L is allowed to expand against a constant external pressure of 2.00 atm at a constant temperature. Calculate the work in units of kJ for the gas expansion. (Hint: Boyle's law applies)
Please explain thoroughly! thank you!
To calculate the work done during the gas expansion, we need to use the formula:
Work = -Pext * ΔV
Where:
- Work represents the work done by the gas during expansion
- Pext is the external pressure exerted on the gas during expansion
- ΔV is the change in volume of the gas during expansion
In this case, the external pressure, Pext, is 2.00 atm, and the initial volume of the gas is 10.0 L. However, to calculate ΔV, we need to know the final volume of the gas.
To find the final volume, we can use Boyle's law, which states that for a given amount of gas at constant temperature, the product of pressure and volume is constant:
P1 * V1 = P2 * V2
Where:
- P1 is the initial pressure of the gas
- V1 is the initial volume of the gas
- P2 is the final pressure of the gas
- V2 is the final volume of the gas
In this case, the initial pressure, P1, is 15.0 atm, and the final pressure, P2, is 2.00 atm. We know that V1 is 10.0 L, and we want to find V2.
Rearranging the equation to solve for V2, we get:
V2 = (P1 * V1) / P2
Plugging in the values, we find:
V2 = (15.0 atm * 10.0 L) / 2.00 atm
V2 = 75.0 L
Now that we know the final volume of the gas, we can calculate the change in volume, ΔV:
ΔV = V2 - V1
ΔV = 75.0 L - 10.0 L
ΔV = 65.0 L
Finally, we can substitute the values of Pext and ΔV into the equation for work:
Work = -Pext * ΔV
Work = -(2.00 atm) * (65.0 L)
Multiplying these values, we find:
Work = -130.0 atm·L
To convert this from atm·L to kJ, we need to use the conversion factor 1 atm·L = 101.3 J:
Work = -130.0 atm·L * (101.3 J / 1 atm·L)
Work = -13,169 J
To express the work in kilojoules (kJ), we divide the value by 1000:
Work = -13,169 J / 1000
Work ≈ -13.17 kJ
Therefore, the work done by the gas during expansion is approximately -13.17 kJ. The negative sign indicates that work is done on the gas, as opposed to the gas doing work.