Thermodynamics

Thermodynamics studies the relationship between heat, work, and internal energy. It applies to systems in thermal equilibrium and provides the laws governing energy transformations.

Basic Concepts

A thermodynamic system is a collection of matter we study. The surroundings is everything else.

State variables: Pressure (P), Volume (V), Temperature (T), Internal energy (U) — these define the state of a system.

Thermodynamic process: A change of state described by a path.
Isothermal: Constant temperature (T = const); for ideal gas, PV = const.
Adiabatic: No heat exchange (Q = 0); PV^gamma = const.
Isobaric: Constant pressure (P = const).
Isochoric (Isovolumetric): Constant volume (V = const, W = 0).

Zeroth Law of Thermodynamics

If body A is in thermal equilibrium with body C, and body B is also in thermal equilibrium with body C, then A and B are in thermal equilibrium with each other. This defines temperature and is the basis for thermometers.

First Law of Thermodynamics

delta_Q = delta_U + delta_W

Heat supplied (Q) to a system equals the increase in its internal energy (delta_U) plus the work done (W) by the system.

Work done by the system: W = P delta_V (for isobaric process), or integral of P dV in general.

For an ideal gas: Internal energy depends only on temperature (U = n C_v T).

Specific Heats of a Gas

C_v = molar heat capacity at constant volume.
C_p = molar heat capacity at constant pressure.
C_p - C_v = R (Mayer's relation, where R = 8.314 J/mol/K)
Ratio gamma = C_p / C_v (used in adiabatic processes)

For monatomic gas: C_v = (3/2)R, C_p = (5/2)R, gamma = 5/3
For diatomic gas: C_v = (5/2)R, C_p = (7/2)R, gamma = 7/5

Second Law of Thermodynamics

Kelvin-Planck statement: No heat engine operating in a cycle can convert all the heat absorbed from a hot reservoir entirely into work.

Clausius statement: Heat cannot spontaneously flow from a colder body to a hotter body.

Both statements are equivalent and reflect the irreversibility of natural processes.

Heat Engines and Efficiency

A heat engine absorbs heat Q₁ from a hot reservoir, does work W, and rejects heat Q₂ to a cold reservoir.
Efficiency: eta = W/Q₁ = (Q₁ - Q₂)/Q₁ = 1 - Q₂/Q₁

Carnot Engine

The Carnot cycle is the most efficient reversible cycle operating between two temperatures T₁ (hot) and T₂ (cold):
eta_Carnot = 1 - T₂/T₁ (temperatures in Kelvin)

The Carnot engine is a theoretical ideal — no real engine can exceed its efficiency between the same temperatures.

Refrigerators and Heat Pumps

A refrigerator does work W to move heat Q₂ from cold reservoir to hot reservoir.
Coefficient of performance: COP = Q₂/W = T₂/(T₁ - T₂) for a Carnot refrigerator.

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Example 1

500 J of heat is supplied to a gas; it does 200 J of work. Find increase in internal energy.
delta_U = Q - W = 500 - 200 = 300 J

Example 2

An ideal gas (n = 1 mol) expands isobarically at P = 10⁵ Pa; volume changes from 1 L to 3 L. Find work done.
W = P delta_V = 10⁵ x (3-1) x 10^-3 = 10⁵ x 2 x 10^-3 = 200 J

Example 3

A Carnot engine operates between 800 K and 300 K. Find efficiency.
eta = 1 - T₂/T₁ = 1 - 300/800 = 1 - 0.375 = 0.625 = 62.5%

Example 4

A heat engine absorbs 1000 J from hot source and rejects 400 J to the cold sink. Find efficiency and work done.
W = Q₁ - Q₂ = 1000 - 400 = 600 J; eta = 600/1000 = 60%

Example 5

For a diatomic gas (n = 2 mol), temperature rises by 50 K at constant volume. Find heat added.
Q = n C_v delta_T = 2 x (5/2 x 8.314) x 50 = 2 x 20.785 x 50 = 2078.5 J

Example 6

A gas expands adiabatically from (P₁ = 10⁶ Pa, V₁ = 1 L) to V₂ = 8 L. gamma = 5/3. Find P₂.
P₁ V₁^gamma = P₂ V₂^gamma => P₂ = P₁ x (V₁/V₂)^gamma = 10⁶ x (1/8)^5/3 = 10⁶ x 2^-5 = 10⁶/32 = 3.125 x 10⁴ Pa

Example 7

Calculate C_p for a monatomic ideal gas.
C_v = (3/2) R = (3/2) x 8.314 = 12.47 J/mol/K; C_p = C_v + R = 12.47 + 8.314 = 20.78 J/mol/K

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Common mistakes

In the first law, W is work done BY the system. If the surroundings do work on the system, delta_U = Q + W_on = Q - W_by.
Do not confuse isothermal (T constant, PV = constant) with adiabatic (Q = 0, PV^gamma = constant).
Carnot efficiency is the maximum possible efficiency — never a guarantee in real engines.
Always use Kelvin temperatures in Carnot efficiency formula.

Summary

The zeroth law defines temperature; the first law is conservation of energy; the second law sets the direction of natural processes. The Carnot engine provides the upper limit of efficiency. Thermodynamic processes differ in which state variable is held constant.