Thermodynamics Vocabulary

Thermodynamics

Introduction (11.1)

• Thermodynamics studies the laws governing thermal energy, focusing on the conversion between work and heat.

• Historically, heat was viewed as a fluid called 'caloric' that flowed between bodies of different temperatures.

• Benjamin Thomson (Count Rumford) demonstrated that heat generated from boring a brass cannon depended on work done, not the drill's sharpness, suggesting heat is a form of energy.

Topics Covered

• 11.1 Introduction

• 11.2 Thermal equilibrium

• 11.3 Zeroth law of Thermodynamics

• 11.4 Heat, internal energy, and work

• 11.5 First law of thermodynamics

• 11.6 Specific heat capacity

• 11.7 Thermodynamic state variables and equation of state

• 11.8 Thermodynamic processes

• 11.9 Second law of thermodynamics

• 11.10 Reversible and irreversible processes

• 11.11 Carnot engine

Thermodynamics (11.2)

• Thermodynamics is a macroscopic science dealing with heat, temperature, and energy conversion, without focusing on the molecular constitution of matter.

• Thermodynamic description uses macroscopic variables like pressure, volume, temperature, mass, and composition.

• Mechanics studies motion under forces, while thermodynamics studies the internal macroscopic state of a body.

• When a bullet hits wood, its kinetic energy converts to heat, increasing the bullet's and wood's temperature; temperature relates to internal (disordered) motion.

Thermal Equilibrium (11.2)

• Equilibrium in mechanics means zero net external force and torque.

• In thermodynamics, a system is in equilibrium if its macroscopic variables remain constant over time.

• A gas in a closed, insulated container with fixed pressure, volume, temperature, mass, and composition is in thermodynamic equilibrium.

• The state of equilibrium depends on the surroundings and the nature of the wall separating the system from the surroundings.

• Gases A and B in separate containers have pressure and volume (PA, VA) and (PB, VB), respectively.

• An adiabatic wall prevents heat flow; any pair of values (PA, VA) will be in equilibrium with any (PB, VB).

• A diathermic wall allows heat flow until both systems reach equilibrium states (PA′, VA′) and (PB′, VB′).

• In thermal equilibrium, the temperatures of the two systems are equal.

Zeroth Law of Thermodynamics (11.3)

• Systems A and B are separated by an adiabatic wall and in contact with system C via a conducting wall until A and B reach thermal equilibrium with C.

• If the adiabatic wall between A and B is replaced by a conducting wall, no further changes occur; A and B are in thermal equilibrium with each other.

• The Zeroth Law states: If two systems are separately in thermal equilibrium with a third system, they are in thermal equilibrium with each other.

• R.H. Fowler formulated this law in 1931.

• When systems A and B are in thermal equilibrium, there must be a physical quantity (temperature T) that has the same value for both.

• If A and B are separately in equilibrium with C, TA = TC and TB = TC, implying TA = TB, meaning A and B are also in thermal equilibrium.

Heat, Internal Energy, and Work (11.4)

• Temperature indicates a body's 'hotness' and determines the direction of heat flow between two bodies in thermal contact.

• Heat flows from a higher to a lower temperature until temperatures equalize, achieving thermal equilibrium.

• Internal energy (U) is the sum of kinetic and potential energies of all molecules in a system.

• In thermodynamics, internal energy (U) is a macroscopic variable depending only on the state of the system, not how that state was achieved.

• Internal energy depends on pressure, volume, and temperature, but not on the path taken to reach that state.

• Pressure, volume, temperature, and internal energy are thermodynamic state variables.

• Internal energy of a gas includes translational, rotational, and vibrational motions of its molecules.

• Ways to change internal energy include heat transfer and work done on/by the system.

• Heat is energy in transit, not a state variable; internal energy characterizes the state of a system.

• Heat and work are modes of energy transfer, whereas internal energy is a state variable.

First Law of Thermodynamics (11.5)

• The internal energy U of a system can change through heat and work.

• \Delta Q = Heat supplied to the system

• \Delta W = Work done by the system

• \Delta U = Change in internal energy of the system

• The First Law of Thermodynamics, derived from the conservation of energy, is:

• \Delta Q = \Delta U + \Delta W

• The First Law can also be expressed as:

• \Delta Q - \Delta W = \Delta U

• \Delta U depends only on initial and final states, while \Delta Q and \Delta W depend on the path.

• For a process where \Delta U = 0 (e.g., isothermal expansion of an ideal gas):

• \Delta Q = \Delta W

• Work done by a gas moving a piston is:

• \Delta W = P \Delta V

• Therefore:

• \Delta Q = \Delta U + P \Delta V

• Example: For 1 g of water changing from liquid to vapor phase, \Delta Q = 2256 J and \Delta W = 1.013 × 10^5 × (1671 × 10^{–6}) = 169.2 J.

• \Delta U = 2256 - 169.2 = 2086.8 J, showing most heat increases internal energy during the phase transition.

Specific Heat Capacity (11.6)

• Heat capacity (S) is defined as:

• S = \frac{\Delta Q}{\Delta T}

• Specific heat capacity (s) is defined as:

• s = \frac{S}{m} = \frac{1}{m} \frac{\Delta Q}{\Delta T}

• Molar specific heat capacity (C) is defined as:

• C = \frac{S}{\mu} = \frac{1}{\mu} \frac{\Delta Q}{\Delta T}

• For solids with N atoms vibrating about their mean position, the total energy is U = 3RT.

• C = \frac{\Delta Q}{\Delta T} = \frac{\Delta U}{\Delta T} = 3R

• Specific heat capacity depends on the process. For gases, there are specific heats at constant volume and constant pressure.

• For an ideal gas:

• Cp – Cv = R

• At constant volume:

• Cv = (\frac{\Delta Q}{\Delta T})v = (\frac{\Delta U}{\Delta T})_v = \frac{\Delta U}{\Delta T}

• At constant pressure:

• Cp = (\frac{\Delta Q}{\Delta T})p = (\frac{\Delta U}{\Delta T})p + P(\frac{\Delta V}{\Delta T})p

• And

• P(\frac{\Delta V}{\Delta T})_p = R

• Which proves: Cp – Cv = R

Thermodynamic State Variables and Equation of State (11.7)

• Every equilibrium state is described by macroscopic state variables like pressure, volume, temperature, and mass.

• Thermodynamic state variables describe equilibrium states of systems, and their interrelation is the equation of state.

• For an ideal gas, the equation of state is:

• PV = \mu RT

• State variables are either extensive (indicating the 'size' of the system, e.g., internal energy U, volume V, total mass M) or intensive (not indicating size, e.g., pressure P, temperature T, density ρ).

Thermodynamic Processes (11.8)

Quasi-static Process (11.8.1)

• A quasi-static process is an idealized process where the system remains in equilibrium with its surroundings at every stage.

Special Thermodynamic Processes (11.8.1)

• Isothermal: Temperature is constant.

• Isobaric: Pressure is constant.

• Isochoric: Volume is constant.

• Adiabatic: No heat flow between the system and surroundings.

Isothermal Process (11.8.2)

• For an isothermal process (T fixed), the ideal gas equation yields:

• PV = constant

• Work done during an isothermal process:

• W = \int{V1}^{V2} P dV = \mu RT \ln \frac{V2}{V_1}

• In an isothermal process, the heat supplied to the gas equals the work done by the gas: Q = W.

Adiabatic Process (11.8.3)

• In an adiabatic process, the system is insulated from the surroundings; no heat is exchanged (Q = 0).

• Adiabatic process of an ideal gas:

• P V^\gamma = const

• Where

• \gamma = \frac{Cp}{Cv}

• Work done in an adiabatic process:

• W = \frac{P1V1 - P2V2}{\gamma - 1} = \frac{\mu R(T1 - T2)}{\gamma - 1}

Isochoric Process (11.8.4)

• In an isochoric process, Volume is constant.

• No work is done on or by the gas.

• The heat absorbed by the gas goes entirely to change its internal energy and its temperature.

Isobaric Process (11.8.5)

• In an isobaric process, Pressure is constant.

• Work done by the gas is:

• W = P (V2 – V1) = µ R (T2 – T1)

Cyclic Process (11.8.6)

• In a cyclic process, the system returns to its initial state; the total change in internal energy is zero (\Delta U = 0).

• The total heat absorbed equals the work done by the system.

Second Law of Thermodynamics (11.9)

• The Second Law sets a fundamental limit to the efficiency of heat engines and the coefficient of performance of refrigerators: efficiency of a heat engine can never be unity.

• Kelvin-Planck statement: No process is possible whose sole result is the absorption of heat from a reservoir and the complete conversion of the heat into work.

• Clausius statement: No process is possible whose sole result is the transfer of heat from a colder object to a hotter object.

Reversible and Irreversible Processes (11.10)

• A thermodynamic process (state i → state f ) is reversible if the process can be turned back such that both the system and the surroundings return to their original states, with no other change anywhere else in the universe.

• A reversible process is quasi-static and has no dissipative effects.

Carnot Engine (11.11)

• A reversible heat engine operating between two temperatures is called a Carnot engine.

• The Carnot cycle consists of the following steps, for an ideal gas:

  • (a) Step 1 → 2: Isothermal expansion at temperature T_1:

    • W{1 \rightarrow 2} = Q1 = \mu R T1 \ln \frac{V2}{V_1}

  • (b) Step 2 → 3: Adiabatic expansion from (P2, V2, T1) to (P3, V3, T2)

    • W{2 \rightarrow 3} = \frac{\mu R (T1 - T_2)}{\gamma - 1}

  • (c) Step 3 → 4: Isothermal compression at temperature T_2:

    • W{3 \rightarrow 4} = \mu R T2 \ln \frac{V3}{V4}

  • (d) Step 4 → 1: Adiabatic compression from (P4, V4, T2) to (P1, V1, T1)

    • W{4 \rightarrow 1} = \frac{\mu R (T2 - T_1)}{\gamma - 1}

• Efficiency (η) of the Carnot engine:

• η = 1 - \frac{T2}{T1}

• Final remark shows:

• \frac{T2}{T1} = \frac{Q2}{Q1}