(II) Estimate the time needed for a glycine molecule (see Table 13–4) to diffuse a distance of \(25\;\mu {\rm{m}}\) in water at 20°C if its concentration varies over that distance from \(1.00\;{\rm{mol/}}{{\rm{m}}^{\rm{3}}}\) to \(0.50\;{\rm{mol/}}{{\rm{m}}^{\rm{3}}}\)? Compare this “speed” to its rms (thermal) speed. The molecular mass of glycine is about 75 u.

The higher concentration is \({C_1} = 1.00\;{\rm{mol/}}{{\rm{m}}^{\rm{3}}}\).

The lower concentration is \({C_2} = 0.50\;{\rm{mol/}}{{\rm{m}}^{\rm{3}}}\).

The distance is \(\Delta x = 25\;\mu {\rm{m}} = 25 \times {10^{ - 6}}\;{\rm{m}}\).

The temperature is \(T = {20^ \circ }{\rm{C}} = 293\;{\rm{K}}\).

The molar mass of the glycine is \(m = 75\;u = 75 \times 1.66 \times {10^{ - 27}}\;{\rm{kg}}\).

The diffusivity of the glycine is \(D = 95 \times {10^{ - 11}}\;{{\rm{m}}^{\rm{2}}}{\rm{/s}}\).

The rms thermal speed of any gas is\({v_{{\rm{rms}}}} = \sqrt {\frac{{3kT}}{m}} \).

The time for the diffusion is\(t = \frac{{\bar C}}{{\Delta C}}\frac{{{{\left( {\Delta x} \right)}^2}}}{D}\).

The average concentration is \(\bar C = \frac{1}{2}\left( {{C_1} + {C_2}} \right)\).

The difference between the concentrations is \(\Delta C = {C_1} - {C_2}\).

The time for the diffusion is:

\(\begin{aligned}{c}t &= \frac{{\bar C}}{{\Delta C}}\frac{{{{\left( {\Delta x} \right)}^2}}}{D}\\t &= \frac{{\frac{1}{2}\left( {{C_1} + {C_2}} \right)}}{{{C_1} - {C_2}}}\frac{{{{\left( {\Delta x} \right)}^2}}}{D}\\t &= \frac{{\frac{1}{2}\left( {1.00\;{\rm{mol/}}{{\rm{m}}^{\rm{3}}} + 0.50\;{\rm{mol/}}{{\rm{m}}^{\rm{3}}}} \right)}}{{1.00\;{\rm{mol/}}{{\rm{m}}^{\rm{3}}} - 0.50\;{\rm{mol/}}{{\rm{m}}^{\rm{3}}}}}\frac{{{{\left( {25 \times {{10}^{ - 6}}\;{\rm{m}}} \right)}^2}}}{{95 \times {{10}^{ - 11}}\;{{\rm{m}}^{\rm{2}}}{\rm{/s}}}}\\t &= 0.99\;{\rm{s}}\\\end{aligned}\)

Hence, the required time for the diffusion is 0.99 s.

The diffusion speed is:

\(\begin{aligned}{c}{v_d} &= \frac{{\Delta x}}{t}\\ &= \frac{{25 \times {{10}^{ - 6}}\;{\rm{m}}}}{{0.99\;{\rm{s}}}}\\ &= 2.5 \times {10^{ - 5}}\;{\rm{m/s}}\end{aligned}\)

The rms speed of the glycine molecules is:

\(\begin{aligned}{c}{v_{{\rm{rms}}}} &= \sqrt {\frac{{3kT}}{m}} \\ &= \sqrt {\frac{{3 \times \left( {1.38 \times {{10}^{ - 34}}\;{\rm{J/K}}} \right) \times \left( {293\;{\rm{K}}} \right)}}{{75 \times 1.66 \times {{10}^{ - 27}}\;{\rm{kg}}}}} \\ &= 312\;{\rm{m/s}}\end{aligned}\)

Therefore, the rms speed is much greater than the diffusion speed.