A gas transfers energy by doing pressure-volume work. At constant external pressure, work done by the gas is W = PΔV, and the area under a pressure-volume graph represents the work.
Learning Outcomes
Pressure force on a piston is F = PA.
Piston displacement changes volume by ΔV = AΔy.
At constant pressure, W = PΔV.
Expansion gives ΔV > 0 and positive work by the gas.
Compression gives ΔV < 0 and negative work by the gas under the textbook convention.
Area under a P-V curve equals work done by the gas.
Intermediate Level
Pressure-Volume Work
Consider gas in a cylinder with a movable frictionless piston of area A. External pressure P produces force F = PA. If the piston moves distance Δy, the volume changes by AΔy.
Key equation
F=PA
Key equation
Δ V=AΔ y
Key equation
W=FΔ y=PAΔ y
Key equation
W=PΔ V
Here P is the external pressure and ΔV = V₂ − V₁. This simple form applies when external pressure is constant.
P-V Graph
On a graph of pressure against volume, a narrow strip has area P dV. Adding all strips gives the total work.
Key equation
W=∫V1V2PdV
For constant pressure, the area is a rectangle: W = P(V₂ − V₁).
Textbook Sign Convention
Process
ΔV
Work done by gas W
Energy description
Expansion
positive
positive
gas does work on surroundings
Compression
negative
negative
surroundings do work on gas
Constant volume
zero
zero
no boundary work
Solved Examples
Constant-Pressure Expansion
A gas expands from 0.020 m³ to 0.050 m³ against 2.0 × 10⁵ Pa. Find work by the gas.
1
Working
ΔV = 0.050 − 0.020 = 0.030 m³
2
Step 2
W = PΔV
3
Step 3
W = (2.0 × 10⁵)(0.030)
Final answer:W = +6.0 × 10³ J
Compression
A gas is compressed by 4.0 × 10⁻³ m³ at constant pressure 1.5 × 10⁵ Pa. Find W by the gas and work on the gas.
1
Working
ΔV = −4.0 × 10⁻³ m³
2
Step 2
Wby = PΔV
3
Step 3
Wby = (1.5 × 10⁵)(−4.0 × 10⁻³)
Final answer:Work by gas = −600 J; work on gas = +600 J
Common Mistakes
Use absolute temperature in kelvin.
Convert all quantities to SI units.
State the assumptions or sign convention before applying a formula.
Short-Question Preparation
Q. Derive W = PΔV.
A.Use F = PA, W = FΔy and ΔV = AΔy.
Q. How is work found from a P-V graph?
A. It is the area under the curve between the initial and final volumes.
Q. State the textbook sign convention.
A. Expansion gives positive work by the gas; compression gives negative work by the gas.
Q. When is W = PΔV directly applicable?
A. When the external pressure is constant.
Sticky Notes
Pressure force on a piston is F = PA.
Piston displacement changes volume by ΔV = AΔy.
At constant pressure, W = PΔV.
Expansion gives ΔV > 0 and positive work by the gas.
Compression gives ΔV < 0 and negative work by the gas under the textbook convention.
Area under a P-V curve equals work done by the gas.
60-Second Revision
A gas transfers energy by doing pressure-volume work. At constant external pressure, work done by the gas is W = PΔV, and the area under a pressure-volume graph represents the work.
Summary
A gas transfers energy by doing pressure-volume work. At constant external pressure, work done by the gas is W = PΔV, and the area under a pressure-volume graph represents the work.
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Exam preparation
Exam Focus
Important points
Pressure force on a piston is F = PA.
Piston displacement changes volume by ΔV = AΔy.
At constant pressure, W = PΔV.
Expansion gives ΔV > 0 and positive work by the gas.
Compression gives ΔV < 0 and negative work by the gas under the textbook convention.
Area under a P-V curve equals work done by the gas.
Common mistakes
Use the stated model and convention.
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.
Thermodynamic Work and P-V Diagrams
Sticky Notes · Quick Revision
1.
Key Point 1
Pressure force on a piston is F = PA.
2.
Key Point 2
Piston displacement changes volume by ΔV = AΔy.
3.
Key Point 3
At constant pressure, W = PΔV.
4.
Key Point 4
Expansion gives ΔV > 0 and positive work by the gas.
