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Diffusion Time Calculator

When considering the diffusion of ions and molecules in solutions, it is generally useful to be able to estimate the time required for diffusion over a given distance. From a physiological perspective, this knowledge helps us better understand how long it takes molecules and ions to travel physiologically relevant distances by diffusion alone. For example, it is of value to know how long it takes molecular O_{2} to reach metabolically active cells 10 μm away from a capillary. As another example, it is important to know how long it takes a neurotransmitter molecule released from the pre-synaptic neuron to diffuse across the synaptic cleft of 20-50 nm to reach target receptors on the post-synaptic plasma membrane. Likewise, numerous other physiological examples can be considered.

While in solutions, diffusing solutes move in three dimensions down a concentration gradient from an area of higher concentration to an area of lower concentration, a simple equation may be used to approximate the time it takes a given molecule to diffuse an average distance in one dimension (see equation below). This equation is also referred to as the Einstein's approximation equation. The important determinants of diffusion time (*t*) are the distance of diffusion (*x*) and the diffusion coefficient (*D*). Diffusion time increases with the square of diffusion distance. The diffusion coefficient is unique for each solute and must be determined experimentally. It is a function of a number factors including molecular weight of the diffusing species, temperature, and viscosity of the medium in which diffusion occurs. For ions in aqueous solutions, the ion charge density influences the size of the hydration shell around the ion which, in turn, influences the value of the diffusion coefficient for that ion. For very large molecules (macromolecules), molecular shape also plays a role. Diffusion time is inversely proportional to the diffusion coefficient (*D*). Table 1 provides the diffusion coefficient for a few selected ions and molecules.

Approximation equation for diffusion time

Table 1. Diffusion coefficient values for selected ions and small and large molecules.

Ion/Molecule | Atomic/Molecular Weight (g/mol) |
Diffusion Coefficient (cm ^{2}/s) |

H^{+} |
1.008 | 9.31 × 10^{-5} |

Na^{+} |
22.990 | 1.33 × 10^{-5} |

K^{+} |
39.098 | 1.96 × 10^{-5} |

Ca^{2+} |
40.078 | 0.79 × 10^{-5} |

Cl^{-} |
35.453 | 2.03 × 10^{-5} |

Ammonia (NH_{3}) |
17.031 | 1.51 × 10^{-5} |

Oxygen (O_{2}) |
31.999 | 2.10 × 10^{-5} |

Carbon dioxide (CO_{2}) |
44.01 | 1.97 × 10^{-5} |

Urea | 60.055 | 1.38 × 10^{-5} |

Glucose | 180.156 | 5 × 10^{-6} |

Sucrose | 342.296 | 5.23 × 10^{-6} |

Hemoglobin | 68,000 | 6.9 × 10^{-7} |

DNA | ≈ 6,000,000 | 1.3 × 10^{-8} |

Note: The diffusion coefficient varies with temperature and is also a function of the medium in which diffusion occurs. The values shown are for diffusion in water (H_{2}O) at 25 °C.

is the diffusion coefficient of a solute in free solution. The diffusion coefficient determines the time it takes a solute to diffuse a given distance in a medium.*D**D*has the units of area/time (typically cm^{2}/s). Its value is unique for each solute and must be determined empirically.*D*is a function of both the physical characteristics of the solute and those of the medium. It is inversely related to the molecular weight of the solute. For ions, the size of the hydration shell influences the value of*D*. For large molecules, molecular shape also plays a role in determining the value of*D*. Temperature also influences*D*. A few selected examples are shown in Table 1.is the mean distance traveled by the diffusing solute in one direction along one axis after elapsed time*x*.*t*is the elapsed time since diffusion began. Diffusion time increases with the square of diffusion distance. Diffusion time is inversely proportional to the diffusion coefficient (*t**D*).

Diffusion time calculator

Each calculator cell shown below corresponds to a term in the formula presented above. Enter appropriate values in all cells except the one you wish to calculate. *Therefore, at least two cells must have values, and no more than one cell may be blank*. The value of the blank cell will be calculated based on the other values entered. After a calculation is performed, the calculated cell will be highlighted and subsequent calculations will calculate the value of the highlighted cell (with no requirement to have a blank cell). However, a blank cell has priority over a highlighted cell.

Please note that while the distance and time may be expressed in any of the available units, the diffusion coefficient must be expressed, or will be calculated, in cm^{2}/s. Please also note that the calculator below does not accept values expressed using the scientific notation (e.g., 1 × 10^{-5}). Instead, please use either the standard decimal notation or the E notation. For example, 1 × 10^{-5} may be expressed as 0.00001 (standard decimal notation) or 1e-5 (E notation).

Interpretation and physiological significance

Table 2. Time required for diffusion of O_{2} over a range of distances.

Distance of Diffusion | Approximate Time Required |

10 nm | 23.8 ns |

50 nm | 595 ns |

100 nm | 2.38 μs |

1 μm | 238 μs |

10 μm | 23.8 ms |

100 μm | 2.38 s |

1 mm | 3.97 min |

1 cm | 6.61 hours |

10 cm | 27.56 days |

The above calculator allows us to evaluate the effectiveness of diffusion over physiologically relevant distances. The diffusion time values shown in Table 2 were obtained by considering the diffusion of O_{2} over a range of distances. Looking at these values, it can be concluded that while diffusion is adequate for the movement of ions/molecules over short distances, diffusion times are unrealistically long for movement over long distances. Thus, for large multicellular organisms, diffusion alone is grossly inadequate for ensuring the delivery of nutrients to metabolically active cells deep within the tissues. It is for this reason that in large organisms such as humans, a cardiovascular system is present to transport nutrients to all cells of the body. This system is composed of blood vessels and the heart. The heart functions as a pump to move blood in the blood vessels. The blood vessels serve as a conduit for the transport of molecules to the tissues, however, exchange of materials between the circulatory system and tissues takes place only across the capillary endothelium. In mammals, the circulatory system is such that no cell is more than approximately 10 μm from a capillary. This ensures proper nourishment and waste removal for all cells of the body. This system is also responsible for collection of waste and transport to organs involved in waste elimination (lungs and kidneys).

As another example, we can consider the diffusion of acetylcholine across the neuromuscular junction where the synaptic cleft is approximately 50 nm. Using 4 × 10^{-6} cm^{2}/s for the diffusion coefficient of acetylcholine in the extracellular space of the neuromuscular junction, it can be calculated that it takes acetylcholine approximately 3.1 μs to diffuse from its pre-synaptic release site to reach the post-synaptic nicotinic acetylcholine receptors. This diffusion time is only a minor fraction of the total synaptic delay of approximately 500 μs observed at the neuromuscular junction.

References

- Atkins, P.W. (1994)
*Physical Chemistry*. 5^{th}Edition. W. H. Freeman, New York. - Hille, B. (2001)
*Ion Channels of Excitable Membranes*. 3^{rd}Edition. Sinauer Associates, Inc., Sunderland. - Robinson, R.A., Stokes, R.H. (1968)
*Electrolyte Solutions*. Revised 2^{nd}Edition. Butterworths, London. - Sperelakis, N., Editor. (2001)
*Cell Physiology Sourcebook: A Molecular Approach*. 3^{rd}Edition. Academic Press, San Diego. - Van Winkle, L.J. (1999)
*Biomembrane Transport*. Academic Press, San Diego. - Weiss, T.F. (1996)
*Cellular Biophysics: Transport (Vol. 1)*. MIT Press, Cambridge.

Posted: Sunday, April 2, 2006

Last updated: Sunday, March 2, 2014

Last updated: Sunday, March 2, 2014