Comprehensive Guide to Important Thermodynamic Formulas

Comprehensive Guide to Important Thermodynamic Formulas

Thermodynamics is the study of heat, energy, and their transformations. This guide covers essential formulas in thermodynamics that govern energy transfer, entropy, and efficiency in various systems.

1. First Law of Thermodynamics (Energy Conservation)

$\Delta U = Q - W$ where:

$\Delta U$ = Change in internal energy (J)
$Q$ = Heat added to the system (J)
$W$ = Work done by the system (J)

This principle states that energy cannot be created or destroyed, only transferred or converted.

2. Second Law of Thermodynamics (Entropy Increase)

$\Delta S = \frac{Q_{rev}}{T}$ where:

$\Delta S$ = Change in entropy (J/K)
$Q_{rev}$ = Reversible heat transfer (J)
$T$ = Temperature (K)

This law implies that all natural processes tend to move toward a state of increased entropy or disorder.

3. Carnot Efficiency (Maximum Efficiency of Heat Engines)

$\eta = 1 - \frac{T_C}{T_H}$ where:

$\eta$ = Efficiency
$T_C$ = Temperature of the cold reservoir (K)
$T_H$ = Temperature of the hot reservoir (K)

This equation determines the theoretical efficiency of an ideal heat engine operating in a reversible Carnot cycle.

4. Ideal Gas Law

$PV = nRT$ where:

$P$ = Pressure (Pa)
$V$ = Volume (m³)
$n$ = Number of moles (mol)
$R$ = Universal gas constant (8.314 J/mol·K)
$T$ = Temperature (K)

This fundamental equation relates the pressure, volume, and temperature of an ideal gas.

5. Gibbs Free Energy (Spontaneity of a Process)

$\Delta G = \Delta H - T\Delta S$ where:

$\Delta G$ = Gibbs free energy change (J)
$\Delta H$ = Enthalpy change (J)
$T$ = Temperature (K)
$\Delta S$ = Entropy change (J/K)

A negative $\Delta G$ indicates a spontaneous reaction.

6. Enthalpy Change

$H = U + PV$ where:

$H$ = Enthalpy (J)
$U$ = Internal energy (J)
$P$ = Pressure (Pa)
$V$ = Volume (m³)

This formula defines the total heat content of a system.

7. Work Done in a Polytropic Process

$W = \frac{P_2V_2 - P_1V_1}{1 - n}$ where:

$W$ = Work done (J)
$P_1, P_2$ = Initial and final pressures (Pa)
$V_1, V_2$ = Initial and final volumes (m³)
$n$ = Polytropic index

This equation calculates work done in a process where pressure and volume change under a constant heat capacity ratio.

8. Helmholtz Free Energy

$F = U - TS$ where:

$F$ = Helmholtz free energy (J)
$U$ = Internal energy (J)
$T$ = Temperature (K)
$S$ = Entropy (J/K)

Helmholtz free energy is useful in describing closed systems where volume remains constant.

9. Rankine Cycle Efficiency

$\eta = 1 - \frac{h_4 - h_1}{h_3 - h_2}$ where:

$h_1, h_2, h_3, h_4$ = Specific enthalpies at different stages (J/kg)

The Rankine cycle describes the operation of steam power plants, where heat energy is converted into work.

10. Black Body Radiation (Stefan-Boltzmann Law)

$E = \sigma T^4$ where:

$E$ = Radiant energy emitted per unit area (W/m²)
$\sigma$ = Stefan-Boltzmann constant (5.67 × 10⁻⁸ W/m²K⁴)
$T$ = Temperature (K)

This equation governs the energy radiated by an ideal black body.

Table of Key Thermodynamic Formulas

Formula Name	Equation
First Law of Thermodynamics	$\Delta U = Q - W$
Second Law of Thermodynamics	$\Delta S = \frac{Q_{rev}}{T}$
Carnot Efficiency	$\eta = 1 - \frac{T_C}{T_H}$
Ideal Gas Law	$PV = nRT$
Gibbs Free Energy	$\Delta G = \Delta H - T\Delta S$
Enthalpy	$H = U + PV$
Work in a Polytropic Process	$W = \frac{P_2V_2 - P_1V_1}{1 - n}$
Helmholtz Free Energy	$F = U - TS$
Rankine Cycle Efficiency	$\eta = 1 - \frac{h_4 - h_1}{h_3 - h_2}$
Stefan-Boltzmann Law	$E = \sigma T^4$

Conclusion

Understanding these thermodynamic formulas is crucial for engineers, scientists, and students working with energy systems. These principles form the foundation for studying power plants, refrigeration cycles, and chemical reactions.

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An infographic summarizing key thermodynamic formulas, with labeled variables and real-world applications such as heat engines, refrigeration cycles, and gas laws.