Laws and Equations

This appendix is a list of the key laws and equations common to many topics:

General Laws

• Fick's Law of Diffusion
  Diffusion of a substance across a membrane is given by:
  $\dot{V} = {A \times D \times \Delta P \over T}$ , where:
  • $A$ = Area of the sheet
  • $D$ = Diffusion constant, which is proportional to the solubility of the gas and inversely proportional to the square root of the molecular weight, i.e. $D \propto {Sol \over \sqrt{MW}}$
  • $\Delta P$ = Pressure gradient
  • $T$ = Thickness of the sheet

• Hagan-Poiseuille Equation
  Calculates the flow for a given pressure different of a particular fluid. May also be rearranged to calculate pressure or resistance.
  • Given by the equation:
    $Q = {\pi P r^4 \over 8 \eta l}$, where:
    • Q is the flow
    • P is the driving pressure
    • η is the dynamic viscosity
    • L is the length of tubing
    • r is the radius
  • Has several limitations:
    • Only models laminar flow
    • Fluid must be incompressible
      Not technically valid for air, but provides a good approximation when used clinically.
    • Fluid must be Newtonian
    • Fluid must be in a cylindrical pipe of uniform cross-section

• Reynolds Number
  Reynolds Number is a dimensionless index used to predict the likelihood of turbulent flow. R < 2000 is likely to be laminar, R > 2000 is likely to be turbulent. Given by the equation:
  • $Reynolds \ Number = {2rvd \over n}$, where:
    • v is the linear velocity of fluid in $m/s$
    • d is the fluid density in $Pa S$
    • r is the radius in $m$
    • n is the viscosity in $m^2 / s$

Cell Physiology

• Nernst Equation
  Calculates the electrochemical equilibrium for a given ion:
  $E \ (mV) = {R.T \over z.F}ln{[ion]_{outside} \over [ion]_{inside}}$, where:
  • $E$ is the equilibrium potential for the ion
  • $R$ is the gas constant (8.314 J.deg^-1.mol^-1)
  • $T$ is the temperature in Kelvin
  • $F$ is Faraday's Constant
  • $z$ is the ionic valency (e.g. +2 for Mg⁺², -1 for Cl^-)

• Goldman-Hodgkin-Katz Equation
  Calculates the membrane potential for given values of intracellular and extracellular ionic concentrations:
  $E \ (mV) = {R.T \over F }ln{P_K[K^+]_o \ + \ P_{Na}[Na^+]_o \ + \ P_{Cl}[Cl^-]_i \over P_K[K^+]_i + \ P_{Na}[Na^+]_i \ + \ P_{Cl}[Cl^-]_o}$, where:
  • $P_x$ is the permeability constant for the ion, $x$
    If the membrane is impermeable to $x$, then $P_x = 0$.

• Henderson-Hasselbalch
  Calculates the pH of a buffer solution:
  $pH = pK_a + log{[A^-] \over [HA]}$, where:
  • $pH$ is the pH of the solution
  • $pK_a$ is the pKa of the buffer
  • $A^-$ is the concentration of base
  • $HA$ is the concentration of acid

Respiratory Laws

• Modified Bohr Equation
  The ratio of dead space to tidal volume ventilation equations the arterial - mixed-expired CO2 difference, over the arterial CO2. ${V_D \over V_T} = {PaCO_2 - PE'CO_2 \over PaCO_2}$

• Laplace's Law
  The larger the vessel radius, the larger the wall tension required to withstand a given internal fluid pressure. For a thin-walled sphere, Wall Tension (T) is half the product of pressure and radius, i.e. $T = {P.r \over 2}$

• Alveolar Gas Equation
  The alveolar PO2 is equal to the PiO2 minus the alveolar CO2/the respiratory quotient, i.e.: $PA_{O_2} = Pi_{O_2} - {PA_{CO_2} \over R}$

Gas Laws

• Boyle's Law
  $PV = K$, i.e. pressure and volume are inversely related at constant temperature and pressure.
  • Boyles Law can be used to work out how many litres of gas are remaining in gas cylinder, e.g.:
    • A standard C cylinder is 1.2L in size
    • Normal cylinder pressure is ~137bar, and atmospheric pressure is ~1bar
      $\begin{align*} P_1 . V_1 &= P_2 . V_2 \\ 137 . 1.2 &= 164 \\ \end{align*}$
    • Therefore, the cylinder contains ~164L of oxygen
    • This can be used to calculate the volume of gas remaining in the cylinder during use, using the volume of the cylinder (fixed) and the current pressure as measured at the regulator

• Charle's Law
  $V = KT$, i.e. volume and temperature are linearly related when pressure is constant.

• Gay-Lussac's Law/The Third Gas Law $P = k.T$, i.e. pressure and temperature are linearly related when volume is constant.

• The Universal Gas Equation
  $PV = nRT$, i.e. combination of Boyle's, Charle's law combining each variable and the universal gas constant, R (8.13).

• Henry's Law
  The number of molecules of dissolved gas is proportional to the partial pressure of the gas at the surface of the liquid

• Graham's Law of Diffusion
  Diffusion rates through orifices are inversely proportional to the square root of the molecular weight

• Dalton's Law of Partial Pressures
  In a mixture of gases, each gas exerts the pressure that it would exert if it occupied the volume alone.

Cardiovascular Equations

• Fick's Principle
  Flow of blood through an organ equals the uptake of a tracer substance by the organ divided by the concentration difference of the substance across it, i.e.:
  $\dot{Q} = {\dot{V}_{O_2} \over Ca_{O_2} - C\bar{v}_{O_2}}$

• Starling's Law of Fluid Exchange
  Flow of fluid across the capillaries is proportional to the hydrostatic pressure difference and the oncotic pressure difference (times the reflection coefficient), all times by the filtraiton coefficient, i.e.:
  $Net \ flow \ out = K[(P_c - P_i) - \sigma (\pi_c - \pi_i)]$

• Venous Admixture
  Calculates the shunt fraction by identifying how much mixed venous blood must be added to ideal pulmonary capillary blood to produce the identified arterial oxygen content. ${\dot{Q}_S \over \dot{Q}_T} = {Cc'_{O_2} - Ca'_{O_2} \over Cc'_{O_2} - C\bar{v}'_{O_2}}$

Equipment

• Doppler equation
  Calculates the velocity of an object based on the change in observed frequency when a wave is reflected off (or emitted from) the object:
  $V = { \Delta Fs \over 2F_0cos\theta}$ where:
  • $V$ = Velocity of object
  • $F$ = Frequency shift
  • $s$ = Speed of sound (in blood)
  • $F_0$ = Frequency of the emitted sound
  • $\theta$ = Angle between the sound wave and the object

References

1. Davis & Kenny. Basic Physics and Measurement in Anaesthesia, 5th Edition.
2. Gorman. RAH Diving and Hyperbaric Medicine. Chapter 3: The physics of diving.