Laws of Thermodynamics

The Carnot cycle consists of four processes:

1. Isothermal expansion (high temperature T₁): System absorbs heat Q₁ while maintaining constant temperature, entropy increases from S₁ to S₂
2. Adiabatic expansion: Temperature decreases from T₁ to T₂ with no heat transfer, entropy remains constant at S₂
3. Isothermal compression (low temperature T₂): System releases heat Q₂ while maintaining constant temperature, entropy decreases from S₂ to S₁
4. Adiabatic compression: Temperature increases from T₂ to T₁ with no heat transfer, entropy remains constant at S₁

This cycle represents the most efficient possible heat engine operating between two temperature reservoirs.

Work done by a gas in an isothermal process: $W = \int_{V_1}^{V_2} p\,dV = nRT\ln\frac{V_2}{V_1}$

This differs from other processes because:

• Temperature remains constant, so internal energy doesn't change ($\Delta U = 0$)
• All heat added becomes work ($\Delta Q = W$)
• For ideal gas, $pV = nRT$ can be substituted directly into the integral
• The natural logarithm appears because we're integrating $1/V$
• Work depends only on volume ratio, not on specific path between states

Compare to isobaric: $W = p(V_2-V_1)$ Compare to isochoric: $W = 0$

Internal vs External Combustion Engines:

Internal Combustion Engines:

• Fuel burns inside the main cylinder (no separate furnace)
• Higher efficiency (typically 20-40%)
• Compact design suitable for vehicles
• Two main types:
  • Otto (Petrol) Engine: Spark ignition, fuel-air mixture compressed
  • Diesel Engine: Air compressed first, fuel injected at peak compression

External Combustion Engines (Steam):

• Fuel burns externally to heat working fluid (typically water)
• Lower efficiency (typically 3-10%)
• Larger, heavier design
• Heat transfer across temperature gradients causes efficiency losses

Key differences in operation:

• Otto: Four strokes (charging, compression, working, exhaust)
• Diesel: Higher compression ratio, no spark plug, self-ignition
• Steam: Separate boiler, continuous flow system, simpler mechanism

The efficiency advantage of internal combustion engines comes from higher operating temperatures and reduced heat transfer losses.

Clausius statement of the Second Law: "It is impossible to design a refrigerator which works in a cyclic process and whose only result is to transfer heat from a body to a hotter body."

Why perfect refrigerators are impossible:

• Heat naturally flows from hot to cold bodies, never spontaneously in reverse
• To move heat against its natural gradient requires external work
• For every unit of heat Q₂ extracted from a cold reservoir at temperature T₂, at least W = Q₂(T₁/T₂ - 1) work must be done
• A refrigerator without work input would violate the second law by decreasing total entropy
• Such a device (if possible) could be used with a heat engine to create a perpetual motion machine

This limitation is fundamental and applies to all cooling systems, from home refrigerators to industrial cooling plants and air conditioners.

When T₂ > T₁ (wall temperature exceeds gas temperature):

• Before collision: Gas molecules have average velocity corresponding to temperature T₁
• During collision: Gas molecules rebound with increased velocity
• Mechanism: Molecules in hotter walls have higher average kinetic energy, which transfers partially to gas molecules during elastic collisions
• Result: The post-collision velocity (v₂) of a gas molecule is greater than its pre-collision velocity (v₁)
• System effect: The average kinetic energy of all gas molecules increases, raising the gas temperature

This explains thermodynamic equilibration at the molecular level - individual molecular collisions drive the macroscopic process of heat transfer until the gas and wall reach the same temperature.

The work done by a gas during expansion from volume V₁ to V₂ is calculated by:

$W = \int_{V_1}^{V_2} p \, dV$

On a p-V diagram, this work equals the area under the pressure-volume curve bounded by:

• The p-V curve itself
• The volume axis (p=0)
• The vertical lines at V=V₁ and V=V₂

For special cases:

• Isothermal process (ideal gas): $W = nRT\ln(V_2/V_1)$
• Isobaric process (constant p): $W = p(V_2-V_1)$
• Isochoric process (constant V): $W = 0$

Note: When the gas expands (V₂ > V₁), work is positive (done by the gas). When compressed (V₂ < V₁), work is negative (done on the gas).

The power transmission system in a steam engine converts motion through:

1. Linear-to-rotary conversion sequence:

   • Piston creates reciprocating (back-and-forth) linear motion in the cylinder
   • Piston-rod transfers this motion to the crosshead (C₁)
   • Crank-rod (connecting rod) connects crosshead to the crank
   • Crank translates linear force into rotational force on the shaft
   • Mechanism follows slider-crank principle: constrained linear motion drives rotation

2. Key mechanical relationships:

   • When piston is at top/bottom position (dead center), rotational torque is minimal
   • Maximum torque occurs when crank is perpendicular to connecting rod
   • Complete piston cycle (up and down) produces one full shaft rotation

3. Flywheel function:

   • Stores kinetic energy during high-torque portions of cycle
   • Releases energy during low-torque portions (dead centers)
   • Smooths out rotational velocity variations
   • Maintains momentum through low-power portions of cycle
   • Provides consistent rotational energy output despite pulsating input
   • Helps overcome static inertia when starting the engine

This integrated system effectively converts the intermittent force of expanding steam into the smooth rotational motion needed for practical mechanical applications.

The Second Law of Thermodynamics limits heat engine efficiency by requiring:

• No heat engine operating between two temperature reservoirs can exceed the Carnot efficiency

• Maximum theoretical efficiency: η = 1 - T₂/T₁

  • Where T₁ = high temperature (Kelvin)
  • And T₂ = low temperature (Kelvin)

• Physical meaning: Some heat must always be rejected to a lower temperature reservoir

• As T₂ approaches 0 K, efficiency approaches 1 (100%)

• As T₂ approaches T₁, efficiency approaches 0

Example: A heat engine operating between 600 K and 300 K has maximum theoretical efficiency of 1 - 300/600 = 0.5 or 50%.

Note: This limit exists because heat naturally flows from hot to cold; complete conversion to work would violate entropy principles.

A quasi-static process in thermodynamics is:

• A process occurring infinitely slowly such that the system remains in thermodynamic equilibrium at every instant
• Characterized by a well-defined path on thermodynamic diagrams (p-V, T-S, etc.)
• Representable as a continuous curve where each point corresponds to a unique equilibrium state

Differences from non-quasi-static processes:

1. Representation:

   • Quasi-static: Can be fully represented on thermodynamic coordinates
   • Non-quasi-static: Cannot be fully represented as the system passes through non-equilibrium states

2. Parameters:

   • Quasi-static: Pressure, temperature, etc. are uniform throughout the system at each moment
   • Non-quasi-static: System properties vary spatially (non-uniform pressure, temperature gradients)

3. Reversibility:

   • Quasi-static: Can be reversible if non-dissipative
   • Non-quasi-static: Always irreversible

Example: A gas expanding very slowly against a piston (quasi-static) versus a rapid, uncontrolled expansion after puncturing a membrane (non-quasi-static).

In a p-V diagram of a Carnot cycle:

Work:

• The area enclosed by the entire cycle represents the net work output (W) of the engine
• Clockwise cycles produce positive work (heat engines)
• During expansion processes (a→b and b→c), work is done by the system (positive)
• During compression processes (c→d and d→a), work is done on the system (negative)

Heat:

• During isothermal expansion (a→b): Heat Q₁ enters the system at temperature T₁
• During isothermal compression (c→d): Heat Q₂ leaves the system at temperature T₂
• During adiabatic processes (b→c and d→a): No heat exchange (Q = 0)
• Net heat input = Q₁ - Q₂

Entropy changes:

• Isothermal expansion (a→b): ΔS = Q₁/T₁ (positive)
• Isothermal compression (c→d): ΔS = -Q₂/T₂ (negative)
• Adiabatic processes (b→c and d→a): ΔS = 0
• For the complete cycle: ΔS = 0 (entropy is a state function)

Mathematical relationships:

• Efficiency: η = W/Q₁ = 1 - Q₂/Q₁ = 1 - T₂/T₁
• The ratio Q₁/T₁ = Q₂/T₂ (a fundamental property of the Carnot cycle)

The enclosed area's significance extends beyond work done - it represents the maximum possible conversion of heat to work for any engine operating between temperatures T₁ and T₂, as established by the Second Law of Thermodynamics.