13.13 A gas in equilibrium has uniform density and pressure throughout its volume. This is strictly true only if there are no external influences. A gas column under gravity, for example, does not have uniform density (and pressure). As you might expect, its density decreases with height. The precise dependence is given by the so-called law of atmospheres

n₂ = n₁ exp [ -mg (h₂ – h₁)/ k_BT]

where n₂, n₁ refer to number density at heights h₂ and h1 respectively. Use this relation to derive the equation for sedimentation equilibrium of a suspension in a liquid column:

n₂ = n₁ exp [ -mg N_A ( $\mathit{\rho }$ - ${\mathit{\rho }}^{\mathit{\text{'}}}$ ) (h₂ –h₁₎/ ( $\mathit{\rho }$ RT)]

where $\mathit{\rho }$ is the density of the suspended particle, and  that of surrounding medium. [N_A is Avogadro’s number, and R the universal gas constant.] [Hint : Use Archimedes principle to find the apparent weight of the suspended particle.]

<p><strong>13.13&nbsp;</strong>According to law of atmospheres, we have</p><p><em>n</em>₂ = <em>n</em>₁ exp [ -<em>mg </em>(<em>h</em>₂ – <em>h</em>₁)/ <em>k</em>_B<em>T</em>] ….(i)</p><p>where&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>n</mi></mrow></mrow><mrow><mrow><mn>1</mn></mrow></mrow></msub></math></span> is the number of density at height&nbsp; <span title="Click to copy mathml"><math><msub><mrow><mrow><mi>h</mi></mrow></mrow><mrow><mrow><mn>1</mn></mrow></mrow></msub></math></span>and <span title="Click to copy mathml"><math><msub><mrow><mrow><mi>n</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msub></math></span> is the number of density at height <span title="Click to copy mathml"><math><msub><mrow><mrow><mi>h</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msub></math></span></p><p>mg is the weight of the particle suspended in the gas column</p><p>Density of the medium = <span title="Click to copy mathml"><math><mi>?</mi></math></span></p><p>Density of the suspended particle = <span title="Click to copy mathml"><math><msup><mrow><mrow><mi>?</mi></mrow></mrow><mrow><mrow><mi>'</mi></mrow></mrow></msup></math></span></p><p>Mass of one suspended particle = m’</p><p>Mass of medium displaced = m</p><p>Volume of the suspended particle = V</p><p>According to Archimedes’s principle for a particle suspended in a liquid column, the effective weight of the suspended particle is given as:</p><p>Weight of the medium displaced – weight of the suspended particle = mg – m’g</p><p>= mg- V <span title="Click to copy mathml"><math><msup><mrow><mrow><mi>?</mi></mrow></mrow><mrow><mrow><mi>'</mi></mrow></mrow></msup><mi>g</mi></math></span> = mg – (<span title="Click to copy mathml"><math><mfrac><mrow><mrow><mi>m</mi></mrow></mrow><mrow><mrow><mi></mi><mi>?</mi></mrow></mrow></mfrac></math></span>&nbsp;) <span title="Click to copy mathml"><math><msup><mrow><mrow><mi>?</mi></mrow></mrow><mrow><mrow><mi>'</mi></mrow></mrow></msup><mi>g</mi></math></span></p><p>= mg(1- <span title="Click to copy mathml"><math><mfrac><mrow><mrow><msup><mrow><mrow><mi>?</mi></mrow></mrow><mrow><mrow><mi>'</mi></mrow></mrow></msup></mrow></mrow><mrow><mrow><mi></mi><mi>?</mi></mrow></mrow></mfrac></math></span>&nbsp;) ……(ii)</p><p>Gas constant R = <span title="Click to copy mathml"><math><msub><mrow><mrow><mi>k</mi></mrow></mrow><mrow><mrow><mi>B</mi></mrow></mrow></msub></math></span>&nbsp;N</p><p><span title="Click to copy mathml"><math><msub><mrow><mrow><mi>k</mi></mrow></mrow><mrow><mrow><mi>B</mi></mrow></mrow></msub><mo>=</mo><mi></mi><mfrac><mrow><mrow><mi>R</mi></mrow></mrow><mrow><mrow><mi>N</mi></mrow></mrow></mfrac></math></span>&nbsp;…(iii)</p><p>Substituting from (ii) and (iii) in (i), we get</p><p><em>n</em>₂ = <em>n</em>₁ exp [ -<em>mg </em>(<em>h</em>₂ – <em>h</em>₁)/ <em>k</em>_B<em>T</em>]</p><p>= n1 exp [<span title="Click to copy mathml"><math><mfenced open="[" close="]" separators="|"><mrow><mrow><mo>-</mo><mi>m</mi><mi>g</mi><mfenced separators="|"><mrow><mrow><mo>-</mo><mfrac><mrow><mrow><msup><mrow><mrow><mi>?</mi></mrow></mrow><mrow><mrow><mi>'</mi></mrow></mrow></msup></mrow></mrow><mrow><mrow><mi>?</mi></mrow></mrow></mfrac></mrow></mrow></mfenced><mo>(</mo><mi>h</mi><mn>2</mn><mi mathvariant="normal"></mi><mo>–</mo><mi mathvariant="normal"></mi><mi>h</mi><mn>1</mn><mo>)</mo><mfrac><mrow><mrow><mi>N</mi></mrow></mrow><mrow><mrow><mi>R</mi><mi>T</mi></mrow></mrow></mfrac></mrow></mrow></mfenced></math></span>]<span title="Click to copy mathml"><br></span></p><p>= n1 exp[<span title="Click to copy mathml"><math><mfenced open="[" close="]" separators="|"><mrow><mrow><mo>-</mo><mi>m</mi><mi>g</mi><mfenced separators="|"><mrow><mrow><mi>?</mi><mo>-</mo><msup><mrow><mrow><mi>?</mi></mrow></mrow><mrow><mrow><mi>'</mi></mrow></mrow></msup><mi></mi></mrow></mrow></mfenced><mo>(</mo><mi>h</mi><mn>2</mn><mi mathvariant="normal"></mi><mo>–</mo><mi mathvariant="normal"></mi><mi>h</mi><mn>1</mn><mo>)</mo><mfrac><mrow><mrow><mi>N</mi></mrow></mrow><mrow><mrow><mi>R</mi><mi>T</mi><mi>?</mi></mrow></mrow></mfrac></mrow></mrow></mfenced></math></span>]</p>

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