1.00L piston with 1.00 mole of an ideal gas at 298.0K and 1.00bar is isothermally and reversibly compressed to a final volume of 0.200L, the irreversibly expanded in first step to its original volume with an applied pressure of 1.00bar. Calculate the change in heat for the system for the compression and expansion step
To calculate the change in heat for the compression and expansion steps, we can use the first law of thermodynamics, which states that the change in heat (ΔQ) is equal to the work done on or by the system (ΔW) plus the change in internal energy (ΔU).
1. Calculation for the compression step:
Since the process is isothermal, the temperature remains constant at 298.0K. Therefore, the change in internal energy (ΔU) for an ideal gas under isothermal conditions is zero.
We can calculate the work done (ΔW) during the compression step using the formula:
ΔW = -PΔV
Where P is the pressure and ΔV is the change in volume.
Given:
P_initial = 1.00 bar
P_final = 1.00 bar
V_initial = 1.00 L
V_final = 0.200 L
Using the values:
ΔV = V_final - V_initial = 0.200 L - 1.00 L = -0.800 L = -0.800 dm³
Converting L to dm³:
ΔW = -PΔV = -(1.00 bar)(-0.800 dm³) = 0.800 bar dm³
Since 1 bar dm³ = 1 J (Joule), we have:
ΔW = 0.800 J
Since ΔU is zero, the change in heat (ΔQ) for the compression step is equal to the work done (ΔW):
ΔQ_compression = ΔW = 0.800 J
2. Calculation for the expansion step:
In this step, the gas is irreversibly expanded back to its original volume of 1.00 L. Again, since the temperature remains constant, the change in internal energy (ΔU) is zero.
To calculate the work done (ΔW), we use the same formula as before:
ΔW = -PΔV
In this case:
P_applied = 1.00 bar
V_initial = 0.200 L
V_final = 1.00 L
ΔV = V_final - V_initial = 1.00 L - 0.200 L = 0.800 L = 0.800 dm³
ΔW = -PΔV = -(1.00 bar)(0.800 dm³) = -0.800 bar dm³
Converting to Joules:
ΔW = -0.800 J
Again, since ΔU is zero, the change in heat (ΔQ) for the expansion step is equal to the work done (ΔW):
ΔQ_expansion = ΔW = -0.800 J
Therefore, the change in heat for the whole process is the sum of the change in heat for the compression and expansion steps:
ΔQ_total = ΔQ_compression + ΔQ_expansion
ΔQ_total = 0.800 J + (-0.800 J)
ΔQ_total = 0 J
Hence, the change in heat for the system for the compression and expansion steps is zero.