Gas mixtures and psychrometrics
Nothing you breathe, burn or condition is a pure substance. Air is a mixture before anything interesting happens to it, and the moment it picks up water vapour it becomes a mixture whose second component is having a phase change while the first one just sits there. Both facts have to be handled, and neither is difficult once the bookkeeping is straight.
Every property relation used so far in this course was written for a single, pure substance: water, a refrigerant, air treated as one gas. Real air is nitrogen, oxygen, argon and a handful of trace gases at minimum, and the moment it goes anywhere near a cooling coil it is also carrying water vapour, which condenses while the rest of the mixture does nothing of the sort. This topic is the machinery for handling several substances occupying the same space at once, applied to the one mixture that shows up in nearly every building on the planet.
Describing a mixture
A gas mixture's composition can be reported two ways, and the two give different numbers for the same jar. Count particles and each component's share is a mole fraction; weigh the jar component by component and each share is a mass fraction. Dry air is close to \(21\)% oxygen and \(79\)% nitrogen by mole, a convenient approximation this whole course leans on, ignoring the roughly one per cent of argon and trace gases that make no practical difference to an energy balance. By mass the same jar is \(23.3\)% oxygen and \(76.7\)% nitrogen, because oxygen is the heavier molecule and claims more of the total weight than its head count alone would suggest.
Both fractions sum to one over the whole mixture, by definition rather than by any physical fact worth proving:
and the two are tied together by each species' molar mass. Since \(m_i = N_i M_i\), dividing through by the mixture's total mass gives the conversion in either direction:
where the mixture's own apparent molar mass is the mole-weighted average of the components' molar masses, \(M_m = \sum_i y_i M_i\). For air, \(M_m = 0.21(32) + 0.79(28) = 28.84\) kg/kmol, close enough to the more careful value of \(28.97\) that includes argon to be used interchangeably in this course.
Neither fraction is more correct than the other; they answer different questions. A chemical reaction, worked one molecule at a time in combustion, is naturally kept on a mole basis, because reaction stoichiometry counts particles, not kilograms. An energy balance, which needs a mass to multiply a specific enthalpy by, is naturally kept on a mass basis. Converting between the two with the relation above, rather than guessing which was meant, is worth the extra line every time a mixture problem hands you one and asks for the other.
One jar of air, two gases, and two entirely different ways to say how much of each.
Dalton's law and mixture properties
For an ideal-gas mixture, each component behaves as if the others were not there. Imagine gas \(i\) alone, occupying the full volume \(V\) of the mixture at the mixture's temperature \(T\); the pressure it would exert on its own is its partial pressure,
and Dalton's law of additive pressures is the statement that these partial pressures sum to the actual total: \(P_m = \sum_i P_i\). It follows immediately from the ideal gas equation applied to each species and to the mixture as a whole, since \(N_m = \sum_i N_i\) is simply a count. A twin statement, Amagat's law of additive volumes, does the same trick holding pressure fixed instead of volume, and for an ideal-gas mixture the two give the same mole fraction either way: \(y_i = P_i/P_m = V_i/V_m\).
Extensive properties of the mixture (internal energy, enthalpy, and the two specific heats) are mass-weighted sums of the components' own values, evaluated at the mixture temperature:
which is no more than saying that the total energy of the mixture is the sum of the energy each component would have carried on its own, because internal energy and enthalpy of an ideal gas depend on temperature alone and every component sits at the same temperature as the mixture.
Entropy breaks that pattern, and it is worth knowing exactly where and why, because the mistake of treating it like the others is common and quiet. Each component's entropy must be evaluated at its own partial pressure, not at the total mixture pressure, since entropy depends on pressure through a term that does not cancel the way internal energy's does:
Mixing two gases that started out separated is itself an irreversible process. It is exactly the unrestrained-expansion idea from the second law, run once for each component as it expands from its own volume into the shared one, and the entropy generated by mixing shows up automatically once each \(s_i\) is charged its own lower partial pressure rather than the higher mixture pressure. Skip that step and mixing looks, wrongly, like it costs nothing.
Two gases share one volume at one temperature, at one total pressure.
Atmospheric air and the humidity ratio
Atmospheric air, the working fluid of this whole topic, is a two-component ideal-gas mixture: dry air, itself already a mixture but treated here as one component, plus water vapour in whatever small amount happens to be present. The vapour's partial pressure is low enough, almost always, that it can be treated as an ideal gas even while sitting close to its own saturation dome, a point raised without proof in ideal and real gases and used without further comment from here on.
The amount of water carried is reported as the humidity ratio (also called the specific humidity), the mass of vapour per unit mass of dry air (dry air, not total mixture mass, which is the detail that trips people up first):
Writing each mass through the ideal gas equation at the same volume and temperature turns this into a ratio of partial pressures, using the two components' molar masses, \(18.015\) for water and \(28.97\) for dry air:
where \(P\) is the total pressure and \(P_a = P - P_v\) by Dalton's law. Take ordinary room air at \(101.325\) kPa carrying a vapour partial pressure of \(1.2\) kPa, and \(\omega = 0.622(1.2)/(101.325 - 1.2) \approx 0.00746\) kilograms of water per kilogram of dry air. A few grams per kilogram is the ordinary range indoors, and the number looks small because it is small; the mass of water in a room is a tiny fraction of the mass of air, and it is nonetheless the whole subject of the rest of this topic, because that tiny fraction governs comfort, condensation and mould growth far out of proportion to its size.
The enthalpy of moist air is reported per unit mass of dry air for the same bookkeeping reason mass fractions were reported per kilogram of mixture earlier: dry air's mass does not change as water condenses or evaporates within a process, so it makes a stable basis to divide by, whereas the total mass of moist air does. With dry air's specific heat and water vapour's enthalpy both nearly linear in temperature over the ordinary range,
with \(T\) in °C and both enthalpies in kJ per kilogram of dry air, the constant \(2500\) being water's latent heat of vaporisation near \(0\) °C. That constant is doing more work in the formula than the temperature-dependent terms around it, which is the first hint of where the next two sections are headed.
Ordinary air is dry air plus a small, variable amount of water vapour.
Relative humidity and the dew point
Humidity ratio says how much water is present. It says nothing about how close that amount is to the most the air could hold, which is the number that actually predicts condensation, mould and the sense of stuffiness a room gets when it is humid. That number is relative humidity, the ratio of the vapour's actual partial pressure to the saturation pressure a pure water sample would have at the same temperature:
where \(P_g(T)\) is read from the same saturation tables used throughout properties of pure substances. Water's saturation pressure does not care whether the water is alone in the vessel or diluted into a sea of nitrogen and oxygen, only that it is at temperature \(T\). At \(\phi = 100\)% the air is holding all the vapour it can at that temperature; push any more in, or drop the temperature at fixed \(P_v\), and liquid water starts to appear.
That second route, cooling rather than adding moisture, is the more common way saturation gets reached in practice, and it defines the dew point: the temperature at which air, cooled at constant total pressure and constant humidity ratio, first reaches \(\phi = 100\)%. Since \(\omega\) fixes \(P_v\) and \(P_v\) does not change as the air cools (no water has left yet), the dew point is simply the saturation temperature that corresponds to the vapour's own partial pressure:
A cold drink sweats, a bathroom mirror fogs, and a single-glazed window streams with water on a winter morning for the identical reason each time: a surface sitting below the dew point of the air touching it forces some of that air's vapour to condense on contact, whether or not the room as a whole feels humid. Insulating the surface, or lowering the room's humidity ratio, are the only two remedies, because the dew point of the bulk air is fixed once \(P_v\) and \(P\) are, and no amount of stirring the air changes it.
Relative humidity and humidity ratio measure genuinely different things and neither substitutes for the other. Cold winter air brought indoors and warmed to room temperature keeps its humidity ratio (no water was added), but its relative humidity collapses, because \(P_g(T)\) rises steeply with \(T\) while \(P_v\) has not moved at all. That is the entire reason heated buildings feel dry in winter without a drop of water having left the air, and it is why humidification, taken up in the next section, adds moisture rather than warmth to fix it.
The saturation curve gives the most vapour pressure the air could hold at each T.
The psychrometric chart
Two independent properties fix the state of moist air at a given total pressure, the same state postulate as ever, and the psychrometric chart is simply that fact turned into a picture: dry-bulb temperature along the bottom, humidity ratio up the side, and every other property of interest (relative humidity, enthalpy, specific volume, the wet-bulb temperature a thermometer with a wet wick would read) drawn as a family of curves across the same two axes. Once you can read one state off it, every other property associated with that state comes free, which is the entire reason the chart is worth learning rather than just computing everything from the formulas above directly.
The \(\phi = 100\)% curve is the chart's most important line and its left-hand boundary: no state can sit to its left, because that would mean more vapour present than the air could hold at that temperature without condensing. Every other relative humidity is a curve of the same shape scaled down towards the temperature axis, and the dew point of any state is read by the horizontal move already described in the previous section: across, at constant \(\omega\) and therefore constant \(P_v\), until the saturation curve is reached.
A cooling and dehumidifying process, the ordinary job of an air conditioner on a humid day, reads as a line that drops both \(T\) and \(\omega\) together. Air is drawn across a coil held below its dew point; it cools along a path that eventually meets the saturation curve, condenses some of its water onto the coil, and leaves near saturation at a lower temperature and a lower humidity ratio than it entered with. Left alone, that air would feel clammy rather than merely cool, because near-saturated air at a low temperature still reads as a high relative humidity even though it is carrying less water in absolute terms than it started with.
The fix, standard in commercial air conditioning, is a reheat coil downstream that raises the temperature back up at constant humidity ratio (a horizontal move on the chart, since sensible heating alone adds no water), landing on a supply condition that is both cooler and considerably drier than the outdoor air, at a comfortable relative humidity rather than a merely lower one. A winter process runs the opposite trade: heating alone, a horizontal move to the right, drops relative humidity exactly as described above, and a humidifier (spraying water or steam directly into the airstream) adds \(\omega\) back in, a vertical or near-vertical move depending on whether the water added is cold or is steam. Every one of these processes is a straight line, or close to one, on a chart built for exactly this purpose, which is the whole reason it survives in an era when the underlying equations could just as easily be run through a calculator.