BIOPHYSICS FORMULAE

First derivative of a function f(x) x small change in dx

dy = f'(x)dx

To find slope or gradient

tan ⍺ = Δy/Δx

Partial differentiation

Chemical potential

dG = Tot differential

p, T, n, j = Constant variable

dni = Differential with this K

Work

W = F x dx

F = Force

dx = Displacement

First law of T. for isolated system

dU = 0

Mathematical formulation of 1° law of T. for closed system

dU = dQ + dW

Q = Heat

U = Internal energy

W = Work

Internal energy

Ek = Kinetic energy

Ep = Potential energy

Thermodynamic work

dW = -PdV

P = Pressure

V = Volume

Electric work

dW = ɸdq

ɸ = Electrical potential

dq = Amount of charge

Osmotic work

dW = μdn

2° law for reversible process

dS = S2 - S1 = dQ/T

S = Entropy

Q = Heat

T = Temperature

2° law for irreversible process

dS > dQ/T

S = Entropy

Q = Heat

T = Temperature

2° law during reversible process (isolated system)

dS = 0

2° law during irreversible process (isolated system)

dS > 0

2° law for closed and open system

dS < 0 dQ < 0

Boltzmann equation for entropy

S = k ln W

S = Entropy

k = Boltzmann constant

W = Thermodynamic probability

Boltzmann constant (k)

k = R/Na

R = Constant of perfect gas

Na = Avogadro's number

Perfect gas law

PV = nRT

P = Pressure

V = Volume

n = Number of mole

R = Universal gas constant

T = Temperature

Shannon equation of information

I = k ln P

I = Information

k = Boltzmann constant

P = Mathematical probability

Mathematical probability

P = n. of favorables cases/ Greatest n. of cases

Heat of reaction

ΔH = H (products) - H (reagents)

Potential energy

Ep = mgh

m = Mass

g = Acceleration of gravity

h = Height

Work

W = -dEp

Ep = Potential energy

Force

Fdx = -dEp (Force = - gradient of Ep)

F = -(dEp/dx)

Electrical potential

ɸ = Ep/q

Driving force (electrical force)

F = -q x dɸ/dx

q = Charge

dɸ/dx = Electrical potential gradient

Combined Law of Thermodynamics

dU = TdS - PdV

U = Internal energy

S = Entropy

P = Pressure

V= Volume

1° law for an open system

dU = dQ + dW + μdn (dU = TdS - PdV + μdn)

U =Internal energy

Q = Heat

W = Work

μ = Chemical potential

n = number of particles

Chemical potential

μ = μ0 + RT ln C

Change in free energy

dG = μdn

μdn = Change in electrochemical potential

Electrochemical potential

dμ* = dμ0 + RT ln C2/C1 + z x F x (ɸ2 - ɸ1)

F = Faraday's constant

z = Ion valency

(ɸ2 - ɸ1) = Potential difference

Faraday's constant

F = eNa

e = Generic charge

Na = Avogadro's number

Charge

q = e x z

e = Elementary charge

z = Atomic number

Flux

J = Cux

C = Concentration

u = Mobility

x = Foce

Flux of matter

J^m = -D x dC/dx

D = Diffusion coefficient

dC/x = Concentration gradient

Flux of heat

J^Q = - k x dT/dx

K = Constant

dT/dx = Temperature gradient

Force

F = m x a

m = Mass

a = Acceleration

Mobility (uncharged particles)

u = v/x

v = Velocity

x = Force

Relationship btw Force and Flux (Onsager coefficient)

J = Lx

L = Onsager coefficient

x = Force

Phenomenological equation

Jm = Lm1x1 + lm2x2 + ......... + Lmnxn

Jm = Number of flux

n = Number of driving force

Total entropy balance

Criteria of coupling flux

Stock's law (Frictional force)

Fd = 6πηrv

η = Viscosity

r = Radius

v = Velocity

Fick's law for diffusion

dm/dt = -DA dC/dx

dm/dt = Mass flow

D = diffusion coefficient

A = Area

dC/dx = Coefficient gradient

Flux of matter (Fick's)

dm/dtdA

m = Mass

t = Time

A = Area

Coefficient of diffusion

D = uRT

u = mobility

R = Universal gas constant

T = Temperature

Fick's for free diffusion of non-charged particles

J = -D dC/dx ==> J = -uRT dC/dx

D = Diffusion coefficient

u = mobility

R = Universal gas constant

T = Temperature

Electric field

E = F/q

F = Force

q = Charge

Electric potential gradient

E = - dɸ/dx

1 mol of ions

F = - z x F x dɸ/dx

z = Ion valency

F =Faraday's constant

dɸ/dx = Electric potential gradient

Free diffusion of charged particles (Drift)

J = - cuzF dɸ/dx

c = Concentration

u = Mobility

z = Ion valency

dɸ/dx = Electrical potential gradient

Nernst-Plank molar flux equation

J = - uRT dC/dx - cuzF dɸ/dx

u = Mobility

R = Universal gas constant

T = Temperature

dC/dx = Concentration gradient

c = Concentration

z = Ion valency

F = Faraday's constant

dɸ/dx = Electrical potential gradient

Partition coefficient

k = Cme/Ce = Cmi/Ci

Concentration intracellular and extracellular

Permeability

P = Dk/d

D= Diffusion coefficient

k = Partition coefficient

d = Membrane thickness

Fick's law for simple diffusion

J^m = -PdC

P = Permeability

C = Difference in concentration

Pousille's law

πr^4dP/8ηl

r = Radius

P = Pressure

η = Viscosity

l = Length

Michealis-Menten equation

J = Jmax x C/Km + C

J = Flux

Jmax = Maximum flux

C = Concentration

Km = Michealis-Menten constant

Henderson equation (Diffusion potential)

u = Mobility

R = Universal gas constant

T = Temperature

z = Ion valency

C = Concentration

Nernst equation (Equilibrium potential)

Em = - RT/zF ln C2/C1 ==> (Em = - 60 log10 C2/C1)

R = Universal gas constant

T = Temperature

z = Ion valency

C = Concentration

Donnan equilibrium

[K+]A x [Cl-]A = [K+]B x [Cl-]B

Resting membrane potential (Goldman equation)

Thomas equation

Electric current

I = 1/R x V = gV

R = Resistance

V = Voltage

g = Conductance

Conductance

g = 1/R

R = Resistance

Ohm's law

V = I x R

I = Electric current

R = Resistance

Fourier's law

dQ/dt = - k dT/dx

k = constant

dT/dx = Temperature gradient

mobility (charged partcles)

u = v/E

Osmotic pressure

Posm = cRT

c = Concentration

R = Universal gas constant

T = Temperature

Einstein equation

E = mc^2

m = Gravitational mass

c = velocity of the light

Newton's law for viscosity

τ = μ x du/dy

τ = Shear stress

μ = Viscosity

du/dy = Rate of shear deformation