Work Done in Thermodynamic

In thermodynamics, work is one of the two primary ways (along with heat) that energy is transferred between a system and its surroundings. Work plays a fundamental role in engines, compressors, turbines, and refrigerators. Understanding the work done during various thermodynamic processes is essential for analyzing energy systems and predicting system behavior.

This post provides a detailed explanation of work in thermodynamics, formulas for different processes, graphical interpretations, real-life examples, and applications.

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1. Introduction to Work in Thermodynamics

Work is defined as:

The energy transferred when a force acts over a distance or when a system expands or contracts against external pressure.

Key points:

• Symbol: WWW
• SI unit: Joule (J)
• Work can be done by the system on the surroundings or by the surroundings on the system
• Work is path-dependent, meaning it depends on how a process is carried out, not just on the initial and final states.

In thermodynamics, work is associated with energy transfer due to macroscopic motion or changes in system boundaries, such as:

• Gas expansion in a piston-cylinder device
• Rotation of turbines
• Movement of fluids in pumps

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2. Work as a Path Function

Unlike internal energy, which is a state function (depends only on initial and final states), work (WWW) is a path function: W=∫pathP dVW = \int_{\text{path}} P \, dVW=∫path​PdV

Where:

• PPP = pressure
• dVdVdV = infinitesimal change in volume

The total work done depends on the process path. For example, expanding a gas from volume ViV_iVi​ to VfV_fVf​ along different pressure paths yields different work values.

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3. Types of Work in Thermodynamic Systems

Thermodynamic work can occur in various forms:

1. Boundary Work: Work done by a system as its boundary moves (e.g., piston movement)
2. Shaft Work: Work done by rotating devices (turbines, fans)
3. Electrical Work: Work done when electric charges move in a system
4. Other Forms: Magnetic, elastic, or chemical work

Most common type in classical thermodynamics is boundary work, expressed as W=∫P dVW = \int P \, dVW=∫PdV.

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4. Work in Different Thermodynamic Processes

Work done depends on the nature of the process the system undergoes. The common thermodynamic processes are isobaric, isochoric, isothermal, adiabatic, and polytropic.

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4.1 Isobaric Process (Constant Pressure)

• Definition: Pressure remains constant (P=constantP = \text{constant}P=constant)
• Work formula:

W=PΔV=P(Vf−Vi)W = P \Delta V = P (V_f – V_i)W=PΔV=P(Vf​−Vi​)

• Graphical interpretation: Area under the PV curve (rectangle, since PPP is constant)

Example:

• Heating water in an open container: Water expands at atmospheric pressure
• Gas in a piston-cylinder apparatus expands at constant pressure

Key Point: Work is positive if volume increases (expansion) and negative if volume decreases (compression).

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4.2 Isochoric Process (Constant Volume)

• Definition: Volume remains constant (ΔV=0\Delta V = 0ΔV=0)
• Work formula:

W=∫P dV=0W = \int P \, dV = 0W=∫PdV=0

• Graphical interpretation: Vertical line on PV diagram
• Implication: No work is done because the system boundary does not move

Example:

• Heating gas in a rigid, sealed container

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4.3 Isothermal Process (Constant Temperature)

• Definition: Temperature remains constant (T=constantT = \text{constant}T=constant)
• Work formula for ideal gas:

W=∫ViVfP dVW = \int_{V_i}^{V_f} P \, dVW=∫Vi​Vf​​PdV

Since PV=nRTP V = n R TPV=nRT (ideal gas law): P=nRTV⇒W=nRT∫ViVfdVV=nRTln⁡VfViP = \frac{nRT}{V} \quad \Rightarrow \quad W = nRT \int_{V_i}^{V_f} \frac{dV}{V} = nRT \ln \frac{V_f}{V_i}P=VnRT​⇒W=nRT∫Vi​Vf​​VdV​=nRTlnVi​Vf​​

• Graphical interpretation: Area under a hyperbolic PV curve

Example:

• Slow expansion of gas in a piston while keeping it in thermal contact with a heat reservoir

Key Point: Heat added equals work done (Q=WQ = WQ=W) in isothermal expansion of ideal gas because ΔU=0\Delta U = 0ΔU=0.

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4.4 Adiabatic Process (No Heat Exchange)

• Definition: No heat transfer occurs (Q=0Q = 0Q=0)
• Work formula for ideal gas:

W=∫ViVfP dVW = \int_{V_i}^{V_f} P \, dVW=∫Vi​Vf​​PdV

Using adiabatic relation PVγ=constantP V^\gamma = \text{constant}PVγ=constant: W=PiVi−PfVfγ−1W = \frac{P_i V_i – P_f V_f}{\gamma – 1}W=γ−1Pi​Vi​−Pf​Vf​​

Where:

• γ=Cp/Cv\gamma = C_p / C_vγ=Cp​/Cv​ = ratio of specific heats
• Graphical interpretation: Area under steeper PV curve than isothermal

Example:

• Rapid compression or expansion in a piston-cylinder system (no time for heat exchange)

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4.5 Polytropic Process

• Definition: Follows the relation PVn=constantP V^n = \text{constant}PVn=constant
• Work formula:

W=PfVf−PiVi1−n(n≠1)W = \frac{P_f V_f – P_i V_i}{1 – n} \quad (n \neq 1)W=1−nPf​Vf​−Pi​Vi​​(n=1)

• Special Cases:
  • n=0n = 0n=0 → Isobaric
  • n=1n = 1n=1 → Isothermal
  • n=γn = \gamman=γ → Adiabatic

Example:

• Compression in real gas systems where heat transfer is partially allowed

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5. Graphical Representation of Work

• PV Diagrams:
  • Work done = area under the curve
  • Different processes have characteristic curves:
    • Isobaric: rectangle
    • Isochoric: vertical line → zero area
    • Isothermal: hyperbola
    • Adiabatic: steeper hyperbola
• TS Diagrams:
  • Area under curve represents heat transfer, not work

Graphical understanding is crucial for analyzing engines, compressors, and turbines.

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6. Work in Cyclic Processes

A cyclic process returns a system to its initial state: ΔU=0⇒Q=W\Delta U = 0 \quad \Rightarrow \quad Q = WΔU=0⇒Q=W

• Total work done equals area enclosed by the cycle on a PV diagram
• Applications: Heat engines, refrigerators, and air conditioners

Examples of cycles:

1. Carnot cycle
2. Otto cycle (petrol engines)
3. Diesel cycle
4. Rankine cycle (steam turbines)

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7. Sign Conventions

Correct sign convention is essential:

Process	Work (W)	Description
Work done by the system	Positive	System expands against surroundings
Work done on the system	Negative	System is compressed by surroundings

Consistency is critical when applying the first law of thermodynamics: ΔU=Q−W\Delta U = Q – WΔU=Q−W

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8. Work and Energy Transfer

• Work is energy transfer, not a property of the system
• Work increases or decreases internal energy depending on the process
• Relation to heat (First Law):

ΔU=Q−W\Delta U = Q – WΔU=Q−W

• Example:
  • Gas expansion: Does work on surroundings → internal energy may decrease unless heat is added
  • Gas compression: Work done on system → internal energy increases

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9. Practical Examples of Work Done

1. Piston-Cylinder Assembly:
   • Gas expands → does boundary work on piston
   • Work calculated via W=∫P dVW = \int P \, dVW=∫PdV
2. Steam Turbines:
   • Steam expands → rotates turbine → shaft work
3. Refrigerators:
   • Compressor does work on refrigerant → increases pressure
4. Internal Combustion Engines:
   • Expansion stroke: Gas does work on piston

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10. Work in Non-Mechanical Forms

• Electrical work: W=∫V dQW = \int V \, dQW=∫VdQ
• Shaft work: W=τθW = \tau \thetaW=τθ
• Elastic work: W=12kx2W = \frac{1}{2} k x^2W=21​kx2 in springs
• Magnetic work: W=∫H dBW = \int H \, dBW=∫HdB

Thermodynamics primarily deals with boundary work, but energy can transfer via multiple forms.

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11. Mathematical Examples

Example 1: Isobaric Work

• Gas expands at 2 atm from 0.01 m³ → 0.03 m³
• Work:

W=PΔV=2×105×(0.03−0.01)=4000 JW = P \Delta V = 2 \times 10^5 \times (0.03 – 0.01) = 4000 \, \text{J}W=PΔV=2×105×(0.03−0.01)=4000J

Example 2: Isothermal Expansion

• 1 mole ideal gas at 300 K expands from 10 L → 20 L
• Work:

W=nRTln⁡VfVi=1×8.314×300ln⁡2≈1728 JW = nRT \ln \frac{V_f}{V_i} = 1 \times 8.314 \times 300 \ln 2 \approx 1728 \, \text{J}W=nRTlnVi​Vf​​=1×8.314×300ln2≈1728J

Example 3: Adiabatic Expansion

• 1 mole ideal gas, γ=1.4\gamma = 1.4γ=1.4, initial P = 2 atm, V = 0.01 m³, final V = 0.02 m³
• Work:

W=PiVi−PfVfγ−1(calculate Pf=Pi(Vi/Vf)γ)W = \frac{P_i V_i – P_f V_f}{\gamma – 1} \quad (\text{calculate } P_f = P_i (V_i/V_f)^\gamma)W=γ−1Pi​Vi​−Pf​Vf​​(calculate Pf​=Pi​(Vi​/Vf​)γ)

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12. Importance of Work in Thermodynamics

• Design of engines: Calculating work done helps determine efficiency
• Refrigeration and air conditioning: Work input needed for compressors
• Power generation: Work done by steam or gas drives turbines
• Energy analysis: Quantifying energy transfer in heating, cooling, and chemical reactions

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13. Relation Between Work and Heat Capacities

• For processes where volume changes, specific heats and work done are interrelated:
• Isochoric (constant volume): W=0W = 0W=0 → Q=ΔU=mCvΔTQ = \Delta U = m C_v \Delta TQ=ΔU=mCv​ΔT
• Isobaric (constant pressure): W=PΔVW = P \Delta VW=PΔV → Q=ΔU+W=mCpΔTQ = \Delta U + W = m C_p \Delta TQ=ΔU+W=mCp​ΔT

Key Concept: Difference between CpC_pCp​ and CvC_vCv​ arises due to work done by the system at constant pressure.