The formula reference

Every entry's LaTeX is generated from the SymPy expression it was verified against — no transcription typos. Units are dimensionally checked; declared derivations are machine-proven. Also browsable as a concept graph.

Mechanics

Centripetal acceleration

regime 1

a_c = \frac{v^{2}}{r}

v	m/s	speed
r	m	radius of the circular path

Valid when: uniform circular motion

Centripetal force

regime 1

F_c = \frac{m v^{2}}{r}

m	kg	mass
v	m/s	speed
r	m	radius of the circular path

Valid when: uniform circular motion

Damped angular frequency

regime 2

\omega_d = \sqrt{\frac{k}{m} - \frac{b^{2}}{4 m^{2}}}

k	N/m	spring constant
m	kg	mass
b	kg/s	linear damping coefficient

Valid when: underdamped (b < 2√(km)); linear damping

Damping ratio

regime 2

\zeta = \frac{b}{2 \sqrt{k m}}

b	kg/s	linear damping coefficient
k	N/m	spring constant
m	kg	mass

Valid when: linear damping

Terminal velocity (linear drag)

regime 2

v_{\text{term}} = \frac{m g}{b}

m	kg	mass
g	m/s²	gravitational acceleration (negative, up positive)
b	kg/s	linear drag coefficient

Valid when: drag linear in speed; constant gravity

Velocity under linear drag

regime 2

v = \frac{m g}{b} + \left(- \frac{m g}{b} + v_{0}\right) e^{- \frac{b t}{m}}

m	kg	mass
g	m/s²	gravitational acceleration (negative, up positive)
b	kg/s	linear drag coefficient
$v_{0}$	m/s	initial velocity
t	s	time

Valid when: drag linear in speed; constant gravity

Kinetic friction

regime 1

f_k = \mu_{k} F_{N}

$\mu_{k}$	1	coefficient of kinetic friction (dimensionless)
$F_{N}$	N	normal force

Valid when: surfaces sliding; Coulomb friction model

Hooke's law (spring force)

regime 1

F = - k x

k	N/m	spring constant
x	m	displacement from equilibrium

Valid when: ideal spring; within the elastic limit

Newton's second law

regime 1

F_{\text{net}} = m a

m	kg	mass
a	m/s²	acceleration

Valid when: F is the net force; m constant

Weight (gravitational force near Earth)

regime 1

F_g = m g

m	kg	mass
g	m/s²	gravitational acceleration (signed; -10 in the house convention)

Valid when: uniform gravitational field; g is the signed acceleration (up positive)

Kinetic energy

regime 1

K = \frac{m v^{2}}{2}

m	kg	mass
v	m/s	speed

Valid when: speeds well below light speed

Mechanical energy (kinetic + gravitational potential)

regime 1

E = \frac{m v^{2}}{2} + m g h

m	kg	mass
v	m/s	speed
g	m/s²	gravitational field strength
h	m	height above a reference level

Valid when: conserved when only conservative forces (gravity) do work; no friction or other dissipation

Mechanical power (constant force)

regime 1

P = F v

F	N	force
v	m/s	speed

Valid when: force parallel to velocity

Elastic potential energy

regime 1

U_s = \frac{k x^{2}}{2}

k	N/m	spring constant
x	m	displacement from equilibrium

Valid when: ideal spring (Hooke's law)

Work (constant force along displacement)

regime 1

W = F d

F	N	force component along the motion
d	m	displacement

Valid when: constant force parallel to displacement

Bernoulli's equation (total pressure)

regime 1

P_0 = P + \frac{\rho v^{2}}{2} + \rho g h

P	Pa	static pressure
$\rho$	kg/m³	fluid density
v	m/s	flow speed
g	m/s²	gravitational field strength
h	m	height

Valid when: steady, incompressible, non-viscous flow along a streamline; energy conservation per unit volume; P₀ is constant along the streamline

Buoyant force (Archimedes' principle)

regime 1

F_b = \rho g V_{\text{disp}}

$\rho$	kg/m³	fluid density
g	m/s²	gravitational field strength
$V_{\text{disp}}$	m³	volume of fluid displaced

Valid when: body fully or partly submerged in a fluid of density ρ; buoyant force equals the weight of displaced fluid

Continuity (volume flow rate)

regime 1

Q = A v

A	m²	cross-sectional area
v	m/s	flow speed

Valid when: incompressible fluid; Q = Av is conserved along a streamtube, so a narrower pipe means faster flow

Hydrostatic force on a vertical wall

regime 2

F = \frac{\rho g w H^{2}}{2}

$\rho$	kg/m³	fluid density
g	m/s²	gravitational field strength
w	m	wall width
H	m	water depth

Valid when: incompressible fluid of constant density; flat vertical wall of constant width, top edge at the surface; gauge pressure (atmospheric cancels across the wall); the area under the pressure–depth profile, ∫P w dh

Hydrostatic pressure with depth

regime 1

P = \rho g h

$\rho$	kg/m³	fluid density
g	m/s²	gravitational field strength
h	m	depth below the surface

Valid when: incompressible fluid of constant density; gauge pressure (above the surface); add atmospheric for absolute

Newton's law of universal gravitation

regime 1

F = \frac{G m_{1} m_{2}}{r^{2}}

G	N·m²/kg²	gravitational constant
$m_{1}$	kg	first mass
$m_{2}$	kg	second mass
r	m	separation between centers

Valid when: point masses (or spherical bodies); magnitude of the attractive force

Constant acceleration

regime 1

a = a

The seed of the integral ladder: hold $a$ constant and the velocity $v=\int a\,dt$ and position $x=\int v\,dt$ accumulate from it — the constant-acceleration formulas are those integrals, evaluated.

m/s²

the (constant) acceleration

Valid when: acceleration is constant

Velocity under constant acceleration

regime 1

v = v_{0} + a t

$v_{0}$	m/s	initial velocity
a	m/s²	acceleration (constant)
t	s	time

Valid when: acceleration is constant

integral-of Constant acceleration (d/dt), proven by SymPy

Position under constant acceleration

regime 1

x = x_{0} + v_{0} t + \frac{t^{2} a}{2}

$x_{0}$	m	initial position
$v_{0}$	m/s	initial velocity
a	m/s²	acceleration (constant)
t	s	time

Valid when: acceleration is constant

integral-of Velocity under constant acceleration (d/dt), proven by SymPy

Impulse (constant force)

regime 1

J = F t

F	N	force
t	s	time interval

Valid when: constant force over the interval

Perfectly inelastic collision (common velocity)

regime 1

v_f = \frac{m_{1} v_{1} + m_{2} v_{2}}{m_{1} + m_{2}}

$m_{1}$	kg	mass of body 1
$m_{2}$	kg	mass of body 2
$v_{1}$	m/s	initial velocity of body 1
$v_{2}$	m/s	initial velocity of body 2

Valid when: isolated 1D collision (momentum conserved); bodies move off together (e = 0)

Linear momentum

regime 1

p = m v

m	kg	mass
v	m/s	velocity

Valid when: speeds well below light speed

Kepler's third law (circular orbit)

regime 1

T = 2 \pi \sqrt{\frac{R^{3}}{G M}}

G	N·m²/kg²	gravitational constant
M	kg	mass of the central body
R	m	orbital radius

Valid when: circular orbit; the orbiting mass cancels; T² = 4π²R³/GM — the period squared scales as the radius cubed

Circular orbital speed

regime 1

v = \sqrt{\frac{G M}{R}}

G	N·m²/kg²	gravitational constant
M	kg	mass of the central body
R	m	orbital radius

Valid when: circular orbit; central body of mass M; the orbiting mass cancels; gravity supplies the centripetal force, GM/R² = v²/R

Angular momentum (rigid body)

regime 1

L = I \omega

I	kg·m²	moment of inertia
$\omega$	1/s	angular velocity

Valid when: rotation about a fixed axis

Rotational kinetic energy

regime 1

K_{\text{rot}} = \frac{I \omega^{2}}{2}

I	kg·m²	moment of inertia
$\omega$	1/s	angular velocity

Valid when: rotation about a fixed axis

Moment of inertia of a disk

regime 2

I = \frac{M R^{2}}{2}

M	kg	disk mass
R	m	disk radius

Valid when: uniform solid disk (or cylinder), axis through the centre; from I = ∫r² dm with mass spread over area

Moment of inertia of a rod (about one end)

regime 2

I = \frac{M L^{2}}{3}

M	kg	rod mass
L	m	rod length

Valid when: uniform thin rod, axis through one end; derived from I = ∫r² dm; ⅓ML² about the centre is ML²/12

Parallel-axis theorem

regime 1

I = I_{\text{cm}} + M d^{2}

$I_{\text{cm}}$	kg·m²	moment of inertia about the centre of mass
M	kg	total mass
d	m	distance between the two parallel axes

Valid when: axis parallel to one through the centre of mass, offset by d; Icm is the moment about the centre-of-mass axis

Torque (force at a lever arm)

regime 1

\tau = r F

r	m	lever arm
F	N	force

Valid when: force perpendicular to the lever arm

Rotational work (constant torque through an angle)

regime 1

W = \tau \theta

$\tau$	N·m	torque
$\theta$	rad	angle turned

Valid when: constant torque about a fixed axis; the area under the torque–angle curve, ∫τ dθ

Angular frequency of a spring–mass oscillator

regime 2

\omega = \sqrt{\frac{k}{m}}

k	N/m	spring constant
m	kg	mass

Valid when: ideal Hooke's-law spring; no damping

Period of a spring–mass oscillator

regime 2

T = \frac{2 \pi}{\omega}

\omega

1/s

angular frequency

Valid when: ideal Hooke's-law spring; no damping

Position in simple harmonic motion

regime 2

x = x_{0} \cos{\left(\omega t \right)} + \frac{\sin{\left(\omega t \right)} v_{0}}{\omega}

$x_{0}$	m	initial displacement
$v_{0}$	m/s	initial velocity
$\omega$	1/s	angular frequency
t	s	time

Valid when: ideal Hooke's-law spring; no damping

Electricity & magnetism

Peak EMF of an AC generator

regime 2

\mathcal{E}_0 = B A \omega

B	T	magnetic flux density
A	m²	coil area
$\omega$	1/s	angular rotation rate ω

Valid when: single-turn coil rotating at constant ω in a uniform field B; peak of ℰ(t) = BAω sin(ωt)

Magnetic field of a long straight wire

regime 1

B = \frac{\mu_{0} I}{2 \pi r}

$\mu_{0}$	T·m/A	permeability of free space μ₀
I	A	current
r	m	distance from the wire

Valid when: infinitely long, thin straight wire; field circles the wire; magnitude falls as 1/r

Capacitance

regime 1

C = \frac{Q}{V}

Q	C	charge stored on one plate
V	V	voltage across the plates

Valid when: linear capacitor: charge proportional to voltage; C set by geometry, not by Q or V

Energy stored in a capacitor

regime 2

U = \frac{Q^{2}}{2 C}

Q	C	charge stored
C	F	capacitance

Valid when: ideal capacitor; equals ½CV² = ½QV via Q = CV; the area under the voltage–charge line (a triangle)

Coulomb's law

regime 1

F = \frac{k q_{1} q_{2}}{r^{2}}

k	N·m²/C²	Coulomb constant 1/(4πε₀)
$q_{1}$	C	first charge
$q_{2}$	C	second charge
r	m	separation between charges

Valid when: point charges (or spherical charge distributions); magnitude of the force; like charges repel, unlike attract

Cyclotron frequency

regime 1

f = \frac{q B}{2 \pi m}

q	C	charge
B	T	magnetic field strength
m	kg	mass of the charged particle

Valid when: a charge circling in a uniform magnetic field; remarkably, the frequency does not depend on the speed or the radius

Cyclotron radius

regime 1

r = \frac{m v}{q B}

m	kg	mass of the charged particle
v	m/s	speed perpendicular to the field
q	C	charge
B	T	magnetic field strength

Valid when: a charge moving perpendicular to a uniform magnetic field; the magnetic force qvB supplies the centripetal force mv²/r

Displacement current

regime 2

I_d = \varepsilon_0 \frac{d\Phi_E}{dt}

$\varepsilon_0$	F/m	permittivity of free space ε₀
$\frac{d\Phi_E}{dt}$	V·m/s	rate of change of electric flux dΦ_E/dt

Valid when: Maxwell's correction to Ampère's law; a changing electric flux acts like a current and sources a magnetic field — e.g. between capacitor plates

Electric field of a point charge

regime 1

E = \frac{k q}{r^{2}}

k	N·m²/C²	Coulomb constant 1/(4πε₀)
q	C	source charge
r	m	distance from the charge

Valid when: point charge (or spherical charge); magnitude of the radial field; points away from a positive charge

Faraday's law of induction

regime 2

\mathcal{E} = - \frac{d\Phi}{dt}

\frac{d\Phi}{dt}

Wb/s

rate of change of magnetic flux dΦ/dt

Valid when: single-turn loop (multiply by N turns); the minus sign is Lenz's law: the EMF opposes the change in flux

Magnetic flux

regime 1

\Phi = B A

B	T	magnetic flux density
A	m²	area threaded by the field

Valid when: uniform field perpendicular to a flat area A; general case Φ = ∫B·dA

Gauss's law (electric flux)

regime 1

\Phi_E = \frac{Q}{\varepsilon_0}

Q	C	charge enclosed by the surface
$\varepsilon_0$	F/m	permittivity of free space ε₀

Valid when: the total electric flux through any closed surface; depends only on the enclosed charge, not on its arrangement or the surface shape

Inductor EMF

regime 2

\mathcal{E}_L = L \frac{dI}{dt}

L	H	inductance
$\frac{dI}{dt}$	A/s	rate of change of current dI/dt

Valid when: ideal inductor; the self-induced EMF opposes the change in current (Lenz's law)

Energy stored in an inductor

regime 1

U = \frac{L I^{2}}{2}

L	H	inductance
I	A	current

Valid when: ideal inductor of inductance L carrying current I; energy is held in the magnetic field and returned when the current falls

LC angular frequency

regime 2

\omega = \frac{1}{\sqrt{L C}}

L	H	inductance
C	F	capacitance

Valid when: ideal LC circuit, no resistance; free oscillation L Q'' + Q/C = 0

LC period

regime 2

T = 2 \pi \sqrt{L C}

L	H	inductance
C	F	capacitance

Valid when: ideal LC circuit, no resistance; free oscillation at ω = 1/√(LC)

Magnetic field at the centre of a current loop

regime 1

B = \frac{\mu_0 I}{2 R}

$\mu_0$	T·m/A	permeability of free space μ₀
I	A	current
R	m	loop radius

Valid when: a single circular loop, field evaluated at its centre; for N turns, multiply by N

Torque on a current loop

regime 1

\tau = N I A B \sin{\left(\theta \right)}

N	1	number of turns
I	A	current
A	m²	area enclosed by the loop
B	T	magnetic field strength
$\theta$	1	angle between the field and the loop normal

Valid when: a flat coil of N turns and area A in a uniform field; θ is the angle between the field and the loop's normal; the torque is the motor principle

Magnetic force on a charge

regime 1

F = q v B

q	C	charge
v	m/s	speed
B	T	magnetic flux density

Valid when: magnitude for velocity perpendicular to the field; force is perpendicular to both v and B (F = qv×B)

Magnetic energy density

regime 1

u = \frac{B^{2}}{2 \mu_0}

B	T	magnetic field strength
$\mu_0$	T·m/A	permeability of free space μ₀

Valid when: energy per unit volume stored in a magnetic field; integrating it over the field volume recovers the total stored energy

Motional EMF

regime 2

\mathcal{E} = B L v

B	T	magnetic flux density
L	m	length of the moving conductor
v	m/s	speed across the field

Valid when: straight conductor of length L moving at speed v across a field B; a special case of Faraday's law, EMF = −dΦ/dt, with Φ = BLx

Mutual inductance EMF

regime 2

\mathcal{E}_2 = - M \frac{dI_1}{dt}

M	H	mutual inductance
$\frac{dI_1}{dt}$	A/s	rate of change of current in coil 1 dI₁/dt

Valid when: two magnetically coupled coils; a changing current in coil 1 induces an EMF in coil 2 (the transformer principle)

Ohm's law

regime 1

V = I R

I	A	current
R	ohm	resistance

Valid when: ohmic conductor: resistance R independent of current; steady current

Force per length between parallel wires

regime 1

\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2 \pi d}

$\mu_0$	T·m/A	permeability of free space μ₀
$I_1$	A	current in wire 1
$I_2$	A	current in wire 2
d	m	separation between the wires

Valid when: two long parallel wires a distance d apart; attracts if the currents are parallel, repels if anti-parallel

Electric potential energy of two charges

regime 2

U = \frac{k q_{1} q_{2}}{r}

k	N·m²/C²	Coulomb constant 1/(4πε₀)
$q_{1}$	C	first charge
$q_{2}$	C	second charge
r	m	separation between charges

Valid when: point charges; energy taken zero at infinite separation; the work to assemble the charges, ∫F dr

Electric potential of a point charge

regime 2

V = \frac{k q}{r}

k	N·m²/C²	Coulomb constant 1/(4πε₀)
q	C	source charge
r	m	distance from the charge

Valid when: point charge; potential taken zero at infinity; the area under the field to infinity converges (1/r²)

Electrical power

regime 1

P = I V

I	A	current through the element
V	V	voltage across the element

Valid when: power delivered to a circuit element; equals I²R = V²/R for a resistor (Ohm's law)

Charge on a charging capacitor (RC circuit)

regime 2

Q = C \left(1 - e^{- \frac{t}{C R}}\right) V

C	F	capacitance
V	V	battery EMF
R	ohm	series resistance
t	s	time since the switch closed

Valid when: series RC circuit charged from a constant EMF; capacitor initially uncharged

RC time constant

regime 2

\tau = R C

R	ohm	series resistance
C	F	capacitance

Valid when: series resistor and capacitor; the charge approaches its final value as 1 − e^{−t/τ}

RL current growth

regime 2

I = \frac{V \left(1 - e^{- \frac{R t}{L}}\right)}{R}

V	V	battery EMF
R	ohm	series resistance
t	s	elapsed time
L	H	inductance

Valid when: a series RL circuit switched onto a battery of EMF V at t = 0; the inductor opposes the change, so the current rises gradually to V/R

RL time constant

regime 2

\tau = \frac{L}{R}

L	H	inductance
R	ohm	series resistance

Valid when: a resistor and inductor in series; after one τ the current has risen to 1−1/e ≈ 63% of its final value

Magnetic field inside a solenoid

regime 1

B = \mu_0 n I

$\mu_0$	T·m/A	permeability of free space μ₀
n	1/m	turns per unit length
I	A	current

Valid when: long, tightly-wound solenoid (length much greater than radius); field is uniform inside and ~zero outside

Inductance of a solenoid

regime 1

L = \mu_0 n^{2} A l

$\mu_0$	T·m/A	permeability of free space μ₀
n	1/m	turns per unit length
A	m²	cross-sectional area
l	m	length of the solenoid

Valid when: long, tightly-wound solenoid (uniform interior field); n is turns per length; the total turns are N = n·l

Magnetic force on a current-carrying wire

regime 1

F = B I L

B	T	magnetic flux density
I	A	current
L	m	length of wire in the field

Valid when: straight wire perpendicular to the field; uniform field over the length L

Modern physics

de Broglie wavelength

regime 3

\lambda = \frac{h}{p}

h	J·s	Planck's constant
p	kg·m/s	momentum

Valid when: matter has a wavelength set by its momentum; h is Planck's constant

Radioactive decay law

regime 2

N = N_{0} e^{- \lambda t}

$N_{0}$	1	initial number of nuclei
$\lambda$	1/s	decay constant λ
t	s	elapsed time

Valid when: a large number of identical nuclei, each decaying independently; λ is the decay constant (probability of decay per unit time)

Half-life

regime 2

t_{1/2} = \frac{\ln{\left(2 \right)}}{\lambda}

\lambda

1/s

decay constant λ

Valid when: exponential decay with decay constant λ

Mass–energy equivalence

regime 3

E = m c^{2}

m	kg	rest mass
c	m/s	speed of light

Valid when: rest energy of a mass m; c is the speed of light

Photoelectric effect

regime 3

K_{\max} = h f - \phi

h	J·s	Planck's constant
f	Hz	frequency of the light
$\phi$	J	work function of the surface

Valid when: one photon ejects one electron; no current below the threshold frequency f = φ/h, regardless of intensity

Photon energy

regime 3

E = h f

h	J·s	Planck's constant
f	Hz	frequency of the light

Valid when: light comes in quanta of energy hf; h is Planck's constant

Waves & optics

Magnification (thin lens or mirror)

regime 3

m = - \frac{d_i}{d_o}

$d_i$	m	image distance
$d_o$	m	object distance

Valid when: thin lens or mirror, paraxial rays; negative m means an inverted image; |m| > 1 is enlarged

Snell's law of refraction

regime 3

\theta_2 = \arcsin{\left(\frac{n_{1} \sin{\left(\theta_{1} \right)}}{n_{2}} \right)}

$n_{1}$	1	refractive index of the first medium
$\theta_{1}$	rad	angle of incidence (from the normal)
$n_{2}$	1	refractive index of the second medium

Valid when: light crossing a flat boundary between two media; n₁ sin θ₁ = n₂ sin θ₂; angles measured from the normal

Thin-lens equation

regime 3

\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}

$d_o$	m	object distance
$d_i$	m	image distance

Valid when: thin lens, paraxial rays; sign convention: real images at positive dᵢ

Period and frequency

regime 3

T = \frac{1}{f}

frequency

Valid when: periodic motion or wave; period is the time for one cycle; frequency is cycles per second

Wave speed

regime 3

v = f \lambda

f	Hz	frequency
$\lambda$	m	wavelength λ

Valid when: periodic wave; v set by the medium; frequency and wavelength trade off at fixed v

Standing-wave harmonics (string fixed at both ends)

regime 3

f_n = \frac{n v}{2 L}

n	1	harmonic number (1 = fundamental)
v	m/s	wave speed on the string
L	m	length of the string

Valid when: string fixed at both ends (a node at each end); n = 1, 2, 3, … selects the harmonic

Standing-wave wavelengths (string fixed at both ends)

regime 3

\lambda_n = \frac{2 L}{n}

L	m	length of the string
n	1	harmonic number (1 = fundamental)

Valid when: string fixed at both ends (a node at each end); an integer number of half-wavelengths fits the length

Speed of a wave on a string

regime 3

v = \sqrt{\frac{F_T}{\mu}}

$F_T$	N	tension in the string
$\mu$	kg/m	linear mass density

Valid when: transverse wave on a uniform string; small-amplitude (linear) waves

Thermodynamics

Adiabatic relation (PVγ = const)

regime 3

P_2 = P_{1} \left(\frac{V_{1}}{V_{2}}\right)^{\gamma}

$P_{1}$	Pa	initial pressure
$V_{1}$	m³	initial volume
$V_{2}$	m³	final volume
$\gamma$	1	heat-capacity ratio Cp/Cv

Valid when: reversible adiabatic process: PVγ = const; ideal gas with constant heat-capacity ratio γ = Cp/Cv

Adiabatic work

regime 3

W = n C_V \left(T_{1} - T_{2}\right)

n	mol	amount of substance
$C_V$	J/(mol·K)	molar heat capacity at constant volume
$T_{1}$	K	initial temperature
$T_{2}$	K	final temperature

Valid when: adiabatic process: no heat exchanged (Q = 0), so W = −ΔU = nCv(T₁−T₂); equivalently W = (P₁V₁ − P₂V₂)/(γ−1); the gas cools as it does work

Carnot efficiency

regime 3

\eta = 1 - \frac{T_C}{T_H}

$T_C$	K	cold-reservoir temperature
$T_H$	K	hot-reservoir temperature

Valid when: reversible (Carnot) engine between two reservoirs; the maximum efficiency any heat engine can reach between Tc and Th

First law of thermodynamics

regime 3

\Delta U = Q - W

Q	J	heat added to the system
W	J	work done by the system

Valid when: energy conservation for a closed system; sign convention: Q added to the gas positive, W done by the gas positive

Heat and specific heat

regime 3

Q = m c \Delta T

m	kg	mass
c	J/(kg·K)	specific heat capacity
$\Delta T$	K	temperature change ΔT

Valid when: specific heat c constant over the temperature range; no phase change

Ideal-gas law (pressure form)

regime 3

P = \frac{n R T}{V}

n	mol	amount of substance
R	J/(mol·K)	ideal-gas constant
T	K	absolute temperature
V	m³	volume

Valid when: ideal gas (no intermolecular forces, point particles)

Internal energy of a monatomic ideal gas

regime 3

U = \frac{3 n R T}{2}

n	mol	amount of substance
R	J/(mol·K)	ideal-gas constant
T	K	absolute temperature

Valid when: ideal gas; monatomic (3 translational degrees of freedom); use 5/2 for diatomic

Isobaric work (constant pressure)

regime 3

W = P \left(V_{2} - V_{1}\right)

P	Pa	constant pressure
$V_{1}$	m³	initial volume
$V_{2}$	m³	final volume

Valid when: constant pressure (isobaric process); work is the area under the P–V curve, W = ∫P dV — here a rectangle

Isothermal work (ideal gas)

regime 3

W = n R T \ln{\left(\frac{V_{2}}{V_{1}} \right)}

n	mol	amount of substance
R	J/(mol·K)	ideal-gas constant
T	K	absolute temperature
$V_{1}$	m³	initial volume
$V_{2}$	m³	final volume

Valid when: ideal gas at constant temperature (isothermal, reversible expansion); work is the area under the P–V curve, W = ∫P dV