Gaseous System

Excerpts

Gaseous System
*The entropy of a mixture of gases is the sum of the entropies which the individual gases would have, if each at the same temperature occupied a volume equal to the total volume of the mixture.* This proposition was first established by Gibbs.
Gaseous System
Experience shows that a gas on both sides of a membrane permeable to it is in equilibrium when its partial pressures (18) are the same on both sides, quite independent of the other gases present.
Gaseous System
Platinum foil at a white heat is permeable to hydrogen, but impermeable to air. If a vessel having a platinum wall be filled with pure hydrogen, and hermetically sealed, and the platinum be then heated, the hydrogen must completely diffuse out against atmospheric pressure. As the air cannot enter, the vessel must finally become completely exhausted.
Gaseous System
It also appears that the increase of the entropy depends solely on the number of the molecules $n_{1}$, $n_{2}$, and not on the nature---*e.g.* the molecular weight, of the diffusing gases. The increase of the entropy does not depend on whether the gases are chemically alike or not.
Gaseous System
It follows that the chemical difference of two gases, or, in general, of two substances, cannot be represented by a continuous variable; but that here we can speak only of a discontinuous relation, either of equality or inequality. This fact involves a fundamental distinction between chemical and physical properties, since the latter may always be regarded as continuous.
Gaseous System
Where the volume remains unchanged, as, *e.g.*, in the dissociation of hydriodic acid, considered below, the equilibrium is independent of the pressure.
Gaseous System
The term containing $b$ refers to the heat spent in the increase of the internal energy; the term containing $\theta$ to that spent in external work.
Gaseous System
There is always present a finite, though perhaps a very small number of all possible kinds of molecules. Thus, in water vapour at any temperature at least a trace of oxygen and hydrogen must be present (see also 259).

Equations

This equation is in System of any Number of Independent Constituents (System of any Number of Independent Constituents)
Gaseous System
V = \frac{R\theta}{p} (n_{1} + n_{2} + \dots) = \frac{R\theta}{p} \tsum n_{1}
The volume of a mixture of perfect gases equals R times temperature over pressure times the total number of molecules.
Gaseous System
U_{1} = \tsum n_{1} (c_{v_{1}}\theta + h_{1})
The energy of a perfect gas depends only on temperature and equals the number of molecules times (molecular heat at constant volume times temperature plus a constant).
Gaseous System
U = \tsum n_{1} (c_{v_{1}}\theta + h_{1})
The total energy of a mixture of perfect gases is the sum of the energies of its constituents.
This equation is in Proof (Proof)
Gaseous System
\Phi = \tsum n_{1} \left(c_{v_{1}} \log \theta + R \log \frac{\theta}{p}\right) + C
The entropy of a perfect gas mixture is a sum over constituents plus a constant of integration that depends only on composition.
Gaseous System
n (c_{v} \log \theta + R \log \frac{\theta}{p} + k)
The entropy of a perfect gas with n molecules is n times (c_v log temperature + R log(temperature/pressure) + k).
Gaseous System
n = \dfrac{M}{m}
The number of molecules equals the mass divided by the molecular weight.
Gaseous System
\Phi = \tsum n_{1} (c_{v_{1}} \log \theta + R \log \frac{\theta}{p_{1}} + k_{1})
The entropy of a gas mixture is the sum of the entropies each gas would have alone at the same temperature in the total volume, using each gas's partial pressure.
Gaseous System
\tsum p_{1} = p
The pressure of a gas mixture is the sum of the partial pressures of its constituents.
Gaseous System
c_{1} = \frac{n_{1}}{n_{1} + n_{2} + \dots}
The concentration of a gas in a mixture is the number of its molecules divided by the total number of molecules.
Gaseous System
p_{1} = c_{1} p
The partial pressure of a gas equals its concentration times the total pressure.
Gaseous System
\Phi = \tsum n_{1} (c_{v_{1}} \log \theta + R \log \frac{\theta}{pc_{1}} + k_{1})
The entropy of a gas mixture as a function of temperature, pressure and numbers of molecules, with concentrations included.
Gaseous System
C = \tsum n_{1} (k_{1} - R \log c_{1})
The constant of integration of the mixture entropy equals the sum over constituents of n_1 times (k_1 minus R log c_1).
Gaseous System
-n_{1} R \log c_{1} - n_{2} R \log c_{2}
The entropy change on diffusion of two gases at constant temperature and pressure is minus R times n log c summed over the gases, which is positive.
Gaseous System
\Psi = \tsum n_{1} (\varphi_{1} - R \log c_{1})
The characteristic function of a gas mixture is the sum over constituents of n_1 times (phi_1 minus R log c_1).
Gaseous System
c_{v_{1}} \log \theta - \frac{h_{1}}{\theta} + R \log \frac{\theta}{p} + k_{1} - c_{v_{1}} - R = \varphi_{1}
Defines phi_1 as a function of temperature and pressure only, independent of the number of molecules.
This equation is in System of any Number of Independent Constituents (System of any Number of Independent Constituents)
Gaseous System
\tsum (\varphi_{1} - R \log c_{1})\, \delta n_{1} + \tsum n_{1}\, \delta(\varphi_{1} - R \log c_{1}) = 0
The variation of Psi under a chemical change vanishes, expanded into terms in phi, concentrations and molecule numbers.
Gaseous System
\delta n_{1} : \delta n_{2} : \dots = \nu_{1} : \nu_{2} : \dots
The simultaneous changes in molecule numbers of a reaction are in the ratio of the integers nu.
Gaseous System
\tsum (\varphi_{1} - R \log c_{1}) \nu_{1} = 0
The equilibrium condition of a chemical reaction in the gas mixture.
Gaseous System
\nu_{1} \log c_{1} + \nu_{2} \log c_{2} + \dots = \frac{\nu_{1} \varphi_{1} + \nu_{2} \varphi_{2} + \dots}{R}
At equilibrium the weighted sum of log concentrations equals the weighted sum of phi functions divided by R.
Gaseous System
c_{1} + c_{2} + \dots = 1
The concentrations of all kinds of molecules in a mixture sum to one.
Gaseous System
\frac{\tsum \nu_{1} (k_{1} - c_{v_{1}} - R)}{R} = \log a
Defines the constant a as the exponential of the sum of nu times (k_1 minus c_v1 minus R), divided by R.
Gaseous System
\frac{\tsum \nu_{1} h_{1}}{R} = b
Defines the constant b from the constants h_1 of the reacting gases.
Gaseous System
\frac{\tsum \nu_{1} c_{v_{1}}}{R} = c
Defines the constant c from the molecular heats of the reacting gases.
Gaseous System
\nu_{1} \log c_{1} + \nu_{2} \log c_{2} + \dots = \log a + (\nu_{1} + \nu_{2} + \dots) \log \frac{\theta}{p} - \frac{b}{\theta} + c \log \theta
The equilibrium condition written in logarithms of concentrations, temperature and pressure.
Gaseous System
\prod c_{1}^{\nu_{1}} = a\left(\frac{\theta}{p}\right)^{\tsum \nu_{1}} e^{-\efrac{b}{\theta}} \theta^{c}
The equilibrium condition as a product of concentrations powered by nu equals a constant times temperature over pressure to the total nu, times an exponential in b over temperature, times temperature to c.
Gaseous System
\prod c_{1}^{\nu_{1}} = a e^{-\efrac{b}{\theta}} \left(\frac{\theta}{p}\right)^{\tsum \nu_{1}}
With c = 0 (constant atomic heats in reactions) the equilibrium condition of a gaseous reaction reduces to this form.
Gaseous System
\prod c_{1}^{\nu_{1}} = ae^{-\efrac{b}{\theta}} \left(\frac{\theta}{p}\right)^{\tsum \nu_{1}}
General equilibrium condition for any gaseous chemical change: the product of concentrations raised to the nu's equals a times exponential of minus b over temperature times (temperature over pressure) to the total nu.
Gaseous System
Q = \delta U + p\, \delta V
The heat received equals the change of energy plus pressure times the change of volume (first law).
Gaseous System
Q = \tsum (c_{v_{1}} \theta + h_{1} + R\theta)\, \delta n_{1}
The heat absorbed at constant temperature and pressure in an infinitesimal reaction.
Gaseous System
L = \tsum (c_{v_{1}} \theta + h_{1} + R\theta) \nu_{1}
The heat absorbed in a finite reaction at constant temperature and pressure.
Gaseous System
L = Rb + R\theta \tsum \nu_{1}
The heat absorbed in a reaction equals R b plus R times temperature times the total nu.
Gaseous System
L = 1.97 (b + \theta \tsum \nu_{1})
The heat absorbed in a reaction in calories, with R expressed as 1.97 cal per degree.
Gaseous System
L = 1.97 \{b + (\nu_{1} + \nu_{2} + \dots) \theta\}
The heat absorbed at constant temperature and pressure written with the sum of nu's explicit.
Gaseous System
c_{1}^{-2} c_{2}^{1} c_{3}^{1} = ae^{-\efrac{b}{\theta}}
Equilibrium condition for the dissociation of hydriodic acid (2 HI giving H2 and I2).
Gaseous System
a = 0.120
The constant a for hydriodic acid dissociation, from Bodenstein's measurements, has the value 0.120.
Gaseous System
b = 1300
The constant b for hydriodic acid dissociation has the value 1300.
Gaseous System
L = 1.97 (14690 + \theta) = 28900 + 1.97\theta
The heat of dissociation of a molecule of iodine as a function of temperature, in calories.
Gaseous System
\frac{c_{2}c_{3}}{c_{1}^{2}} = \frac{n_{2}n_{3}}{n_{1}^{2}} = ae^{-\efrac{b}{\theta}}
Equilibrium of the hydriodic acid reaction within the graded dissociation system, with iodine dissociation included.

Problems

No exercises in this chapter.