Dilute Solutions

Excerpts

• Dilute Solutions

  *the concentration of the dissolved gas is proportional to the pressure of the free gas on the solution* (Henry’s law).

• Dilute Solutions

  Van’t Hoff was the first to calculate $L$ by means of this equation from the solubility of succinic acid at $0°$ C.

• Dilute Solutions

  Conversely, the dissociation of the sodium acetate increases on the addition of water, but the concentration of the free ions decreases, because they are distributed over a larger quantity of water.

• Dilute Solutions

  Since the addition of silver nitrate increases the number of the Ag+-ions, it diminishes the number of the BrO3--ions, and thereby the solubility of the bromate, which is evidently measured by the sum $c_{1} + c_{4}$.

• Dilute Solutions

  In order to distribute the solvent so that the concentration of the common ion CH3- . COO may be the same in both solutions, some water must be withdrawn from the less dissociated electrolyte (acetic acid), and added to the more strongly dissociated (Na-acetate).

• Dilute Solutions

  The concentration of the Ag+-ions is inversely proportional to the concentration of the BrO3--ions. Since the addition of silver nitrate increases the number of the Ag+-ions, it diminishes the number of the BrO3--ions, and thereby the solubility of the bromate, which is evidently measured by the sum $c_{1} + c_{4}$.

• Dilute Solutions

  or, *the two solutions are isohydric if the concentration of the common ion *CH3- COO* is the same in both*.

• Dilute Solutions

  It follows, then, that when two equally diluted solutions of binary electrolytes are mixed, the dissociation of the more weakly dissociated recedes, while that of the more strongly dissociated increases still further.

• Dilute Solutions

  It assigns to each kind of molecule in the two phases a constant ratio of distribution, which is independent of the presence of other dissolved molecules.

• Dilute Solutions

  When chemical interchanges between the different substances in solution are possible, as, *e.g.*, in a solution of dissociating salts and acids with common ions, the term *degree of dissociation* has no meaning, for the ions may be combined arbitrarily into dissociated molecules.

• Dilute Solutions

  On account of incomplete experimental data, however, the calculation of $\Psi$ can be performed, besides for a gaseous phase, only for a *dilute solution*, *i.e.* for a phase in which one kind of molecule far outnumbers all the others in the phase.

• Dilute Solutions

  Physically speaking, this means that the properties of a dilute solution, besides depending on the interactions between the molecules of the solvent, necessarily depend only on the interactions between the molecules of the solvent and the molecules of the dissolved substances, but not on the interactions of the dissolved substances among themselves, for these are small quantities of a higher order.

• Dilute Solutions

  We may therefore enunciate the following proposition: *Further dilution of a dilute solution, if no chemical changes accompany the process, produces neither an appreciable change of volume nor an appreciable heat effect*; or, in other words, *any change of volume or any heat effect produced by further dilution of a dilute solution is due to chemical transformations among the molecules of the dissolved substances*.

• Dilute Solutions

  The influence of the temperature on $K$, and therewith on the condition of equilibrium towards a certain chemical reaction, is controlled by the heat effect of that reaction, and the influence of the pressure is controlled by the corresponding change of volume of the system.

• Dilute Solutions

  It also follows from it that absolutely semipermeable membranes are non-existent, for the substance of any membrane would, in time, become saturated with the molecules of all the various kinds of substances in contact with one side of it, and thus give up each kind of substance to the other side.

• Dilute Solutions

  We must not neglect any kind of molecule until we have ascertained by a particular experiment that its quantity is inappreciable.

• Dilute Solutions

  This number represents the ratio of the number of dissociated molecules to the total number of molecules.

• Dilute Solutions

  *i.e.* *the concentration of the dissolved gas is proportional to the pressure of the free gas on the solution* (Henry’s law).

• Dilute Solutions

  *For every kind of molecule, which possesses the saute molecular weight in both phases, there is a constant ratio of distribution, which is independent of the presence of other molecules* (Nernst’s law of distribution).

• Dilute Solutions

  It may be well therefore to emphasize this fact, that nothing concerning the molecular weight of the solvent can be inferred from the relative lowering of the vapour pressure, any more than from its boiling point, freezing point, or osmotic pressure.

• Dilute Solutions

  Should the number calculated from such a measurement disagree with the number calculated from the percentage composition of the solution on the assumption of normal molecules, some chemical change of the dissolved molecules must have taken place by dissociation, association, hydrolysis, or the like.

• Dilute Solutions

  Conversely, a disagreement between the depression of the freezing point as calculated from the conductivity, and as observed, is not in itself an objection to the theory, but rather to the assumptions made in the calculation concerning the kinds of molecules present.

• Dilute Solutions

  Raoult was the first to establish rigorously by experiment the relation between the depression of the freezing point and the number of the molecules of the dissolved substance; and van’t Hoff gave a thermodynamical explanation and generalization of it by means of his theory of osmotic pressure. Application to electrolytes was rendered possible by Arrhenius’ theory of electrolytic dissociation.

• Dilute Solutions

  Thomsen found the heat effect of the absorption of one gram molecule of carbon dioxide to be $5880~\Unit{cal}$.

Equations

• Dilute Solutions

  U &= n_{0} u_{0} + n_{1} u_{1} + n_{2} u_{2} + \dots\Add{,}

  The internal energy of a dilute solution is the sum over molecule kinds of each kind's number times its per-molecule energy, which is linear in the numbers.

• Dilute Solutions

  V &= n_{0} v_{0} + n_{1} v_{1} + n_{2} v_{2} + \dots\Add{.}

  The volume of a dilute solution is the sum over molecule kinds of each kind's number times its per-molecule volume.

• Dilute Solutions

  \frac{U}{n_{0}} = u_{0} + u_{1}\, \frac{n_{1}}{n_{0}} + u_{2}\, \frac{n_{2}}{n_{0}} + \dots\Add{,}

  The energy per solvent molecule is a linear function of the ratios of dissolved to solvent molecule numbers.

• Dilute Solutions

  \frac{U}{n_{0}} = u_{0} + u_{1}\, \frac{n_{1}}{n_{0}} + \dots + u_{11} \left(\frac{n_{1}}{n_{0}}\right)^{2} + 2u_{12}\, \frac{n_{1}}{n_{0}} · \frac{n_{2}}{n_{0}} + u_{22} \left(\frac{n_{2}}{n_{0}}\right)^{2} + \dots\Add{.}

  A more accurate expansion of energy per solvent molecule including quadratic terms for the dissolved substances' mutual interactions.

• Dilute Solutions

  V' = (n_{0} + 1) v_{0} + n_{1} v_{1} + n_{2} v_{2} + \dots

  After adding one solvent molecule, the solution volume becomes the old sum with n_0 increased by one.

• Dilute Solutions

  U' = (n_{0} + 1) u_{0} + n_{1} u_{1} + n_{2} u_{2} + \dots\Add{.}

  After adding one solvent molecule, the energy becomes the old sum with n_0 increased by one.

• Dilute Solutions

  U' - (U + u_{0}) + p \bigl\{V' - (V + v_{0})\bigr\}

  The heat absorbed on adding one solvent molecule at constant temperature and pressure, by the first law; it vanishes for a dilute solution.

• This equation is in Proof (Proof)

• Dilute Solutions

  d\phi_{0} = \frac{du_{0} + p\, dv_{0}}{\theta}

  The function phi_0 of temperature and pressure has differential equal to the solvent's energy plus pressure-volume differential over temperature.

• Dilute Solutions

  \Phi = n_{0} \phi_{0} + n_{1} \phi_{1} + n_{2} \phi_{2} + \dots + C,

  The entropy of a dilute solution is the sum of molecule numbers times their phi functions plus an integration constant depending only on the molecule numbers.

• Dilute Solutions

  C = n_{0} (k_{0} - R \log c_{0}) + n_{1} (k_{1} - R \log c_{1}) + \dots\Add{.}

  The integration constant C is fixed by matching the solution to the ideal-gas mixture, giving a concentration-dependent expression.

• Dilute Solutions

  c_{0} = \frac{n_{0}}{n_{0} + n_{1} + n_{2} + \dots}

  The concentration of the solvent is its molecule number divided by the total number of molecules.

• Dilute Solutions

  \Phi = n_{0} (\phi_{0} + k_{0} - R \log c_{0}) + n_{1} (\phi_{1} + k_{1} - R \log c_{1}) + \dots\Add{.}

  The entropy of a dilute solution written in terms of concentrations.

• Dilute Solutions

  \phi_{0} + k_{0} - \frac{u_{0} + pv_{0}}{\theta} &= \varphi_{0}\Add{,}

  Defines phi_0 (here written varphi_0) as a function of temperature and pressure only.

• Dilute Solutions

  \Psi = n_{0} (\varphi_{0} - R \log c_{0}) &+ n_{1} (\varphi_{1} - R \log c_{1}) \\ &+ n_{2} (\varphi_{2} - R \log c_{2}) + \dots\Add{.}

  The Psi function of a dilute solution: the sum over molecule kinds of number times (phi minus R log concentration). This determines the thermodynamic properties of a dilute solution.

• This equation is in System of any Number of Independent Constituents (System of any Number of Independent Constituents)

• Dilute Solutions

  \tsum \nu_{0} \log c_{0} + \nu_{1} \log c_{1} + \nu_{2} \log c_{2} + \dots &= \frac{1}{R} \tsum \nu_{0} \varphi_{0} + \nu_{1} \varphi_{1} + \dots \\ &= \log K.

  Equilibrium condition for a chemical change: the weighted sum of log concentrations equals log K, a constant independent of molecule numbers.

• Dilute Solutions

  \frac{\dd \log K}{\dd \theta} = \frac{L}{R\theta^{2}}

  The temperature dependence of the equilibrium constant is set by the heat absorbed by the reaction.

• Dilute Solutions

  \frac{\dd \log K}{\dd p} = -\frac{s}{R\theta}\Add{.}

  The pressure dependence of the equilibrium constant is set by the volume change of the reaction.

• Dilute Solutions

  s = \tsum \nu_{0} v_{0} + \nu_{1} v_{1} + \nu_{2} v_{2} + \dots

  The volume increase of a reaction is the weighted sum of molecular volumes.

• Dilute Solutions

  L = \tsum (\nu_{0} u_{0} + \nu_{1} u_{1} + \dots) + p(\nu_{0} v_{0} + \nu_{1} v_{1} + \dots);

  The heat absorbed in the change equals the energy change plus pressure times the volume change, by the first law.

• Dilute Solutions

  \log K = \log a - \frac{b}{\theta} + (\nu_{1} + \nu_{2} + \dots) \log \frac{\theta}{p}.

  Explicit form of log K for a reaction with constants a and b, which recovers the earlier special equations.

• Dilute Solutions

  2\, \frac{\dd \log c_{1}}{\dd \theta} = \frac{1}{R} · \frac{L}{\theta^{2}}.

  For the water dissociation H2O to H+ and OH-, the temperature derivative of log c1 relates to the heat of dissociation.

• Dilute Solutions

  -\log c_{0} + \log c_{1} + \log c_{2} = K

  Equilibrium condition for the dissociation of water into H+ and OH- ions.

• Dilute Solutions

  L = \frac{4045000}{\theta}

  Thomsen's measured heat of neutralization, used as the heat of dissociation of water, as a function of mean temperature (in calories).

• Dilute Solutions

  c_{1} = C e^{-\efrac{513000}{\theta^{2}}}

  Integrated relation giving the dissociation concentration of water as a function of temperature with an integration constant C.

• Dilute Solutions

  c_{1} = 6.1 e^{-\efrac{513000}{\theta^{2}}} × 10^{-7}

  Degree of dissociation of water at any temperature, with the constant fixed by the measured value at 18 degrees C.

• Dilute Solutions

  \frac{c_{2}^{2}}{c_{1}} = K\Add{.}

  Equilibrium of the dissociation of a binary electrolyte: the square of the ion concentration over the undissociated concentration is constant (Ostwald's dilution law).

• Dilute Solutions

  K = \dfrac{\lambda_{v}}{\lambda_{\infty} (\lambda_{\infty} - \lambda_{v})^{v}}

  Ostwald's law of dilution of binary electrolytes in terms of molecular conductivities (from the footnote).

• Dilute Solutions

  c_{1} + c_{2} = c

  The total concentration of undissociated and dissociated acid molecules is given.

• Dilute Solutions

  \frac{c_{2} c_{3}}{c_{1}} = K

  First dissociation equilibrium of sulphuric acid, H2SO4 into H+ and HSO4-.

• Dilute Solutions

  \frac{c_{2} c_{4}}{c_{3}} = K'

  Second dissociation equilibrium of the HSO4- ion into H+ and SO4 ions.

• Dilute Solutions

  2c_{4} + c_{3} = c_{2}

  Conservation of SO4 radicals and hydrogen atoms: the ion count condition for two independent constituents.

• Dilute Solutions

  c_{1} + c_{3} + c_{4} = c

  The quantity of sulphuric acid in the solution is given, summing its undissociated and dissociated forms.

• Dilute Solutions

  \frac{\dd \varphi_{0}}{\dd \theta} = \frac{u_{0} + pv_{0}}{\theta^{2}};\quad

  Temperature derivative of the solvent's phi function (stated with its pressure-derivative companion).

• Dilute Solutions

  \frac{\dd \varphi_{0}}{\dd p} = -\frac{v_{0}}{\theta}

  Pressure derivative of the solvent's phi function is minus the molecular volume over temperature.

• Dilute Solutions

  -\log c_{1} = \log K

  At fixed temperature and pressure, the equilibrium condition fixes the concentration of the dissolved gas in the solution.

• Dilute Solutions

  c_{1} = \frac{n_{1}}{n_{0} + n_{1}}

  The concentration of the dissolved gas in the liquid is its number of molecules divided by all molecules in the liquid phase.

• Dilute Solutions

  \frac{\dd \log c_{1}}{\dd p} = \frac{1}{R} · \frac{s}{\theta}

  The change of the logarithm of the dissolved-gas concentration with pressure, at constant temperature.

• Dilute Solutions

  \frac{\dd \log c_{1}}{\dd \theta} = -\frac{1}{R} · \frac{L}{\Erratum{\theta_{2}}{\theta^{2}}}

  The change of the logarithm of the dissolved-gas concentration with temperature. The book marks the denominator as an erratum: printed as theta-2, corrected to theta-squared; the sympy form is withheld because the erratum is not resolved in the source text.

• Dilute Solutions

  c_{1} = Cp

  The concentration of the dissolved gas is proportional to the pressure of the free gas on the solution.

• Dilute Solutions

  L = -\frac{R \theta^{2}}{C} · \frac{\dd C}{\dd \theta}

  The heat effect of absorption of the gas from the solution can be calculated from the temperature variation of the solubility factor C.

• Dilute Solutions

  s = \frac{R\theta}{p}

  The volume of one gram molecule of a perfect gas at temperature theta and pressure p, taken from the gas equation (16).

• Dilute Solutions

  \frac{\dd \log c_{1}}{\dd p} = \frac{1}{p}

  Substituting the perfect-gas volume into the pressure derivative gives the logarithmic derivative of the concentration as one over the pressure.

• Dilute Solutions

  \log c_{1} = \frac{L}{R\theta} + \const

  If the heat effect L is independent of temperature, integrating the temperature relation gives the logarithm of the concentration as a constant plus L over R theta.

• Dilute Solutions

  L = -R\theta^{2} \frac{\dd \log c_{1}}{\dd \theta}

  The heat effect of precipitating one gram molecule of a salt from a saturated solution is obtained from the temperature variation of the solubility (van't Hoff's use on succinic acid).

• Dilute Solutions

  \frac{c_{2}^{2}}{c_{1}} = K'

  In a saturated solution of a dissociating salt, the dissociated and undissociated molecule concentrations satisfy a constant equilibrium relation at given temperature and pressure.

• Dilute Solutions

  \frac{n_{1} + n_{2} + n_{3} + \dots}{n_{0}} = \log K

  For a solvent passing to the vapour phase, the ratio of dissolved molecules to solvent molecules equals log K, which is therefore a small quantity.

• Dilute Solutions

  \frac{n_{1} + n_{2} + n_{3} + \dots}{n_{0}} = \frac{1}{R} \left(\frac{m_{0}}{m_{0}'}\, \varphi_{0}' - \varphi_{0}\right)

  The ratio of dissolved molecules to solvent molecules is given by the difference of the solvent's potential functions in the two phases, scaled by the molecular weights.

• Dilute Solutions

  \theta - \theta_{0} = \frac{R\theta^{2}}{n_{0} L} (n_{1} + n_{2} + n_{3} + \dots)

  The elevation of the boiling point of a dilute solution follows from the number of dissolved molecules, the temperature, and the heat of vaporization.

• Dilute Solutions

  \theta - \theta_{0} = \frac{c \theta^{2} \varphi}{L}

  The elevation of the boiling point from the general theory, in terms of the mass ratio of dissolved substance to solvent and the quantity phi.

• Dilute Solutions

  \varphi = \frac{R(n_{1} + n_{2} + \dots)}{n_{1} m_{1} + n_{2} m_{2} + \dots}

  The two theories agree only if phi takes this molecular value, which relates it to the number of dissolved molecules and their molecular weights.

• Dilute Solutions

  c\varphi = \frac{R(n_{1} + n_{2} + n_{3} + \dots)}{n_{0} m_{0}}

  For dilute solutions the product of the mass ratio and phi reduces to a quantity set by the number of dissolved molecules and the solvent's mass.

• Dilute Solutions

  p_{0} - p = \frac{R\theta}{n_{0}s} (n_{1} + n_{2} + n_{3} + \dots)

  For dilute solutions, the lowering of the vapour pressure equals a quantity proportional to the total number of dissolved molecules.

• Dilute Solutions

  p_{0} - p = \frac{m_{0}'p (n_{1} + n_{2} + \dots)}{n_{0} m_{0}}

  If the solvent vapour is a perfect gas and the solution's volume is negligible, the lowering of the vapour pressure is given in terms of the molecular weights and the number of dissolved molecules.

• Dilute Solutions

  \frac{p_{0} - p}{p} = (n_{1} + n_{2} + n_{3} + \dots)\, \frac{m_{0}'}{n_{0} m_{0}}

  The relative lowering of the vapour pressure of a dilute solution is set by the number of dissolved molecules; the book notes the common form, which holds only when the solvent's molecular weight is the same in liquid and vapour (m_0 = m_0').

• Dilute Solutions

  \theta_{0}' - \theta' = \frac{R \theta^{2}}{n_{0} L'} (n_{1} + n_{2} + n_{3} + \dots)

  The depression of the freezing point of a dilute solution is proportional to the number of dissolved molecules.

• Dilute Solutions

  P = \frac{R\theta}{n_{0} m_{0} v} (n_{1} + n_{2} + n_{3} + \dots)

  The osmotic pressure of a dilute solution is set by the number of dissolved molecules and the temperature.

• Dilute Solutions

  P = \frac{R\theta}{V} (n_{1} + n_{2} + n_{3} + \dots)

  The osmotic pressure has the same form as the characteristic equation of a mixture of perfect gases, with the dissolved molecules as the gas.

• Dilute Solutions

  c = \frac{n_{1} m_{1} + n_{2} m_{2} + \dots}{n_{0} m_{0}}

  The mass ratio of dissolved substance to solvent is expressed in molecule numbers and molecular weights.

• Dilute Solutions

  c_{1} + c_{2} + \dots + \log c_{0}' = \log K

  For a solvent evaporating from a liquid solution, the concentrations of dissolved molecules in the liquid and the solvent's concentration in the vapour are related by the equilibrium constant.

• Dilute Solutions

  \log c_{0}' = \log K

  When the solvent's vapour molecules far outnumber others, the solvent's vapour concentration does not depend on the composition of the solution, so the partial pressure of the solvent equals that of the pure solvent.

• Dilute Solutions

  (c_{1} + c_{2} + \dots) - (c_{1}' + c_{2}' + \dots) = \log K

  When the dissolved substance also passes into the vapour, the boiling-point elevation and vapour-pressure lowering depend on the difference of the concentrations in liquid and vapour, not on the liquid concentrations alone.

• Dilute Solutions

  \frac{c_{1}'}{c_{1}} = K

  For each kind of molecule with the same molecular weight in both phases, the ratio of its concentrations in the two phases is a constant independent of other molecules present.

• Dilute Solutions

  \frac{n_{2}^{2}}{n_{1} n_{0}} = K

  For the dissociation of a weak electrolyte in a dilute solution, the ion count squared over the undissociated count and solvent count is a constant at fixed temperature and pressure.

• Dilute Solutions

  \frac{n_{2}'^{2}}{n_{1}' n_{0}'} = K'

  The same dissociation equilibrium holds for the second solution, with its own counts.

• Dilute Solutions

  \bar{n}_{0} = n_{0} + n_{0}'

  After mixing, the number of water molecules is the sum of the water molecules of the two solutions.

• Dilute Solutions

  \bar{n}_{2} + \bar{n}_{4} = n_{1}' + n_{2}'

  After mixing, the sodium atoms are conserved: their number equals that of the sodium-bearing molecules and ions of the second solution.

• Dilute Solutions

  \bar{n}_{1} + \bar{n}_{3} = n_{1} + n_{2}

  After mixing, the hydrogen atoms are conserved across acetic acid and H+ ions.

• Dilute Solutions

  \bar{n}_{3} + \bar{n}_{4} = \bar{n}_{5}

  Electrical neutrality: the number of positive ions equals the number of negative ions in the mixed solution.

• Dilute Solutions

  \frac{\bar{c}_{3} \bar{c}_{5}}{\bar{c}_{1}} = K

  At equilibrium, the dissociation of acetic acid in the mixed solution satisfies a constant ratio of ion to undissociated concentrations.

• Dilute Solutions

  \frac{\bar{c}_{4} \bar{c}_{5}}{\bar{c}_{2}} = K'

  At equilibrium, the dissociation of sodium acetate in the mixed solution satisfies a constant ratio of ion to undissociated concentrations.

Problems

No exercises in this chapter.