Thermodynamics

Thermodynamics

$Learning Objectives$

Differentiate between temperature, heat, and internal energy.
Calculate thermal expansion and heat required for temperature changes and phase changes.
Explain the three mechanisms of heat transfer: conduction, convection, and radiation.
Understand and apply the laws of thermodynamics (Zeroth, First, Second, and Third).
Analyze thermodynamic processes and calculate the efficiency of heat engines.
Relate macroscopic gas properties to microscopic behavior using the Ideal Gas Law and Kinetic Theory.

Thermodynamics is the branch of physics that deals with heat, work, and temperature, and their relation to energy, radiation, and physical properties of matter. It is fundamental in designing engines, HVAC systems, and power plants.

Temperature and Heat

$Temperature and Heat Concepts$

In everyday language, "heat" and "temperature" are often used interchangeably. In physics, they have very precise and distinct meanings.

$T$

A thermodynamic state variable that determines the direction of spontaneous heat transfer when systems are placed in thermal contact. In the ideal-gas kinetic model, temperature is directly related to average molecular kinetic energy; that microscopic interpretation should not be generalized to every material without qualification. The SI unit is the kelvin (K), while Celsius ( $^\circ$ C) is also widely used in engineering.

Kelvin–Celsius Conversion

Converts Celsius temperature to absolute temperature in kelvins.

T_K = T_C + 273.15

Variables

Symbol	Description	Unit
$T_K$	Absolute temperature	K
$T_C$	Celsius temperature	°C

$Q$

The transfer of energy between a system and its environment due solely to a temperature difference. Heat always flows spontaneously from an object at higher temperature to one at lower temperature. The SI unit is the Joule (J).

$Heat vs. Internal Energy$

A system has state properties such as temperature and internal energy. Heat is not a stored property; it is energy transferred across a system boundary because of a temperature difference.

Multi-panel diagram distinguishing temperature internal energy and heat transfer, a phase-change heating curve, and thermal expansion. — Thermal State, Heating, Phase Change, and Expansion
The panels distinguish state variables from energy transfer, show sensible versus latent heating, and connect temperature change with thermal expansion.

Thermal Expansion and Heat Capacity

$Thermal Expansion and Heat Capacity Concepts$

Most materials expand when heated and contract when cooled. For solids and liquids, this expansion is typically proportional to the temperature change.

Linear Thermal Expansion

Calculates the change in length of a solid due to a change in temperature.

\Delta L = \alpha L_0 \Delta T

Variables

Symbol	Description	Unit
$\Delta L$	Change in length	m
$\alpha$	Coefficient of linear expansion	$K^{-1}$
$L_0$	Initial length	m
$\Delta T$	Change in temperature	K

Volume Thermal Expansion

Calculates the change in volume of a substance due to a change in temperature.

\Delta V = \beta V_0 \Delta T

Variables

Symbol	Description	Unit
$\Delta V$	Change in volume	$m^3$
$\beta$	Coefficient of volume expansion	$K^{-1}$
$V_0$	Initial volume	$m^3$
$\Delta T$	Change in temperature	K

Specific Heat Capacity ( $c$ )

$c$

The amount of heat ( $Q$ ) required to change the temperature of a substance depends on its mass ( $m$ ) and a material property called specific heat capacity.

Specific Heat Equation

Calculates the heat required to change the temperature of a mass.

Q = mc\Delta T

Variables

Symbol	Description	Unit
$Q$	Heat added or removed	J
$m$	Mass	kg
$c$	Specific heat capacity	$J/(kg \cdot K)$
$\Delta T$	Change in temperature	K

$Specific Heat Capacity$

Water has a very high specific heat ( $4186 \text{ J/kg}\cdot\text{K}$ ), meaning it takes a lot of energy to change its temperature.

Latent Heat ( $L$ )

$L$

When a substance undergoes a phase change (like melting or boiling), heat is added or removed without any change in temperature. The energy goes into breaking or forming intermolecular bonds.

Latent Heat

Calculates the heat required to change the phase of a substance without changing its temperature.

Q = mL

Variables

Symbol	Description	Unit
$Q$	Heat added or removed	J
$m$	Mass of the substance	kg
$L$	Latent heat of fusion or vaporization	J/kg

Mechanisms of Heat Transfer

$Mechanisms of Heat Transfer Concepts$

There are three fundamental mechanisms by which heat is transferred:

Conduction: Heat transfer through stationary matter by physical contact (e.g., a metal spoon getting hot in soup).
Convection: Heat transfer by the macroscopic movement of a fluid (e.g., hot air rising, water circulating in a pot). Natural convection is driven by buoyant forces, while forced convection uses pumps or fans.
Radiation: Heat transfer by electromagnetic waves (e.g., feeling the heat from the sun or a fire). Does not require a medium.

Fourier's Law of Heat Conduction

Calculates the rate of heat transfer through a material via conduction.

P = \frac{Q}{t} = kA \frac{\Delta T}{L}

Variables

Symbol	Description	Unit
$P$	Rate of heat transfer (Power)	W
$k$	Thermal conductivity of the material	$W/(m\cdot K)$
$A$	Cross-sectional area	$m^2$
$\Delta T$	Temperature difference across the material	$K \text{ or } ^\circ\text{C}$
$L$	Thickness of the material	m

Stefan-Boltzmann Law of Radiation

Calculates the thermal-radiation power emitted by a diffuse gray surface.

P = \sigma e A T^4

Variables

Symbol	Description	Unit
$P$	Radiated power	W
$\sigma$	Stefan-Boltzmann constant	$5.67 \times 10^{-8} \text{ W}/(\text{m}^2\cdot\text{K}^4)$
$e$	Emissivity of the object	$dimensionless (0 \text{ to } 1)$
$A$	Surface area	$m^2$
$T$	Absolute temperature	K

$Net Radiation Exchange$

The expression above is emitted power. For a small gray surface exchanging radiation with large surroundings at temperature $T_{sur}$ , a common net model is $P_{net}=\epsilon\sigma A(T^4-T_{sur}^4)$ . Geometry and view factors are required for more general radiative exchange.

$Civil Engineering Applications of Heat Transfer$

Insulation Sizing: Determining $L$ and selecting materials with low $k$ to minimize heat loss/gain in buildings.
Urban Heat Island Effect: Accounting for radiation absorbed and re-emitted by asphalt and concrete surfaces.
Thermal Stress Analysis: Calculating temperature gradients that cause structural expansion or contraction.

Conduction through a wall, convection circulation in a fluid, and thermal radiation from a surface with directional energy-transfer arrows. — Conduction, Convection, and Radiation
The three panels distinguish transfer through matter, bulk fluid motion, and electromagnetic radiation; real systems can involve all three at once.

The Laws of Thermodynamics

$The Laws of Thermodynamics Concepts$

These four empirical laws govern all macroscopic interactions involving energy and temperature.

The Zeroth Law: Thermal Equilibrium

$The Zeroth Law: Thermal Equilibrium Concepts$

If system A is in thermal equilibrium with system C, and system B is also in thermal equilibrium with system C, then system A and system B must be in thermal equilibrium with each other.

This law defines temperature: two systems in thermal equilibrium have the same temperature.

The First Law: Conservation of Energy

$The First Law: Conservation of Energy Concepts$

The First Law is the principle of conservation of energy applied to thermal systems. It states that the change in a system's internal energy ( $\Delta U$ ) is equal to the net heat added to the system ( $Q$ ) minus the net work done by the system on its surroundings ( $W$ ).

The First Law of Thermodynamics

Conservation of energy applied to thermal systems.

\Delta U = Q - W

Variables

Symbol	Description	Unit
$\Delta U$	Change in internal energy of the system	J
$Q$	Net heat added to the system	J
$W$	Net work done by the system	J

$Sign Conventions for the First Law$

Sign conventions are critical here:

$+Q$ : Heat is added to the system.
$-Q$ : Heat is removed from the system.
$+W$ : Work is done by the system (it expands).
$-W$ : Work is done on the system (it is compressed).

Work Done by a Gas

Calculates the work done by a gas expanding or compressing against a pressure.

W = \int_{V_i}^{V_f} P \, dV

Variables

Symbol	Description	Unit
$W$	Work done by the gas	J
$P$	Pressure of the gas	Pa
$V_i, V_f$	Initial and final volumes	$m^3$

$Work Done on a P-V Diagram$

For a quasistatic process, the boundary work done by the gas is the signed area under the process path on a pressure–volume diagram: expansion gives positive work with the sign convention used here, while compression gives negative work.

Multi-panel first-law sign convention, shaded pressure-volume work area, and isochoric isobaric isothermal and reversible adiabatic process paths. — First Law, P–V Work, and Process Comparison
Heat into the system is positive, work by the system is positive, P–V area is boundary work, and the idealized process paths show their distinct constraints.

Thermodynamic Processes

$Thermodynamic Processes Concepts$

When a gas changes state (pressure, volume, temperature), it follows a specific process. The First Law applies to all of them, but simplifies uniquely for each:

Isothermal Process (Constant Temperature): $\Delta T = 0$ , so $\Delta U = 0$ (for an ideal gas). Therefore, $Q = W$ . Heat added is entirely converted to work done by the gas.
Isobaric Process (Constant Pressure): $W = P \Delta V$ . Heat added changes both internal energy and does work.
Isochoric (Isovolumetric) Process (Constant Volume): $\Delta V = 0$ , so $W = 0$ . Therefore, $\Delta U = Q$ . All heat added goes into increasing the internal energy.
Adiabatic Process (No Heat Transfer): $Q = 0$ . Therefore, $\Delta U = -W$ . If the gas expands and does work, its internal energy (and thus temperature) must decrease.

$Interactive Simulation$

Select Carnot, Otto, or Brayton. Every state label, P–V coordinate, heat/work value, animation point, and efficiency comes from one canonical ideal-gas state model for the selected cycle.

4.0

1200 K

300 K

1.40

\eta=1-T_C/T_H

1→2 isothermal expansion
Q=13,831.6 J; W=13,831.6 J

Heat input

13,831.6 J

Heat rejected

3,457.9 J

Net work

10,373.7 J

Efficiency

75.0 %

Active P

997.7 kPa

Active V

10.00 L

Active T

1,200.0 K

The Second Law: Entropy and Direction

$The Second Law: Entropy and Direction Concepts$

The First Law says energy is conserved, but it doesn't restrict the direction of energy transfer. The Second Law dictates that direction. There are several equivalent statements of the Second Law:

Clausius Statement: Heat can never pass spontaneously from a colder body to a warmer body.
Kelvin-Planck Statement: It is impossible to construct a heat engine that, operating in a cycle, extracts heat from a single reservoir and converts it entirely into work. Some heat must be expelled to a colder sink. (No engine is 100% efficient).
Entropy Statement: The total entropy ( $S$ ) of an isolated system can never decrease over time. $\Delta S_{total} \ge 0$ .

$S$

A thermodynamic state function related statistically to the number of microscopic configurations compatible with a macroscopic state. For a reversible transfer, $dS=\delta Q_{rev}/T$ ; for an isolated system, the Second Law requires total entropy not to decrease.

Heat Engines and Efficiency

$Heat Engines and Efficiency Concepts$

A heat engine is a device that extracts heat ( $Q_H$ ) from a hot reservoir, uses some of it to do useful work ( $W$ ), and exhausts the remaining waste heat ( $Q_C$ ) to a cold reservoir.

By the First Law, $W = Q_H - |Q_C|$ .

The thermal efficiency ( $e$ ) is the ratio of what you get to what you pay for:

Thermal Efficiency of a Heat Engine

Calculates the efficiency of a heat engine based on work output and heat input.

e = \frac{W}{Q_H} = 1 - \frac{|Q_C|}{Q_H}

Variables

Symbol	Description	Unit
$e$	Thermal efficiency	dimensionless
$W$	Useful work done by the engine	J
$Q_H$	Heat extracted from the hot reservoir	J
$Q_C$	Heat exhausted to the cold reservoir	J

$Heat Engines and Efficiency Concepts$

The Carnot Engine is an idealized, reversible engine that sets the maximum possible theoretical efficiency between two temperatures:

Carnot Efficiency

Sets the maximum theoretical efficiency for any heat engine operating between two temperatures.

e_{Carnot} = 1 - \frac{T_C}{T_H}

Variables

Symbol	Description	Unit
$e_{Carnot}$	Maximum theoretical efficiency	dimensionless
$T_C$	Absolute temperature of the cold reservoir	K
$T_H$	Absolute temperature of the hot reservoir	K

The Third Law: Absolute Zero

$The Third Law: Absolute Zero Concepts$

For a perfect crystal with a unique ground state, entropy approaches zero as temperature approaches absolute zero. A related unattainability statement is that absolute zero cannot be reached by a finite sequence of thermodynamic operations.

The Ideal Gas Law and Kinetic Theory

$Macroscopic vs. Microscopic$

The Ideal Gas Law ( $PV = nRT$ ) relates the macroscopic properties of a gas: pressure, volume, temperature, and amount of substance.

The Kinetic Theory of Gases provides the microscopic explanation for these macroscopic properties. It models a gas as a large number of tiny, rapidly moving particles in random, continuous motion, colliding elastically with each other and the container walls. The pressure exerted by a gas is the macroscopic result of billions of microscopic particle collisions against the container.

The Ideal Gas Law

The equation of state for a hypothetical ideal gas.

PV = nRT = N k_B T

Variables

Symbol	Description	Unit
$P$	Absolute pressure	Pa
$V$	Volume	$m^3$
$n$	Number of moles	mol
$R$	Universal gas constant	$8.314 J/(mol \cdot K)$
$T$	Absolute temperature	K
$N$	Number of molecules	dimensionless count
$k_B$	Boltzmann constant	J/K

$Interactive Simulation$

Change amount, absolute temperature, volume, and gas species. Pressure follows $PV=nRT$ exactly. Particle motion is a deterministic illustration whose speed scale follows the ideal-gas rms trend; it is not a molecular-dynamics solver.

1.0 mol

300 K

0.025 m³

PV=nRT,\qquad v_{\mathrm{rms}}=\sqrt{3RT/M}

Pressure

99.77 kPa

Density

1.121 kg/m³

v rms

517 m/s

Gas

Nitrogen

Key Takeaways

Temperature ( $T$ ) is a thermodynamic state variable; in the ideal-gas kinetic model it is related to average molecular kinetic energy. Heat ( $Q$ ) is energy transferred because of a temperature difference.
Heat transfer mechanisms are Conduction (contact), Convection (fluid motion), and Radiation (electromagnetic waves).
Specific heat ( $c$ ) relates $Q$ to $\Delta T$ ; Latent heat ( $L$ ) relates $Q$ to phase changes without $\Delta T$ .
Zeroth Law: Defines temperature via thermal equilibrium.
First Law: Conservation of energy ( $\Delta U = Q - W$ ).
Second Law: Dictates the direction of processes (heat flows hot to cold) and introduces Entropy; the total entropy of an isolated system cannot decrease. A cyclic heat engine cannot convert all absorbed heat into work.

$References$

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Learning Objectives

Temperature and Heat

Temperature and Heat Concepts

Temperature (TTT)

Kelvin–Celsius Conversion

Heat (QQQ)

Heat vs. Internal Energy

Thermal Expansion and Heat Capacity

Thermal Expansion and Heat Capacity Concepts

Linear Thermal Expansion

Volume Thermal Expansion

Specific Heat Capacity (ccc)

Specific Heat Capacity (ccc) Concepts

Specific Heat Equation

Specific Heat Capacity

Latent Heat (LLL)

Latent Heat (LLL) Concepts

Latent Heat

Mechanisms of Heat Transfer

Mechanisms of Heat Transfer Concepts

Fourier's Law of Heat Conduction

Stefan-Boltzmann Law of Radiation

Net Radiation Exchange

Civil Engineering Applications of Heat Transfer

The Laws of Thermodynamics

The Laws of Thermodynamics Concepts

The Zeroth Law: Thermal Equilibrium

The Zeroth Law: Thermal Equilibrium Concepts

The First Law: Conservation of Energy

The First Law: Conservation of Energy Concepts

The First Law of Thermodynamics

Sign Conventions for the First Law

Work Done by a Gas

Work Done on a P-V Diagram

Thermodynamic Processes

Thermodynamic Processes Concepts

Interactive Simulation

The Second Law: Entropy and Direction

The Second Law: Entropy and Direction Concepts

Entropy (SSS)

Heat Engines and Efficiency

Heat Engines and Efficiency Concepts

Thermal Efficiency of a Heat Engine

Heat Engines and Efficiency Concepts

Carnot Efficiency

The Third Law: Absolute Zero

The Third Law: Absolute Zero Concepts

The Ideal Gas Law and Kinetic Theory

Macroscopic vs. Microscopic

The Ideal Gas Law

Interactive Simulation

References

$Learning Objectives$

$Temperature and Heat Concepts$

$T$

$Q$

$Heat vs. Internal Energy$

$Thermal Expansion and Heat Capacity Concepts$

Specific Heat Capacity ( $c$ )

$c$

$Specific Heat Capacity$

Latent Heat ( $L$ )

$L$

$Mechanisms of Heat Transfer Concepts$

$Net Radiation Exchange$

$Civil Engineering Applications of Heat Transfer$

$The Laws of Thermodynamics Concepts$

$The Zeroth Law: Thermal Equilibrium Concepts$

$The First Law: Conservation of Energy Concepts$

$Sign Conventions for the First Law$

$Work Done on a P-V Diagram$

$Thermodynamic Processes Concepts$

$Interactive Simulation$

$The Second Law: Entropy and Direction Concepts$

$S$

$Heat Engines and Efficiency Concepts$

$Heat Engines and Efficiency Concepts$

$The Third Law: Absolute Zero Concepts$

$Macroscopic vs. Microscopic$

$Interactive Simulation$

$References$