In: Physics
Sketch a PV diagram and find the work done by the gas during the following stages. (Draw the PV diagram on paper. Your instructor may ask you to turn in this work.)
(a) A gas is expanded from a volume of 1.0 L to 2.6 L at a
constant pressure of 3.1 atm.
J
(b) The gas is then cooled at constant volume until the pressure
falls to 1.8 atm.
J
(c) The gas is then compressed at a constant pressure of 1.8 atm
from a volume of 2.6 L to 1.0 L. [Note: Be careful of the
signs.]
J
(d) The gas is heated until its pressure increases from 1.8 atm to
3.1 atm at a constant volume.
J
(e) Find the net work done during the complete cycle.
J
Observe my image:
http://img682.imageshack.us/img682/1225/pvrectangle.gif
I indicate Piston/cylinder assemblies of each state, plus how each
transition is accomplished in in-between piston-cylinder
assemblies. I've also included a P-V sketch of the cycle, although
your instructor may want a nicely drawn graph which has proper
scale on the axes and good lableing.
Identify each type of process:
1 to 2: isobaric heating/expansion (indicated by piston lifting a
lot of sand due to heat of the flame)
2 to 3: isochoric cooling/de-pressurization (indicated by stops
constraining piston and air cooling gas)
3 to 4: isobaric cooling/compression (indicated by piston falling
under sand load and cooled by air)
4 to 1: isochoric heating (indicated by stops and flame)
We are interested in finding work done by gas (sign convention of
energy is correct) in each stage.
Recall how work is found by integrating pressure relative to
volume. In common cases, the integration has already been done for
you. In the isobaric case, it turns in to a simple multiplication
and subtraction, because of the integral of a constant. Think area
under the P-V trajectory. What's the area of a rectangle? For the
isochoric cases, it is trivially equal to zero because the piston
moves nowhere.
In first process:
W12 = P1*(V2 - V1)
In second process:
W23 = 0
In third process:
W34 = P3*(V4 - V3)
In final process
W41 = 0
Data:
P1:=3.1 atm; P3:=1.8 atm; V1:=1.0 L; V2:=2.6 L;
Equating of pressures and volumes
P2:=P1; P4:=P3; V4:=V1; V3:=V2;
Results:
Part A: W12 = 7.68 Liter-atmospheres
Part B: W23 = 0
Part C: W34 = -4.32 Liter-atmospheres
Part D: W41 = 0
Convert to Joules (1 L-atm = 101.325 Joules):
Part A: W12 = 778.176 Joules
Part B: W23 = 0
Part C: W34 = -437.724 Joules
Part D: W41 = 0
Negative sign on W34 indicates work being done on gas.
Net work: W_net = W12 + W23 + W34 + W41
W_net = 340.452 Joules
Should you want an expression:
W_net = P1*(V2 - V1) + P3*(V4 - V3)
Simplified with equating of pressures and volumes:
W_net = (P1 - P3)*(V2 - V1), which is the area enclosed by the
cycle trajectory on the P-V diagram