Mass diffusivity

<h2 id="temperature-dependence-of-the-diffusion-coefficient">Temperature dependence of the diffusion coefficient</h2>
<h3>Solids</h3>
The diffusion coefficient in solids at different temperatures is generally found to be well predicted by the <a href="/facts/Arrhenius_equation/TdAeo3Lf">Arrhenius equation</a>:

 
 
 
 D
 =
 
 D
 
 0
 
 
 exp
 ⁡
 
 (
 
 −
 
 
 
 E
 
 A
 
 
 
 R
 T
 
 
 
 
 )
 
 
 
 {\displaystyle D=D_{0}\exp \left(-{\frac {E_{\text{A}}}{RT}}\right)}

where

<ul><li>D is the diffusion coefficient (in m2/s),</li>
<li>D0 is the maximal diffusion coefficient (at infinite temperature; in m2/s),</li>
<li>EA is the <a href="/facts/Activation_energy/GqkgPvo0">activation energy</a> for diffusion (in J/mol),</li>
<li>T is the absolute temperature (in K),</li>
<li>R ≈ 8.31446 J/(mol⋅K) is the <a href="/facts/Gas_constant/RUXszrMH">universal gas constant</a>.</li></ul>
Diffusion in crystalline solids, termed <a href="/facts/Lattice_diffusion_coefficient/lMaXQn9b">lattice diffusion</a>, is commonly regarded to occur by two distinct mechanisms,<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> <a href="/facts/Interstitial_site/3kvKwQer">interstitial</a> and <a href="/facts/Substitutional_alloy/I7w7WQWR">substitutional</a> or <a href="/facts/Lattice_diffusion_coefficient/lMaXQn9b">vacancy diffusion</a>. The former mechanism describes diffusion as the motion of the diffusing atoms between <a href="/facts/Interstitial_site/3kvKwQer">interstitial sites</a> in the lattice of the solid it is diffusing into, the latter describes diffusion through a mechanism more analogue to that in liquids or gases: Any crystal at nonzero temperature will have a certain number of <a href="/facts/Vacancy_defect/kDeBLJIT">vacancy defects</a> (i.e. empty sites on the lattice) due to the random vibrations of atoms on the lattice, an atom neighbouring a vacancy can spontaneously "jump" into the vacancy, such that the vacancy appears to move. By this process the atoms in the solid can move, and diffuse into each other. Of the two mechanisms, interstitial diffusion is typically more rapid.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> 

<h3>Liquids</h3>
An approximate dependence of the diffusion coefficient on temperature in liquids can often be found using <a href="/facts/Stokes%E2%80%93Einstein_equation/pLALxmow">Stokes–Einstein equation</a>, which predicts that

 
 
 
 
 
 
 D
 
 
 T
 
 1
 
 
 
 
 
 D
 
 
 T
 
 2
 
 
 
 
 
 
 =
 
 
 
 T
 
 1
 
 
 
 T
 
 2
 
 
 
 
 
 
 
 μ
 
 
 T
 
 2
 
 
 
 
 
 μ
 
 
 T
 
 1
 
 
 
 
 
 
 ,
 
 
 {\displaystyle {\frac {D_{T_{1}}}{D_{T_{2}}}}={\frac {T_{1}}{T_{2}}}{\frac {\mu _{T_{2}}}{\mu _{T_{1}}}},}

where

<ul><li>D is the diffusion coefficient,</li>
<li>T1 and T2 are the corresponding absolute temperatures,</li>
<li>μ is the <a href="/facts/Dynamic_viscosity/QLVfrdCz">dynamic viscosity</a> of the solvent.</li></ul>
<h3>Gases</h3>
The dependence of the diffusion coefficient on temperature for gases can be expressed using <a href="/facts/Chapman%E2%80%93Enskog_theory/7fVAWJkQ">Chapman–Enskog theory</a> (predictions accurate on average to about 8%):<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>

 
 
 
 D
 =
 
 
 
 A
 
 T
 
 
 3
 2
 
 
 
 
 
 p
 
 σ
 
 12
 
 
 2
 
 
 Ω
 
 
 
 
 
 
 
 1
 
 M
 
 1
 
 
 
 
 +
 
 
 1
 
 M
 
 2
 
 
 
 
 
 
 ,
 
 
 {\displaystyle D={\frac {AT^{\frac {3}{2}}}{p\sigma _{12}^{2}\Omega }}{\sqrt {{\frac {1}{M_{1}}}+{\frac {1}{M_{2}}}}},}

A
        =
        
          
            3
            8
          
        
        
          k
          
            b
          
          
            
              3
              2
            
          
        
        
          
            
              
                N
                
                  A
                
              
              
                2
                π
              
            
          
        
      
    
    {\displaystyle A={\frac {3}{8}}k_{b}^{\frac {3}{2}}{\sqrt {\frac {N_{A}}{2\pi }}}}

where

<ul><li>D is the diffusion coefficient (cm2/s),<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a><a class="footnote-ref" id="fnref:7" href="#fn:7">7</a></li>
<li>A is approximately 
 
 
 
 1.859
 ×
 
 10
 
 −
 3
 
 
 
 
 
 
 a
 t
 m
 ⋅
 
 
 
 Å
 
 
 
 2
 
 
 ⋅
 
 
 c
 m
 
 
 2
 
 
 
 
 
 K
 
 3
 
 /
 
 2
 
 
 ⋅
 s
 
 
 
 
 
 
 g
 
 m
 o
 l
 
 
 
 
 
 
 
 {\textstyle 1.859\times 10^{-3}\mathrm {{\frac {atm\cdot \mathrm {\AA} ^{2}\cdot {cm}^{2}}{K^{3/2}\cdot s}}{\sqrt {\frac {g}{mol}}}} }
 
 (with Boltzmann constant 
 
 
 
 
 k
 
 b
 
 
 
 
 {\textstyle k_{b}}
 
, and Avogadro constant 
 
 
 
 
 N
 
 A
 
 
 
 
 {\textstyle N_{A}}
 
)</li>
<li>1 and 2 index the two kinds of molecules present in the gaseous mixture,</li>
<li>T is the absolute temperature (K),</li>
<li>M is the molar mass (g/mol),</li>
<li>p is the pressure (atm),</li>
<li>
 
 
 
 
 σ
 
 12
 
 
 =
 
 
 1
 2
 
 
 (
 
 σ
 
 1
 
 
 +
 
 σ
 
 2
 
 
 )
 
 
 {\textstyle \sigma _{12}={\frac {1}{2}}(\sigma _{1}+\sigma _{2})}
 
 is the average collision diameter (the values are tabulated<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> page 545) (Å),</li>
<li>Ω is a temperature-dependent collision integral (the values tabulated for some <a href="/facts/Pair_potential/Xhx2Uwen">intermolecular potentials</a>,<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a> can be computed from correlations for others,<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a> or must be evaluated numerically.) (dimensionless).</li></ul>
The relation 

 
 
 
 D
 ∼
 
 
 
 T
 
 3
 
 /
 
 2
 
 
 
 p
 Ω
 (
 T
 )
 
 
 
 
 
 {\displaystyle D\sim {\frac {T^{3/2}}{p\Omega (T)}}}

is obtained when inserting the ideal gas law into the expression obtained directly from <a href="/facts/Chapman%E2%80%93Enskog_theory/7fVAWJkQ">Chapman-Enskog theory</a>,<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a> which may be written as

 
 
 
 D
 =
 
 D
 
 0
 
 
 
 
 
 T
 
 1
 
 /
 
 2
 
 
 
 n
 Ω
 (
 T
 )
 
 
 
 
 
 {\displaystyle D=D_{0}{\frac {T^{1/2}}{n\Omega (T)}}}

where 
 
 
 
 n
 
 
 {\displaystyle n}
 
 is the molar density (mol / m3) of the gas, and 

 
 
 
 
 D
 
 0
 
 
 =
 
 
 3
 
 8
 
 σ
 
 12
 
 
 2
 
 
 
 
 
 
 
 
 (
 
 
 R
 
 2
 π
 
 
 
 )
 
 
 (
 
 
 
 1
 
 M
 
 1
 
 
 
 
 +
 
 
 1
 
 M
 
 2
 
 
 
 
 
 )
 
 
 
 ,
 
 
 {\displaystyle D_{0}={\frac {3}{8\sigma _{12}^{2}}}{\sqrt {\left({\frac {R}{2\pi }}\right)\left({\frac {1}{M_{1}}}+{\frac {1}{M_{2}}}\right)}},}

with 
 
 
 
 R
 =
 
 k
 
 B
 
 
 
 N
 
 A
 
 
 
 
 {\displaystyle R=k_{B}N_{A}}
 
 the universal gas constant. At moderate densities (i.e. densities at which the gas has a non-negligible <a href="/facts/Excluded_volume/sbK0t25G">co-volume</a>, but is still sufficiently dilute to be considered as gas-like rather than liquid-like) this simple relation no longer holds, and one must resort to <a href="/facts/Revised_Enskog_theory/7fVAWJkQ">Revised Enskog Theory</a>.<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a> Revised Enskog Theory predicts a diffusion coefficient that decreases somewhat more rapidly with density, and which to a first approximation may be written as

 
 
 
 D
 =
 
 D
 
 0
 
 
 
 
 
 T
 
 1
 
 /
 
 2
 
 
 
 n
 g
 (
 σ
 )
 Ω
 (
 T
 )
 
 
 
 
 
 {\displaystyle D=D_{0}{\frac {T^{1/2}}{ng(\sigma )\Omega (T)}}}

where 
 
 
 
 g
 (
 σ
 )
 
 
 {\displaystyle g(\sigma )}
 
 is the <a href="/facts/Radial_distribution_function/zWN3v1Cr">radial distribution function</a> evaluated at the <a href="/facts/Kinetic_diameter/cLhnp0Sl">contact diameter</a> of the particles. For molecules behaving like <a href="/facts/Hard_spheres/91Gfe05N">hard, elastic spheres</a>, this value can be computed from the <a href="/facts/Hard_spheres/91Gfe05N">Carnahan-Starling Equation</a>, while for more realistic intermolecular potentials such as the <a href="/facts/Mie_potential/5wjURUcj">Mie potential</a> or <a href="/facts/Lennard-Jones_potential/yXwcgLmh">Lennard-Jones potential</a>, its computation is more complex, and may involve invoking a thermodynamic perturbation theory, such as <a href="/facts/Statistical_associating_fluid_theory/5sPr3SDd">SAFT</a>.

<h2 id="pressure-dependence-of-the-diffusion-coefficient">Pressure dependence of the diffusion coefficient</h2>
For self-diffusion in gases at two different pressures (but the same temperature), the following empirical equation has been suggested:<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a>

D
              
                P
                1
              
            
            
              D
              
                P
                2
              
            
          
        
        =
        
          
            
              ρ
              
                P
                2
              
            
            
              ρ
              
                P
                1
              
            
          
        
        ,
      
    
    {\displaystyle {\frac {D_{P1}}{D_{P2}}}={\frac {\rho _{P2}}{\rho _{P1}}},}

where

<ul><li>D is the diffusion coefficient,</li>
<li>ρ is the gas mass density,</li>
<li>P1 and P2 are the corresponding pressures.</li></ul>
<h2 id="population-dynamics-dependence-of-the-diffusion-coefficient-on-fitness">Population dynamics: dependence of the diffusion coefficient on fitness</h2>
In population dynamics, <a href="/facts/Kinesis_(biology)/dxtVuW5H">kinesis</a> is the change of the diffusion coefficient in response to the change of conditions. In models of purposeful kinesis, diffusion coefficient depends on fitness (or reproduction coefficient) r:

D
        =
        
          D
          
            0
          
        
        
          e
          
            −
            α
            r
          
        
        ,
      
    
    {\displaystyle D=D_{0}e^{-\alpha r},}

where 
 
 
 
 
 D
 
 0
 
 
 
 
 {\displaystyle D_{0}}
 
 is constant and r depends on population densities and abiotic characteristics of the living conditions. This dependence is a formalisation of the simple rule: Animals stay longer in good conditions and leave quicker bad conditions (the "Let well enough alone" model).

<h2 id="effective-diffusivity-in-porous-media">Effective diffusivity in porous media</h2>
The effective diffusion coefficient describes diffusion through the pore space of <a href="/facts/Porous_media/KTMRISJf">porous media</a>.<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a> It is <a href="/facts/Macroscopic/NLwPDtLe">macroscopic</a> in nature, because it is not individual pores but the entire pore space that needs to be considered. The effective diffusion coefficient for transport through the pores, De, is estimated as follows:

D
          
            e
          
        
        =
        
          
            
              D
              
                ε
                
                  t
                
              
              δ
            
            τ
          
        
        ,
      
    
    {\displaystyle D_{\text{e}}={\frac {D\varepsilon _{t}\delta }{\tau }},}

where

<ul><li>D is the diffusion coefficient in gas or liquid filling the pores,</li>
<li>εt is the <a href="/facts/Porosity/UoyAJU8X">porosity</a> available for the transport (dimensionless),</li>
<li>δ is the <a href="/facts/Constrictivity/Vza3bkLA">constrictivity</a> (dimensionless),</li>
<li>τ is the <a href="/facts/Tortuosity/oq0QNSUW">tortuosity</a> (dimensionless).</li></ul>
The transport-available <a href="/facts/Porosity/UoyAJU8X">porosity</a> equals the total porosity less the pores which, due to their size, are not accessible to the diffusing particles, and less dead-end and blind pores (i.e., pores without being connected to the rest of the pore system). The constrictivity describes the slowing down of diffusion by increasing the <a href="/facts/Viscosity/QLVfrdCz">viscosity</a> in narrow pores as a result of greater proximity to the average pore wall. It is a function of pore diameter and the size of the diffusing particles.

<h2 id="example-values">Example values</h2>
Gases at 1 atm., solutes in liquid at infinite dilution. Legend: (s) – solid; (l) – liquid; (g) – gas; (dis) – dissolved.

Values of diffusion coefficients (gas)<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a><table><tbody><tr><th colspan="2">Species pair</th><th rowspan="2">Temperature (°C)</th><th rowspan="2">D (cm2/s)</th></tr><tr><th>Solute</th><th>Solvent</th></tr><tr><td>Water (g)</td><td>Air (g)</td><td>25</td><td>0.260</td></tr><tr><td>Oxygen (g)</td><td>Air (g)</td><td>25</td><td>0.176</td></tr></tbody></table>
Values of diffusion coefficients (liquid)<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a><table><tbody><tr><th colspan="2">Species pair</th><th rowspan="2">Temperature (°C)</th><th rowspan="2">D (10−5 cm2/s)</th></tr><tr><th>Solute</th><th>Solvent</th></tr><tr><td>Acetone (dis)</td><td>Water (l)</td><td>25</td><td>1.16</td></tr><tr><td>Air (dis)</td><td>Water (l)</td><td>25</td><td>2.00</td></tr><tr><td>Ammonia (dis)</td><td>Water (l)</td><td>12</td><td>1.64</td></tr><tr><td>Argon (dis)</td><td>Water (l)</td><td>25</td><td>2.00</td></tr><tr><td>Benzene (dis)</td><td>Water (l)</td><td>25</td><td>1.02</td></tr><tr><td>Bromine (dis)</td><td>Water (l)</td><td>25</td><td>1.18</td></tr><tr><td>Carbon monoxide (dis)</td><td>Water (l)</td><td>25</td><td>2.03</td></tr><tr><td>Carbon dioxide (dis)</td><td>Water (l)</td><td>25</td><td>1.92</td></tr><tr><td>Chlorine (dis)</td><td>Water (l)</td><td>25</td><td>1.25</td></tr><tr><td>Ethane (dis)</td><td>Water (l)</td><td>25</td><td>1.20</td></tr><tr><td>Ethanol (dis)</td><td>Water (l)</td><td>25</td><td>0.84</td></tr><tr><td>Ethylene (dis)</td><td>Water (l)</td><td>25</td><td>1.87</td></tr><tr><td>Helium (dis)</td><td>Water (l)</td><td>25</td><td>6.28</td></tr><tr><td>Hydrogen (dis)</td><td>Water (l)</td><td>25</td><td>4.50</td></tr><tr><td>Hydrogen sulfide (dis)</td><td>Water (l)</td><td>25</td><td>1.41</td></tr><tr><td>Methane (dis)</td><td>Water (l)</td><td>25</td><td>1.49</td></tr><tr><td>Methanol (dis)</td><td>Water (l)</td><td>25</td><td>0.84</td></tr><tr><td>Nitrogen (dis)</td><td>Water (l)</td><td>25</td><td>1.88</td></tr><tr><td>Nitric oxide (dis)</td><td>Water (l)</td><td>25</td><td>2.60</td></tr><tr><td>Oxygen (dis)</td><td>Water (l)</td><td>25</td><td>2.10</td></tr><tr><td>Propane (dis)</td><td>Water (l)</td><td>25</td><td>0.97</td></tr><tr><td>Water (l)</td><td>Acetone (l)</td><td>25</td><td>4.56</td></tr><tr><td>Water (l)</td><td>Ethyl alcohol (l)</td><td>25</td><td>1.24</td></tr><tr><td>Water (l)</td><td>Ethyl acetate (l)</td><td>25</td><td>3.20</td></tr></tbody></table>
Values of diffusion coefficients (solid)<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a><table><tbody><tr><th colspan="2">Species pair</th><th rowspan="2">Temperature (°C)</th><th rowspan="2">D (cm2/s)</th></tr><tr><th>Solute</th><th>Solvent</th></tr><tr><td>Hydrogen</td><td>Iron (s)</td><td>10</td><td>1.66×10−9</td></tr><tr><td>Hydrogen</td><td>Iron (s)</td><td>100</td><td>124×10−9</td></tr><tr><td>Aluminium</td><td>Copper (s)</td><td>20</td><td>1.3×10−30</td></tr></tbody></table>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Atomic_diffusion/kNtG3tVA">Atomic diffusion</a></li>
<li><a href="/facts/Effective_diffusion_coefficient/yldrwtXB">Effective diffusion coefficient</a></li>
<li><a href="/facts/Lattice_diffusion_coefficient/lMaXQn9b">Lattice diffusion coefficient</a></li>
<li><a href="/facts/Knudsen_diffusion/elw9GfUR">Knudsen diffusion</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">CRC Press Online: CRC Handbook of Chemistry and Physics, Section 6, 91st Edition <a href="http://www.crcpress.com/product/isbn/9781439820773" target="_blank">http://www.crcpress.com/product/isbn/9781439820773</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Diffusion <a href="http://www.cco.caltech.edu/~brokawc/Bi145/Diffusion.html" target="_blank">http://www.cco.caltech.edu/~brokawc/Bi145/Diffusion.html</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Callister, William D.; Rethwisch, David G. (2012). Fundamentals of materials science and engineering: an integrated approach (4 ed.). Hoboken, NJ: Wiley. ISBN 978-1-118-06160-2. <a href="978-1-118-06160-2" target="_blank">978-1-118-06160-2</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Callister, William D.; Rethwisch, David G. (2012). Fundamentals of materials science and engineering: an integrated approach (4 ed.). Hoboken, NJ: Wiley. ISBN 978-1-118-06160-2. <a href="978-1-118-06160-2" target="_blank">978-1-118-06160-2</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Cussler, E. L. (1997). Diffusion: Mass Transfer in Fluid Systems (2nd ed.). New York: Cambridge University Press. ISBN 0-521-45078-0. <a href="0-521-45078-0" target="_blank">0-521-45078-0</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Cussler, E. L. (1997). Diffusion: Mass Transfer in Fluid Systems (2nd ed.). New York: Cambridge University Press. ISBN 0-521-45078-0. <a href="0-521-45078-0" target="_blank">0-521-45078-0</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Welty, James R.; Wicks, Charles E.; Wilson, Robert E.; Rorrer, Gregory (2001). Fundamentals of Momentum, Heat, and Mass Transfer. Wiley. ISBN 978-0-470-12868-8. <a href="978-0-470-12868-8" target="_blank">978-0-470-12868-8</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Hirschfelder, J.; Curtiss, C. F.; Bird, R. B. (1954). Molecular Theory of Gases and Liquids. New York: Wiley. ISBN 0-471-40065-3. <a href="0-471-40065-3" target="_blank">0-471-40065-3</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Hirschfelder, J.; Curtiss, C. F.; Bird, R. B. (1954). Molecular Theory of Gases and Liquids. New York: Wiley. ISBN 0-471-40065-3. <a href="0-471-40065-3" target="_blank">0-471-40065-3</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">"К юбилею Г.И. Канеля". Теплофизика высоких температур (in Russian). 52 (4): 487–488. 2014. doi:10.7868/s0040364414040279. ISSN 0040-3644. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Chapman, Sydney; Cowling, Thomas George; Burnett, David (1990). The mathematical theory of non-uniform gases: an account of the kinetic theory of viscosity, thermal conduction, and diffusion in gases. Cambridge mathematical library (3rd ed.). Cambridge New York Port Chester [etc.]: Cambridge university press. ISBN 978-0-521-40844-8. <a href="978-0-521-40844-8" target="_blank">978-0-521-40844-8</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Cohen, E. G. D. (1993-03-15). "Fifty years of kinetic theory". Physica A: Statistical Mechanics and Its Applications. 194 (1): 229–257. Bibcode:1993PhyA..194..229C. doi:10.1016/0378-4371(93)90357-A. ISSN 0378-4371. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Cussler, E. L. (1997). Diffusion: Mass Transfer in Fluid Systems (2nd ed.). New York: Cambridge University Press. ISBN 0-521-45078-0. <a href="0-521-45078-0" target="_blank">0-521-45078-0</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Grathwohl, P. (1998). Diffusion in natural porous media: Contaminant transport, sorption / desorption and dissolution kinetics. Kluwer Academic. ISBN 0-7923-8102-5. <a href="0-7923-8102-5" target="_blank">0-7923-8102-5</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Cussler, E. L. (1997). Diffusion: Mass Transfer in Fluid Systems (2nd ed.). New York: Cambridge University Press. ISBN 0-521-45078-0. <a href="0-521-45078-0" target="_blank">0-521-45078-0</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Cussler, E. L. (1997). Diffusion: Mass Transfer in Fluid Systems (2nd ed.). New York: Cambridge University Press. ISBN 0-521-45078-0. <a href="0-521-45078-0" target="_blank">0-521-45078-0</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Cussler, E. L. (1997). Diffusion: Mass Transfer in Fluid Systems (2nd ed.). New York: Cambridge University Press. ISBN 0-521-45078-0. <a href="0-521-45078-0" target="_blank">0-521-45078-0</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
</ol>

Mass diffusivity open-in-new

Mass diffusivity