Thermodynamics Formulas

Complete collection of thermodynamics formulas with detailed explanations, notation meanings, units, and real-world applications. Master heat, work, and energy.

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Gas Laws

Ideal Gas Law

PV = nRT

Pressure times volume equals number of moles times gas constant times temperature.

Notation:

P

:Pressure

V

:Volume

n

:Number of moles

R

:Universal gas constant

T

:Temperature (Kelvin)

Units:

P

:Pascal (Pa) or atm

V

:m³ or L

n

:mol

R

:8.314 J/(mol·K)

T

:Kelvin (K)

Applications:

•Gas behavior prediction
•Chemical reactions
•Industrial processes

Limitations:

Ideal gases only, low pressure

Boyle's Law

P_1V_1 = P_2V_2

At constant temperature, pressure and volume are inversely proportional.

Notation:

P_1, P_2

:Initial and final pressure

V_1, V_2

:Initial and final volume

Units:

P_1, P_2

:Pa or atm

V_1, V_2

:m³ or L

Applications:

•Scuba diving
•Pneumatic systems
•Gas storage

Limitations:

Constant temperature, ideal gas

Charles's Law

\frac{V_1}{T_1} = \frac{V_2}{T_2}

At constant pressure, volume and temperature are directly proportional.

Notation:

V_1, V_2

:Initial and final volume

T_1, T_2

:Initial and final temperature

Units:

V_1, V_2

:m³ or L

T_1, T_2

Applications:

•Hot air balloons
•Thermal expansion
•Temperature measurement

Limitations:

Constant pressure, ideal gas

⚖️

Laws of Thermodynamics

First Law of Thermodynamics

\Delta U = Q - W

Change in internal energy equals heat added minus work done by the system.

Notation:

\Delta U

:Change in internal energy

Q

:Heat added to system

W

:Work done by system

Units:

\Delta U

:Joule (J)

Q

W

Applications:

•Heat engines
•Refrigeration
•Energy conservation

Limitations:

Closed system

Second Law of Thermodynamics

\Delta S \geq \frac{Q}{T}

Change in entropy is greater than or equal to heat divided by temperature.

Notation:

\Delta S

:Change in entropy

Q

:Heat transfer

T

:Temperature

Units:

\Delta S

:J/K

Q

T

Applications:

•Heat engine efficiency
•Spontaneous processes
•Entropy calculations

Limitations:

Reversible processes (equality)

Third Law of Thermodynamics

S \rightarrow 0 \text{ as } T \rightarrow 0

Entropy approaches zero as temperature approaches absolute zero.

Notation:

S

:Entropy

T

:Temperature

Units:

S

:J/K

T

Applications:

•Absolute zero studies
•Perfect crystal entropy
•Quantum systems

Limitations:

Perfect crystals only

🌡️

Heat Transfer

Conduction

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

Heat transferred by conduction equals thermal conductivity times area times temperature difference divided by length times time.

Notation:

Q

:Heat transferred

k

:Thermal conductivity

A

:Cross-sectional area

\Delta T

:Temperature difference

L

:Length

t

:Time

Units:

Q

k

:W/(m·K)

A

:m²

\Delta T

L

t

Applications:

•Building insulation
•Heat exchangers
•Thermal management

Limitations:

Steady state, uniform material

Convection

Q = hA(\Delta T)t

Heat transferred by convection equals convective heat transfer coefficient times area times temperature difference times time.

Notation:

Q

:Heat transferred

h

:Convective heat transfer coefficient

A

:Surface area

\Delta T

:Temperature difference

t

:Time

Units:

Q

h

:W/(m²·K)

A

:m²

\Delta T

t

Applications:

•Cooling systems
•Heating systems
•Natural convection

Limitations:

Constant coefficient

Radiation

Q = \sigma\varepsilon AT^4t

Heat transferred by radiation equals Stefan-Boltzmann constant times emissivity times area times temperature to fourth power times time.

Notation:

Q

:Heat transferred

\sigma

:Stefan-Boltzmann constant

\varepsilon

:Emissivity

A

:Surface area

T

:Temperature

t

:Time

Units:

Q

\sigma

:5.67 × 10⁻⁸ W/(m²·K⁴)

\varepsilon

:dimensionless

A

:m²

T

t

Applications:

•Solar radiation
•Infrared heating
•Thermal imaging

Limitations:

Black body approximation

🌀

Entropy and Disorder

Entropy Change

\Delta S = \frac{Q_{\text{rev}}}{T}

Change in entropy equals reversible heat transfer divided by temperature.

Notation:

\Delta S

:Change in entropy

Q_{\text{rev}}

:Reversible heat transfer

T

:Temperature

Units:

\Delta S

:J/K

Q_{\text{rev}}

T

Applications:

•Phase transitions
•Chemical reactions
•Thermodynamic cycles

Limitations:

Reversible processes only

Statistical Entropy

S = k_B \ln W

Entropy equals Boltzmann constant times natural logarithm of number of microstates.

Notation:

S

:Entropy

k_B

:Boltzmann constant

W

:Number of microstates

Units:

S

:J/K

k_B

:1.381 × 10⁻²³ J/K

W

:dimensionless

Applications:

•Statistical mechanics
•Quantum systems
•Information theory

Limitations:

Equilibrium systems

Entropy of Mixing

\Delta S_{\text{mix}} = -R(n_1\ln x_1 + n_2\ln x_2)

Entropy change of mixing equals negative gas constant times sum of moles times natural log of mole fractions.

Notation:

\Delta S_{\text{mix}}

:Entropy change of mixing

R

:Gas constant

n_1, n_2

:Number of moles

x_1, x_2

:Mole fractions

Units:

\Delta S_{\text{mix}}

:J/K

R

:8.314 J/(mol·K)

n_1, n_2

:mol

x_1, x_2

:dimensionless

Applications:

•Solution thermodynamics
•Chemical mixing
•Phase separation

Limitations:

Ideal solutions

⚙️

Heat Engines and Cycles

Carnot Efficiency

\eta = 1 - \frac{T_c}{T_h}

Maximum efficiency equals one minus cold temperature divided by hot temperature.

Notation:

\eta

:Efficiency

T_c

:Cold reservoir temperature

T_h

:Hot reservoir temperature

Units:

\eta

:dimensionless

T_c, T_h

Applications:

•Power plants
•Refrigeration
•Heat pumps

Limitations:

Reversible Carnot cycle

Work Done by Heat Engine

W = Q_h - Q_c

Work done equals heat absorbed from hot reservoir minus heat rejected to cold reservoir.

Notation:

W

:Work done

Q_h

:Heat absorbed from hot reservoir

Q_c

:Heat rejected to cold reservoir

Units:

W

Q_h, Q_c

Applications:

•Steam engines
•Internal combustion
•Thermal power

Limitations:

Cyclic processes

Coefficient of Performance (Heat Pump)

\text{COP} = \frac{Q_h}{W}

Coefficient of performance equals heat delivered divided by work input.

Notation:

\text{COP}

:Coefficient of performance

Q_h

:Heat delivered

W

:Work input

Units:

\text{COP}

:dimensionless

Q_h, W

Applications:

•Heat pumps
•Air conditioning
•Refrigeration

Limitations:

Reversible processes

Practice Problems

📝Calculate pressure of 2 mol gas at 300K in 5L container
📝Find efficiency of Carnot engine between 500K and 300K
📝Calculate heat transfer through 2m² wall with 10K difference

Try Interactive Calculators

Ideal Gas Law First Law Boyle's Law Charles' Law Carnot Cycle Heat Transfer Newton Cooling

Study Tips

💡Always use Kelvin for temperature
💡Pay attention to sign conventions
💡Remember entropy always increases