System of any Number of Independent Constituents
Excerpts
System of any Number of Independent Constituents
An aqueous solution of sulphuric acid forms a system of three chemical elements, S, H, and O, but contains only two independent constituents, for, in each phase (*e.g.* liquid, vapour, solid) the mass of O depends on that of S and H, while the masses of S and H are not in each phase interdependent.
System of any Number of Independent Constituents
The composition of all the phases is then completely determined by a single variable, *e.g.* the temperature or the pressure. This case is generally called *perfect heterogeneous* equilibrium.
System of any Number of Independent Constituents
The question as to the number of the independent constituents has nothing at all to do with the chemical constitution of the substances in the different phases, in particular, with the number of different kinds of molecules.
System of any Number of Independent Constituents
Thus, a quantity of water in any number of states forms but one independent constituent, however many associations and dissociations of H2O molecules may occur (it may be a mixture of hydrogen and oxygen or ions), for the mass of the oxygen in each phase is completely determined by that of the hydrogen, and *vice versâ*. Should, however, an excess of oxygen or hydrogen be present in the vapour, we have then two independent constituents.
System of any Number of Independent Constituents
The number of the phases, therefore, cannot exceed the number of the independent constituents by more than two; or, a system of $\alpha$ independent constituents will contain at most $(\alpha + 2)$ phases.
System of any Number of Independent Constituents
This means that the heat effect in a variation that leaves the composition of all phases unchanged, divided by the change of volume of the system and by the absolute temperature, gives the rate of change of the equilibrium pressure with the temperature. Where application of heat increases the volume, as in the case of evaporation, the equilibrium pressure increases with temperature; in the opposite case, as in the melting of ice, it decreases with increase of temperature.
System of any Number of Independent Constituents
If, finally, we dissolve salt sufficient for saturation in the newly formed unit of water, at constant temperature $\theta$ and constant pressure $p$, the sum of the heat and work is simply the heat of solution
System of any Number of Independent Constituents
This means that the relative decrease of the vapour pressure is proportional to the concentration of the solution (Wüllner’s law).
System of any Number of Independent Constituents
The error committed in putting the rate of diffusion of a salt through such a membrane equal to zero, falls below all measurable limits.
System of any Number of Independent Constituents
This proposition furnishes a means of distinguishing between a solution and an emulsion. In an emulsion the number of particles suspended in the solution has no influence on the vapour pressure.
System of any Number of Independent Constituents
Since $\Delta$ is *small* for small values of $c$ (dilute solutions, 97), then, according to % [eqn:(178)](178)%, the ratio of the vapour pressure of a dilute solution of fixed concentration to the vapour pressure of the pure solvent is practically independent of the temperature (Babo’s law).
System of any Number of Independent Constituents
This process may be accomplished directly, or in two steps, viz. by condensing unit mass of water vapour into pure water, and then dissolving the salt in the water.
System of any Number of Independent Constituents
Since $\varphi$ is always positive (217), the vapour pressure must decrease with increasing concentration. This proposition furnishes a means of distinguishing between a solution and an emulsion. In an emulsion the number of particles suspended in the solution has no influence on the vapour pressure.
System of any Number of Independent Constituents
Our last equations, therefore, connect in a perfectly general way the laws regarding the lowering of the vapour pressure, the elevation of the boiling temperature, the depression of the freezing point, and the change of the saturation point. Only one of these phenomena need be experimentally investigated in order to calculate $\varphi$, and by means of the value thus determined the others may be deduced for the same solution.
System of any Number of Independent Constituents
It is true that for no solution can perfectly *semipermeable* membranes of this character be manufactured. In fact, the further development of this theory (259) will exclude them as a matter of principle, for in every case the dissolved substance will also diffuse through the membrane, though possibly at an extremely slow rate.
System of any Number of Independent Constituents
Since $\varphi$ is positive, the osmotic pressure increases with increasing concentration, and also, since $p' - p''$ vanishes when $c = 0$, the osmotic pressure is necessarily positive.
System of any Number of Independent Constituents
A better insight into the nature of these quantities is gained by extending to the liquid state the idea of the molecule, hitherto applied only to gases.
System of any Number of Independent Constituents
The number of phases as well as the states of aggregation is quite arbitrary, although we at once recognize the fact that a system in equilibrium may consist of any number of solid and liquid phases, but only one single *gaseous* phase, for two different gases in contact are never in equilibrium with one another.
System of any Number of Independent Constituents
We define the number of independent constituents as follows. First find the number of elements contained in the system, and from these discard, as dependent constituents, all those whose quantity is determined in each phase by the remaining ones. The number of the remaining elements will be the number of independent constituents of the system.
System of any Number of Independent Constituents
There are for each independent constituent $(\beta - 1)$ equations, which must be satisfied, and therefore for all the $\alpha$ independent constituents $\alpha (\beta - 1)$ conditions.
System of any Number of Independent Constituents
For water it was shown in 187, that at the triple point the temperature is $0.0074°$ C., and the pressure $4.62~\Unit{mm.}$ of mercury.
Equations
System of any Number of Independent Constituents
\lambda = \frac{R}{m} \theta^{2} · \frac{d \log \dfrac{p}{p_{0}}}{d\theta}The heat evolved when salt sufficient for saturation dissolves in one gram of pure water equals R/m times θ² times the temperature derivative of log(p/p₀).
System of any Number of Independent Constituents
c' = \frac{M_{2}'}{M_{1}'}The concentration of the second constituent in the first phase is the ratio of its mass to the mass of the first constituent in that phase.
System of any Number of Independent Constituents
M_{1}'\, \frac{\dd^{2} \Psi'}{\dd M_{1}'\, \dd M_{2}'} = \varphi'The quantity φ' is defined as M₁' times the mixed second derivative of Ψ' with respect to M₁' and M₂'.
System of any Number of Independent Constituents
\frac{\dd^{2} \Psi'}{\dd M_{1}'^{2}} = -\frac{M_{2}'}{M_{1}'^{2}} · \varphi'The second derivative of Ψ' with respect to M₁' equals minus M₂'/M₁'² times φ'.
System of any Number of Independent Constituents
\delta^{2} \Psi < 0For stable equilibrium at constant temperature and pressure the second variation of Ψ is negative.
System of any Number of Independent Constituents
\frac{L_{1}}{\theta^{2}}\, d\theta - \frac{s_{1}}{\theta}\, dp - \varphi'\, dc' + \varphi''\, dc'' = 0For the first constituent passing between the two phases, the displacement of equilibrium links dθ, dp, dc' and dc''.
System of any Number of Independent Constituents
\frac{L_{2}}{\theta^{2}}\, d\theta - \frac{s_{2}}{\theta}\, dp - \varphi'\, \frac{dc'}{c'} + \varphi''\, \frac{dc''}{c''} = 0For the second constituent passing into the second phase, the displacement of equilibrium links dθ, dp, dc' and dc''.
System of any Number of Independent Constituents
dp = \frac{\left(\dfrac{c''}{c'} - 1\right) \theta \varphi'\, dc'}{s_{1} + c'' s_{2}}Along an isotherm the change of vapour pressure with concentration of the liquid is given by this expression.
System of any Number of Independent Constituents
dc'' = \frac{\left(\dfrac{1}{s_{1}} + \dfrac{1}{c's_{2}}\right)}{\left(\dfrac{1}{s_{1}} + \dfrac{1}{c'' s_{2}}\right)} · \frac{\varphi'}{\varphi''}\, dc'Along an isotherm the change of the vapour concentration is proportional to the change of the liquid concentration, so both concentrations change in the same sense.
System of any Number of Independent Constituents
c'' = 0The second constituent occurs only in the first phase, so its concentration in the second phase is zero.
System of any Number of Independent Constituents
\frac{L}{\theta^{2}}\, d\theta - \frac{s}{\theta}\, dp - \varphi\, dc = 0For a solution in contact with the pure solvent vapour, the displacement of equilibrium links dθ, dp and dc.
System of any Number of Independent Constituents
\left(\frac{\dd p}{\dd \theta}\right)_{c} = \frac{L}{\theta · s}At constant concentration the rate of change of vapour pressure with temperature equals L divided by θ times s.
System of any Number of Independent Constituents
s = v = \frac{R}{m} · \frac{\theta}{p}Assuming Boyle's and Gay-Lussac's laws for the vapour, its specific volume equals R/m times θ/p.
System of any Number of Independent Constituents
\Delta = \frac{R}{m} \theta^{2} \left(\frac{\dd \log \dfrac{p}{p_{0}}}{\dd \theta}\right)_{c}The heat of dilution equals R/m times θ² times the temperature derivative of log(p/p₀) at constant concentration.
System of any Number of Independent Constituents
\left(\frac{\dd p}{\dd c}\right)_{\theta} = -\frac{\theta\varphi}{s}At constant temperature the vapour pressure of a solution falls as its concentration rises.
System of any Number of Independent Constituents
\frac{p - p_{0}}{p} = \frac{cm\varphi}{R}The relative decrease of the vapour pressure is proportional to the concentration of the solution.
System of any Number of Independent Constituents
\left(\frac{\dd \theta}{\dd c}\right)_{p} = \frac{\theta^{2} \varphi}{L}At constant pressure the boiling point rises with the concentration of the solution.
System of any Number of Independent Constituents
\theta - \theta_{0} = \frac{c\theta^{2} \varphi}{L}For dilute solutions the elevation of the boiling point is proportional to the concentration.
System of any Number of Independent Constituents
\left(\frac{\dd \theta'}{\dd c}\right)_{p} = -\frac{\theta^{2} \varphi}{L'}At constant pressure the freezing or saturation point falls as the concentration of the solution rises, when L' is the heat of solidification.
System of any Number of Independent Constituents
\theta_{0}' - \theta' = \frac{c\theta^{2} \varphi}{L'}For dilute solutions the lowering of the freezing point is proportional to the concentration.
System of any Number of Independent Constituents
\frac{L}{\theta^{2}}\, d\theta - \frac{s'}{\theta}\, dp' - \frac{s''}{\theta}\, dp'' - \varphi\, dc = 0For a solution separated from pure solvent by a semipermeable membrane, the displacement of equilibrium links dθ, dp', dp'' and dc.
System of any Number of Independent Constituents
\left(\frac{\dd P}{\dd c}\right)_{\theta} = -\frac{\theta\varphi}{s'}At constant temperature the osmotic pressure increases with the concentration of the solution, since s' is negative.
System of any Number of Independent Constituents
P = \frac{c\theta\varphi}{v}For small concentrations the osmotic pressure equals c times θ times φ divided by the specific volume v of the solution.
System of any Number of Independent Constituents
\Psi = \Phi - \frac{U + pV}{\theta}The function Psi is defined as the entropy minus the sum of energy and pressure-volume term, divided by the absolute temperature.
System of any Number of Independent Constituents
\delta \Psi = 0The condition of thermodynamic equilibrium at constant temperature and pressure is that Psi does not change under any variation compatible with the given conditions.
System of any Number of Independent Constituents
\Psi = \Psi' + \Psi'' + \dots +\Psi^{\beta}The function Psi of the whole system is the sum of the contributions of its beta phases.
System of any Number of Independent Constituents
\Psi' = \Phi' - \frac{U' + pV'}{\theta}For a single phase, Psi' is defined as its entropy minus its energy plus pressure-volume term, divided by the temperature.
System of any Number of Independent Constituents
\Psi' = \frac{\dd \Psi'}{\dd M_{1}'}\, M_{1}' + \frac{\dd \Psi'}{\dd M_{2}'}\, M_{2}' + \dots + \frac{\dd \Psi'}{\dd M_{\alpha}'}\, M_{\alpha}'.Because Psi' is homogeneous of the first degree in the masses, it equals the sum of each mass times its partial derivative with respect to that mass.
System of any Number of Independent Constituents
\frac{\dd \Psi'}{\dd M_{1}'} = \frac{\dd \Psi''}{\dd M_{1}''} = \dots = \frac{\dd \Psi^{\beta}}{\dd M_{1}^{\beta}}In equilibrium, the partial derivative of Psi with respect to the mass of the first constituent takes the same value in every phase.
System of any Number of Independent Constituents
M_{\alpha} = M_{\alpha}' + M_{\alpha}'' + \dots + M_{\alpha}^{\beta}The total mass of each independent constituent is the sum of its masses in all the phases.
System of any Number of Independent Constituents
\alpha\beta + 2The state of the beta phases depends on alpha times beta masses plus temperature and pressure, which are alpha beta + 2 variables.
System of any Number of Independent Constituents
\bigl[(\alpha - 1)\beta + 2\bigr] - \bigl[\alpha (\beta - 1)\bigr] = \alpha - \beta + 2After the internal conditions are satisfied, alpha minus beta plus two internal variables remain undetermined.
System of any Number of Independent Constituents
\beta \leq \alpha + 2The number of phases cannot exceed the number of independent constituents by more than two.
System of any Number of Independent Constituents
\delta\, d\Psi = 0The condition of equilibrium for a neighbouring equilibrium state requires the variation of the infinitesimal change dPsi to vanish.
System of any Number of Independent Constituents
\frac{\dd \Psi}{\dd \theta} = \frac{U + pV}{\theta^{2}}The partial derivative of Psi with respect to temperature equals energy plus pressure-volume term divided by the square of the temperature.
System of any Number of Independent Constituents
\frac{\dd \Psi}{\dd p} = -\frac{V}{\theta}The partial derivative of Psi with respect to pressure equals minus the volume divided by the temperature.
System of any Number of Independent Constituents
d\Phi' = \frac{dU' + p\, dV'}{\theta}For an infinitely small change of a phase without change of its masses, the change in entropy is the heat term divided by temperature.
System of any Number of Independent Constituents
\Delta \Psi = \eps\Psi'When all masses are increased in the ratio 1 + epsilon to 1, Psi increases by epsilon times Psi'.
System of any Number of Independent Constituents
\frac{dp}{d\theta} = \frac{Q}{\theta\, \delta V}The rate of change of the equilibrium pressure with temperature equals the heat absorbed divided by the temperature and by the change of volume.
System of any Number of Independent Constituents
L = \theta\, \frac{dp}{d\theta} (v'' - v')The heat of vaporization per unit mass equals temperature times the slope of the vapour pressure curve times the difference of the specific volumes of vapour and liquid.
System of any Number of Independent Constituents
Q = \theta · \frac{dp}{d\theta} · \delta VFor a salt solution in three phases at constant temperature, pressure and concentration, the heat absorbed equals temperature times the pressure slope times the volume change.
System of any Number of Independent Constituents
\delta V = \bigl[(v'' + cv''') - (1 + c) v'\bigr]\, \delta M_{1}''The volume change on evaporating water from a salt solution while precipitating salt equals the specific-volume combination times the mass of vapour formed.
System of any Number of Independent Constituents
L = \theta\, \frac{dp}{d\theta} \bigl(v'' + cv''' - (1 + c)v'\bigr)The heat needed to evaporate water from a solution and precipitate salt equals temperature times the pressure slope times the specific-volume combination.
System of any Number of Independent Constituents
v'' = \frac{R}{m} · \frac{\theta}{p}The specific volume of the vapour, treated as a perfect gas, equals the gas constant divided by the molecular weight, times temperature over pressure.
System of any Number of Independent Constituents
L = \frac{R}{m} \theta^{2} · \frac{d \log p}{d\theta}With the vapour treated as a perfect gas and the liquid volume neglected, the heat of vaporization equals gas constant over molecular weight times the square of temperature times the derivative of log pressure with respect to temperature.
Problems
No exercises in this chapter.