⚙️ Laws of Thermodynamics

First Law of Thermodynamics

Energy conservation for thermodynamic systems:

$\Delta U = Q - W$

where:

• $\Delta U$ = change in internal energy (J)
• $Q$ = heat added to system (J)
• $W$ = work done BY system (J)

Sign Conventions:

Heat (Q):

• $Q > 0$: Heat added to system (absorbed)
• $Q < 0$: Heat removed from system (released)

Work (W):

• $W > 0$: System does work on surroundings (expansion)
• $W < 0$: Work done on system (compression)

Internal Energy (ΔU):

• $\Delta U > 0$: Energy increases, temperature rises
• $\Delta U < 0$: Energy decreases, temperature falls

💡 Key Insight: Energy can enter as heat or leave as work, but total energy is conserved.

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Internal Energy

Internal energy is the total kinetic and potential energy of all molecules.

For an ideal gas: $U = \frac{3}{2}nRT$

where:

• $n$ = number of moles
• $R$ = 8.31 J/(mol·K)
• $T$ = absolute temperature (K)

For ideal gas: $\Delta U$ depends ONLY on temperature change! $\Delta U = \frac{3}{2}nR\Delta T$

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Work in Thermodynamic Processes

For gas expansion/compression:

$W = P\Delta V$

where:

• $P$ = pressure (Pa)
• $\Delta V = V_f - V_i$ (m³)

On a PV diagram: Work = area under curve

• Expansion ($\Delta V > 0$): $W > 0$ (system does work)
• Compression ($\Delta V < 0$): $W < 0$ (work done on system)

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Thermodynamic Processes

1. Isothermal (Constant Temperature)

• $\Delta T = 0$ → $\Delta U = 0$
• First Law: $Q = W$
• All heat becomes work
• For ideal gas: $PV =$ constant

2. Adiabatic (No Heat Transfer)

• $Q = 0$
• First Law: $\Delta U = -W$
• Work comes from/goes to internal energy
• Temperature changes!
• Example: Rapid compression/expansion

3. Isobaric (Constant Pressure)

• $P =$ constant
• $W = P\Delta V$
• $Q = \Delta U + P\Delta V$
• Example: Piston free to move

4. Isochoric (Constant Volume)

• $\Delta V = 0$ → $W = 0$
• First Law: $Q = \Delta U$
• All heat changes internal energy
• Example: Rigid container

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PV Diagrams

Pressure-Volume (PV) diagrams show thermodynamic processes:

• Horizontal line: Isobaric (constant P)
• Vertical line: Isochoric (constant V)
• Curve: Isothermal or adiabatic

Work = area under curve

• Clockwise cycle: Net work done BY system (W > 0)
• Counter-clockwise: Net work done ON system (W < 0)

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Second Law of Thermodynamics

Heat flows spontaneously from hot to cold, not the reverse (without external work).

Equivalently: Entropy of an isolated system always increases.

Entropy (S): Measure of disorder/randomness

• Natural processes increase total entropy
• Energy becomes more spread out, less useful

Implications:

1. No perfect heat engine: Can't convert all heat to work
2. No perfect refrigerator: Can't transfer heat from cold to hot without work
3. Time's arrow: Explains why time has a direction

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Heat Engines

Convert heat into mechanical work:

Efficiency: $e = \frac{W}{Q_H} = \frac{Q_H - Q_C}{Q_H} = 1 - \frac{Q_C}{Q_H}$

where:

• $Q_H$ = heat absorbed from hot reservoir
• $Q_C$ = heat expelled to cold reservoir
• $W = Q_H - Q_C$ = useful work output

Maximum efficiency (Carnot engine): $e_{max} = 1 - \frac{T_C}{T_H}$

Temperatures in Kelvin!

💡 Key: Efficiency always < 100%. Some heat must be expelled.

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Refrigerators and Heat Pumps

Move heat from cold to hot (requires work):

Coefficient of Performance (COP): $COP_{refrigerator} = \frac{Q_C}{W}$ $COP_{heat pump} = \frac{Q_H}{W}$

Higher COP = more efficient

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Problem-Solving Strategy

1. Identify the process: Isothermal, adiabatic, isobaric, isochoric
2. Apply First Law: $\Delta U = Q - W$
3. Use process-specific relationships:
   • Isothermal: $\Delta U = 0$
   • Adiabatic: $Q = 0$
   • Isobaric: $W = P\Delta V$
   • Isochoric: $W = 0$
4. For ideal gas: $\Delta U = \frac{3}{2}nR\Delta T$
5. Watch signs for Q and W!

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Common Mistakes

❌ Wrong sign for Q or W ❌ Using Celsius instead of Kelvin for Carnot efficiency ❌ Forgetting ΔU = 0 for isothermal process ❌ Confusing heat added vs. heat expelled in engines ❌ Assuming all heat can be converted to work (violates 2nd Law)