From WolfWikis

LECTURE 1:Gas law

Some history

Our main topic is thermodynamics. It has a long, long history. In fact fire is our oldest technology and we have used heat to prepare our food and keep ourselves warm for millennia. To make heat do useful work for us is a more recent trick, although even in antiquity people played with early versions of a steam engine. Greeks had it but they did not use it for anything practical. It was just a curiosity. The Romans seem to have had a canon based on steam, although the Myth Busters recently denied its viability on television.

After the collapse of the western part of the Roman Empire (5th century AD), there was a prolonged period of stagnation in scientific understanding or worse, at least in western Europe. Thermal energy was at most applied to kill heretics who dared to question established doctrine.

Major developments had to wait to the 17th and 18th century:

• Theory of gases. (will review this). Torricelli, Boyle, Gay-Lussac
• Development of the steam engine (James Watt)
• Development of calculus

The three were combined to develop thermodynamics over most of the 19th century.

Calculus is of course a mathematical development, but it proved vital to the later success of thermodynamics. Thermodynamics involves a lot of multivariate calculus. (lots of ∂ symbols. If you don’t remember: please revisit your math books and notes!!)

Notice also that part of the development was in the form of practical engineering. The steam engine was essential to the industrial revolution, but its scientific understanding actually trailed behind.

Later developments:

• non-ideal gases (van der Waals)
• atomic theory
• quantum mechanics

Thermodynamics was initially entirely phenomenological, the atomistic version came much later and is called statistical thermodynamics.

Review of gases

See chapter 16

Much of this you already know. Let's summarize a few important facts.

Pressure can be measured with a column of liquid (usually mercury: hydrostatic pressure) using P=ρgh.

Any gas will behave as an ideal gas provided it is rarified enough (i.e. lim P-> 0)

Ideal gases obey the gas law: PV= nRT, attributed to Boyle, (Charles) and Gay-Lussac

The law can also be written in intensive form by using the molar volume

The law is an equation of state. It uniquely defines the state of the system at the pertaining conditions.

At most ‘ordinary conditions’ the deviations from ideality are quite small.

The gas law is used to define an absolute temperature scale in Kelvins. It was observed that the volume of a gas increases linearly with temperature, but that the straight line always extrapolates to a temperature of -273.15 ^oC. This point was taken as the zero point of the new temperature scale.

Units

Despite efforts dating back to the days of the French revolution to rationalize and standardize units into one coherent system, there are still quite a few different units in current use. Unfortunately that also holds for science. For pressure e.g. we could encounter: psi, Torr, Atm, bar and Pa. (See page 639). Yes we expect you to be able to use them all and convert between them. We'll stick to metric units (not psi e.g.).

Even within the metric (i.e. decimal) units there is quite a variety that stems from various periods of the development of what is now the SI system.

1. 'ad hoc' metric unit: based on decimals but chosen on arbitrary basis (calorie, Torr, Atm etc.)
2. cgs units based on the basic units cm, g and s (bars, ergs, dynes, Gauss, Ørsted etc)
3. mksa units based on m, kg , s, and A (dm, Ångström,)
4. SI- units: only factors of 1000: m, mm, μm, nm, pm, fm etc. (Pa, N, J, Tesla, etc.)

Pressure units

• The only official (IUPAC, SI) unit for pressure P is the Pascal (Pa). It is equal to one Newton per square m. A pressure is a force exerted over a certain surface area.
• Bars are an older (cgs) unit but quite compatible (1 bar = 100,000 Pa) and a little easier to visualize because the earth's atmosphere exerts a pressure of about 1 bar on us all (depending on weather and altitude).
• An even older 'ad hoc' unit was a little larger 1 Atm = 1.013 bar.
• The oldest metric unit is in mm mercury (mmHgp or Torr). 760 mm mercury corresponds to one atmosphere (1 Atm)

Volume units

• The only real SI units are m³, mm³, μm³ etc.
• However, the gap between m³ and mm³ is a factor of 10⁹ (one billion). Therefore, SI also condones liters. (Originally this was based on a metric unit dm³, but the deci- prefix is no longer acceptable in SI).
  • 1 m³ = 1000 l. (I have never seen a 'kiloliter' used).
  • 1 dm³ = 1 l
  • 1 cm³ = 1 ml
  • 1 mm³ = 1 μl

The fat units are official SI but the liter units are so common that the SI people know they cannot be eridicated. Units like centiliters (cl) and deciliters (dl) are not recommended.

Volumes of gases depend entirely on the container. Volumes of liquids are often assumed to be additive, but this is strictly speaking only true if the resulting solution is ideal.

Temperature units

The only unit that you can use safely in thermodynamics is the Kelvin. Please make sure you convert:

The latter follows from simple arithmetic:

T₁ = 30^oC = (273.15 + 30) K

T₂ = 20^oC = (273.15 + 20) K

ΔT = T₁-T₂= (273.15 + 30) K -(273.15 + 20) K = 10 K

The shift in the origin (273.15) drops out.

It is customary to write a temparature in Kelvin as a capital T. Lower case t is typically reserved for either degrees celsius or -more commonly- time in seconds.

Time units

Classical thermodynamics says very little about time. Often it is simply assumed that a process has sufficient time to go to completion or equilibrium. Time dependency is more the domain of kinetics.

The scientific (SI) unit of time is the second [s]. Units like minutes and hours are typically discouraged because of their sexagesimal nature.

Frequency is the reciprocal of time and has a unit s^-1. Note that frequency can either incorporate a factor of 2π or not:

1 (cycle) per second = 2π (radians) per second.

As radians are dimensionless the unit is s^-1 in both cases.

Concentration units

The concept concentration is only applicable to homogeneous mixtures. If the system is not homogeneous we can speak of its overal composition but not of a concentration. (This implies that any theory or measurement that invokes concentration is invalid if homogeneity is not achieved).

A concentration is always a ratio of one substance quantity and another. However, the substance quantity can be expressed in various ways:

• by volume
• by mass
  • in kg,g etc.
  • in moles

In addition, we could take the quantity ration of A versus B or alternatively of A versus total A+B. This leads to a considerable number of combinations, some of which are more in common use than others.

It should be noted that to convert volumes into masses and vice versa, we need knowledge of the density ρ. In the absence of such information it may therefore not always be possible to convert from one concentration unit to the other.

Another point is that some units depart from the assumption that one substance is -by far- a minority component (i.e. we are dealing with dilute solutions). In that case the minority component(s) are called solute, the majority one solvent and the total is called solution

In non-dilute cases the roles of solute and solvent may reverse (e.g. at 50 mole percent of each in a binary case) and it is better not to use this terminology.

The most current concentration units are

• molarity
• molality
• mole fraction

Molarity

Molarity is a mixed unit:

• the solute is measured by mass and expressed in moles
• the (liquid) solution is measured by volume and generally expressed in liters

We say the solution is 0.50 molar in Na⁺ and write 0.50 M Na⁺ Note that the trailing zero implies a precision. It means that the error in this number is in the second decimal (0.0x). A more precise concentration would be written as 0.5000 M or 0.5012 M The uncertainty is then assumed to be 0.000x.

To prepare such a solution you weigh off 0.5000 mole of e.g. NaCl, put it in a volumetric flask and add solvent (e.g. water) to (but not beyond!) the calibrated mark that assures that the total volume is e.g. 1 liter. Often it is necessary to standaridize the solution e.g. by titration and determine the actual concentration by measurement. This is particularly so if the solute is hygroscopic or its purity is not very high.

The reference point for thermodynamic standard states^o is 1 M = 1 mole per liter

Molarity is by far the most commonly used unit, despite a number of drawbacks:

• volumes are usually less precise than masses
• volumes change with temperature (and so does molarity!)
• volumes are not entirely additive (as we shall see later)
• converting to molalities or mole fractions requires knowledge of density

As long as you work with dilute solutions (large volumes relative to solute mass) at room temperature, these problems are negligible, which explains the wide spread use of this concentration unit. Many analytical procedures are based on volumetric measures, like titrations. For such chemistry molarity is clearly the most useful unit.

Because molarities change with temperature even when not a single atom leaves the solution, the temperature is usually further standardized to 25^oC (rather than left to the temperature of interest as is usual in thermodynamics)

Please never rely on the decorative markings of a 'graduated' cylinder or beaker. People who do should never be 'graduated'. Burettes, volumetric flasks, pipettes and syringes are OK if you know how to deal with them and are aware of their limitations.

Molality

Molality is a purely mass based unit:

• the solute is measured in mass and expressed in moles
• the solvent (not: solution) is measured by mass and expressed in kilograms

Molality is measured in moles of solute per kilogram solvent. Is is usually denoted with lower case m.

A 0.543 molal solution is written as 0.543 m. Again the number of digits implicitly shows precision. (The uncertainty in the value would be 0.00x in this case).

The reference point for thermodynamic standard states^o is m=1 mole per kilogram

It does not suffer from the volume induced drawbacks of molarity. It only requires a good analytical balance (to weigh off not just solids but also liquids). No volumetric flasks or pipettes are required. Just weigh off, say 100g of water, weigh off 0.0543 moles of solute and mix thouroughly.

For dilute aqueous solutions at room temperature molality and molarity converge

At room temperature one kilogram water has a volume of ca 1 liter and the addition of the solute does not change the volume by much. Note that for other solvents this is not the case (density!). Also, when the temperature is changed, molality stays the same while molarity does not (volume expansion!).

Mole fractions

Mole fraction is a purely mass based unit:

• Each component is measured in moles
• The total number of moles of the mixture is determined.

The mole fraction x_A of A is moles of A / total moles. The unit is dimensionless. The reference point^o is x_A=1 is the pure substance.

Preparation is usually by weighing off the two (or more) components. Mixing to a homogenous mixture is not always easy if they are both solids. Prolonged joint powdering (tituration) and/or melting and repowdering may be required. If homogeneity is not achieved (if miscibility is partial or negligible) the mole fraction refers to a composition rather than a concentration.

This unit is used if the full range of composition over a large range of temperature is studied, e.g. in phase diagrams. Notice that the distinction solute/solvent is abandoned in the description.

The unit cannot easily be converted to molarity unless density is known. Conversion to molality is easy.

The unit requires that molar masses are known and that may be a problem for polymers that have a molecular weight distribution rather than a fixed molecular weight. In that case a mass fraction is more appropriate. Even volume fractions are used at times, despite the inherent problems on non-additivity of volumes.

Non-ideal behavior

Consider the ideal gas law: PV = nRT If T (and n) are constant the product PV should not depend on P.

Fig 16.2 shows that actually it does, (but note the scale on the left, the deviations are not very large). In the limit P->0 a mole of gas has one and the same volume (the molar volume) regardless of the gas (mixture).

Compressibility

The small deviations from ideal behavior have taught us a lot about gases and thermo. To look at the deviations consider the quantity Z (compressibility factor):

Clearly Z=1 as long as the gas is ideal. At high pressures (dense gases) fig 13.6 shows what happens. In this figure it is seen that Z is almost constant but not quite. Particularly at low temperatures and at high pressures Z deviates from unity. At low pressures and low temperatures there is a negative deviation. At higher pressures a positive one kicks in

Van der Waals

Van der Waals recognized there were two reasons for the deviations:

1. the finite size of a molecule that makes its volume available for another molecule smaller
2. attractive interactions between the molecules that tend to make them stick to each other

He put two correction terms into the gas law. His corrected law has a larger scope of validity than the old one.

Ultimately we will condense the gas to a fluid, possibly a liquid. It would be nice to have one unified description for all conditions, rare and dense, liquid and gas. This is what the van der Waals equation and more successfully so the Redlich-Kwong equation is trying to do.

The van der Waals law notice on orthography can be rewritten as a third order equation, known as an equation of state. (Study example 16.2).

The equation has two added constants (the van der Waals constants: a and b). They are different for every gas but we'll see below that we can generalize that.

If the equation has one root, the solution gives you the molar volume of the one fluid phase that is present.

If there are three roots of the equation, one corresponds to the molar volume of a liquid phase, the other with a gas. Think of liquid water in equilibrium with water vapor. See here for an animation. The molar volume (or density) of the two are quite different.

The middle root has no significance, because it lies in the two-phase region between liquid and solid. This is seen most clear when plotting isotherms in a P-V diagram. (Fig 16.7)

Between the three roots and the one root case is the case where the three roots coincide. This is the critical point. The critical point is really a point, in that it has one fixed value for pressure, molar volume and temperature for a given gas.

As shown in eq 16.11 at the critical point where the three roots coincide, the equation of state reduces to

This yields a number of simple expressions relating the molar volume, pressure and temperature of the critical point. Study the equations 16.12 through 16.15 on page 652 well.

The relationships clearly show that the van der Waals constants a and b are closely related to the critical point of a gas. Or in other words, the two effects: excluded volume and attraction are responsible for the critical behavior of fluids (gases and liquids together).

Corresponding states

Although the critical point can be quite different from gas to gas, we can combine the above expressions to form the quantity

and this quantity should be constant for all gases!

Van der Waals is really on to something here! What it says is that the behavior of all gases (and liquids!!) is pretty much the same, except from some scale factor that is related to the critical point of the substance.

(The value 3/8 is actually not in such good agreement with measurement (see table 16.5), this is why the more complicated Redlich-Kwong and Peng-Robertson expressions are better, but the idea is the same.)

Using the above 'constant' 3/8, we can rewrite the van der Waals expression using reduced variables by dividing P,V and T by the values at the critical point (T_R=T/T_C) etc.

Notice that these new variables are dimensionless. E.g. T_R=2 means that you are at twice the critical temperature (in Kelvins of course).

This leads to the remarkably universal expression 16.19:

Notice that we have gotten rid of a and b and replaced them with 3 and 1/3. The expression is valid for all gases and liquids to reasonable approximation.

All you need to know to describe any fluid's behavior is its critical point. For example, argon behaves much the same at 300K as ethane does at 600K because these temperatures correspond to twice their respective critical temperatures (150.72K and 305.4K, respectively) so TR=2.0

We can rewrite the universal expression to express the compressibility Z in reduced variables and plot measured values as Z versus the reduced pressure (see fig 16.10). As you can see very different gases/liquids like nitrogen and water can be made to coincide if their properties are plotted relative to there critical points rather than in absolute terms.

Van der Waals did not stop here. He went on to describe the root cause of condensation of gases into liquids at lower temperatures: the attractive interactions between the molecules. You may already be familiar with that part of his work (the van der Waals interactions etc.)