Question

In: Physics

Sketch a PV diagram and find the work done by the gas during the following stages....

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

Solutions

Expert Solution

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


Related Solutions

Sketch a PV diagram for a pure substance. Include several isotherms and phase envelopes. Label the...
Sketch a PV diagram for a pure substance. Include several isotherms and phase envelopes. Label the solid, gas/vapor, liquid/vapor, liquid, solid/liquid, and solid/vapor phases. For the liquid, vapor/liquid, and gas/vapor regions, indicate an appropriate equation of state for predicting PVT data in the phase.
An ideal gas that goes through a cyclical process on a PV diagram and returns to...
An ideal gas that goes through a cyclical process on a PV diagram and returns to the location on the PV diagram where it began. analyze changes in temperature and thermal energy of the gas change during the cycle. Analyze transfers of energy through heat and work during various stages of the cycle. note: including formulas in the analysis would be helpful.
An ideal gas that goes through a cyclical process on a PV diagram and returns to...
An ideal gas that goes through a cyclical process on a PV diagram and returns to the location on the PV diagram where it began. analyze changes in temperature and thermal energy of the gas change during the cycle. Analyze transfers of energy through heat and work during various stages of the cycle. note: including formulas in the analysis would be helpful.
A heat engine using a monatomic gas follows the cycle shown in the ??pV diagram. The...
A heat engine using a monatomic gas follows the cycle shown in the ??pV diagram. The gas starts out at point 1 with a volume of ?1=318 cm3,V1=318 cm3, a pressure of ?1=235 kPa,p1=235 kPa, and a temperature of 287 K.287 K. The gas is held at a constant volume while it is heated until its temperature reaches 455 K455 K (point 2). The gas is then allowed to expand adiabatically until its pressure is again 235 kPa235 kPa (point...
What is the maximum amount of non-pV work that can be done by the reaction 2H2...
What is the maximum amount of non-pV work that can be done by the reaction 2H2 + O2 ->2H2O if DG(H2O)= -2237.13 kJ/mol, and DG(H2) = DG (O2) = 0?
3. Sketch a T-v diagram and find the specified state for the following states a. Propane...
3. Sketch a T-v diagram and find the specified state for the following states a. Propane at critical temp & press find T(°C), p(bar) and v(m3 /kg) b. Water v=.05 m3 /kg, T=140°C find p(bar) c. Water at p=30 MPa, T=100°C find v(m3 /kg)
Find the 4th roots of i. Sketch them in an Argand diagram.
Find the 4th roots of i. Sketch them in an Argand diagram.
(¶) Provide a component diagram sketch of a typical gas chromatography instrument setup and discuss the...
(¶) Provide a component diagram sketch of a typical gas chromatography instrument setup and discuss the importance of each component.
Provide a component diagram sketch of a typical gas chromatography instrument setup and discuss the importance...
Provide a component diagram sketch of a typical gas chromatography instrument setup and discuss the importance of each component.
Hypochlorous acid (HOCl) is produced during chlorination of drinking water. a. Sketch the speciation diagram for...
Hypochlorous acid (HOCl) is produced during chlorination of drinking water. a. Sketch the speciation diagram for HOCl as a function of pH, between pH 4 and pH 12. Clearly label the pH at any intersection points. You will need the Ka of the relevant acid-base reaction. b. Considering that HOCl itself (as opposed to its conjugate base) is the more effective form of the disinfectant. What is the pH range that should be maintained for most effective disinfection?
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT