Gases
An analysis of the behavior of gases is a good starting point to begin the study of physical chemistry. Gases are selected mainly because they are easier to understand and simpler to describe mathematically than are solids and liquids.
A gas may be defined as state of matter distinguished from the solid and liquid states by relatively low density and viscosity, relatively great expansion and contraction with changes in pressure and temperature, the ability to diffuse readily, and the spontaneous tendency to become distributed uniformly throughout any container.
In dealing with gases is useful to get familiar with some parameters and their units such as pressure, volume, and temperature.
Pressure (P) is an effect which occurs when a force (F) is applied on a surface of area A.
P=F/A
Some units of pressure and their conversions in other units of pressure are shown in table 1
Table 1

Pascal (pa)

Bar (bar)

Atmosphere(atm)

Torr(Torr)

1pa

≡ 1 N/m^{2}

10^{−5}

1.3332×10^{−3}

7.5006×10^{−3}

1bar

100,000

≡ 10^{6} dyn/cm^{2}

0.98692

750.06

1atm

101,325

0.980665

≡ 1 atm

760

1 torr

133.322

1.3332×10^{−3}

1.3158×10^{−3}

≡ 1 Torr; ≈ 1 mmHg

Volume can be defined as the amount of 3Dspace an object occupies. In the following example the volume is 4x5x10=200units^{3}.
Units of volume include: centimeters cubits (cm^{3})^{,} meters cubit (m^{3}), liters (l).
Temperature is a physical property of a system that underlies the common notions of hot and cold; something that feels hotter generally has the higher temperature. Temperature is one of the principal parameters of thermodynamics.
Three temperature scales are in common use in science and industry Fahrenheit (°F), Celsius (°C), and Kelvin (K).
Thermal Equilibrium
It is observed that a higher temperature object which is in contact with a lower temperature object will transfer heat to the lower temperature object. The objects will approach the same temperature, and in the absence of loss to other objects, they will then maintain a constant temperature. They are then said to be in thermal equilibrium. Thermal equilibrium is the subject of the Zeroth Law of Thermodynamics. In figure 1 T_{A}=T_{B} after reaching thermal equilibrium.
Fig. 1
Zeroth Law of Thermodynamics
The "zeroth law" states that if two systems are at the same time in thermal equilibrium with a third system, they are in thermal equilibrium with each other; figure 2 shows a pictorial example of this principle
Fig.2
If A and C are in thermal equilibrium with B, then A is in thermal equilibrium with B. Practically this means that all three are at the same temperature, and it forms the basis for comparison of temperatures. It is so named because it logically precedes the First and Second Laws of Thermodynamics.
Mechanical Equilibrium
Two systems are in mechanical equilibrium when their pressures are the same. If we have a system in which two separated (with a movable wall) compartments exist at different pressures (P), there is a time when these pressures will be equal as demonstrate in figure 3.
Fig 3
P_{1}›P_{2 }P_{1}=P_{2}
What Is Absolute Zero?
Absolute zero is the point where no more heat can be removed from a system, according to the absolute or thermodynamic temperature scale. This corresponds to 0 K or 273.15°C. In classical kinetic theory, there should be no movement of individual molecules at absolute zero, but experimental evidences shows this isn't the case. Temperature is used to describe how hot or cold an object it. The temperature of an object depends on how fast its atoms and molecules oscillate. At absolute zero, these oscillations are the slowest they can possibly be. Even at absolute zero, the motion doesn't completely stop. It's not possible to reach absolute zero, though scientists have approached it. The NIST achieved a record cold temperature of 700 nK (billionths of a Kelvin) in 1994;MIT researchers set a new record in 2003 .Interesting. Graph 1 shows the zero absolute temperature.
Graph 1
Boyle's Law
At constant temperature, the volume of a confined gas is inversely proportional to the pressure to which it is subjected. According to this law PV = k (eq. 1) as shown in graph 2.
Graph 2
Charles' Law
At constant pressure, the volume of a confined gas is directly proportional to the absolute temperature. According to this law V=kT (eq.2) as shown in graph 3.
Graph 3
Gas mixtures and partial pressures
Dalton’s law of partial pressures: the total pressure of a mixture of gases equals the sum of the pressures that each would exert if it were present alone.
Expressed mathematically:
P_{T} = P_{1} + P_{2} + P_{3} + ……. (eq. 3)
The pressure exerted by an individual component of a mixture of gases is called the partial pressure.
Note that if the components are all in the same container, they all occupy the same volume and are at the same temperature. Therefore if each gases obeys the ideal gas law:
If we relate the partial pressure of an individual component to the total pressure, we obtain the following expression:
_{ }_{ }P_{i}/P_{T} = n_{i}/n_{T} = X_{i} (eq.5)
where Xi = mole fraction of the individual component
Note that the sum of the mole fractions of the individual components in a mixture must equal 1.
Major equations of state
For a given amount of substance contained in a system, the temperature, volume, and pressure are not independent quantities; they are connected by a relationship of the general form:
f(V, T, P) =0 (eq.6)
In the following equations the variables are defined as follows. Any consistent set of units may be used, although SI units are preferred. Absolute temperature refers to use of the Kelvin (K) or Rankine (°R) temperature scales, with zero being absolute zero.
P= pressure (absolute)
V= volume
n= number of moles of a substance
Vm=V/n = molar volume, the volume of 1 mole of gas or liquid
T= absolute temperature
R = ideal gas constant (8.314472 J/(mol·K))
The ideal gas law may be written: PV=nRT (eq. 7)
Prepared by Nelson Castillo (red), Jesse (blue), Andrei (green).
REAL GASES
We have seen how the Ideal Gas Equation of State (eq. 7, above) describes the behavior of the “Ideal Gas”. However, this equation only attempts to describe how real gases would behave “in a perfect world” (But in the “real world” gases are made up of real molecules and real atoms –and as we’ll see, the fundamental cause for why Real Gases act differently from Ideal Gases. So in a sense, eq. 7 describes how any Real Gas would behave if that gas were made of atoms and molecules that did not interact with each other. Furthermore, As long as the Real Gas molecules of any gas remain far apart that gas behaves (mostly) like an Ideal Gas.
This is the reason why Real Gases do not “follow” eq. 7.
As it turns out, Real Gases deviate from the Ideal Gas law the most when the molecules are close to each other. For that reason, whatever P,V,T conditions bring gas molecules closest are also the P,V,T conditions that make that particular Real Gas deviate the most.
*Let us explore what P,V,T conditions bring gas molecules the closest.
There are two ways to bring gas molecules closer together:
(1) You can squeeze molecules together (compression) at a given Temperature.
(2) You can cool the Real Gas molecules down so much that they don’t have the kinetic energy to push each other away.
So, as it turns out, if you want to make Real Gas molecules come close together, then you use a combination of (1) high pressure (squeezing together, compression) and (2) low temperature (slowing down molecules so they don’t really have the energy to bounce around and push each other away).
*Connection between,
A) bringing gas molecules closer and B) Real Gas deviation:
All Real Gas molecules have the ability to either repel OR attract another Real Gas molecule BUT only when the molecules are too close to each other (In other words, be it repulsive deviation or attractive deviation, deviation happens when Real Gas molecules are too close to each other).
These attractive and repulsive events of the Real Gas molecules is what makes the Real Gas’s pressure either lower or higher compared to an Ideal Gas in the same situation (same moles, Temperature and Volume).
Logically, one would consider if whether or not the attractive forces cancel out with the repulsive forces. In fact, they generally do not.
There are two reasons for this:
(1) First of all, the repulsive force is stronger than the attractive force, but
(2) The attractive force is more important when the molecules are farther away (Several molecular diameters) from each other. So, the way it plays out, when the molecules get closer and closer (Less than one molecular diameter) the repulsive force come into play more and more.
So generally, if the Real Gas molecules are a little bit far apart (Several molecular diameters) they will want to interact (Vander walls forces; nucleus of Molecule A is Attracted to electron cloud of molecule B, and vise versa, DipoleDipole, ect.) and so their Real Gas behavior will be lower than the Ideal Gas, in the same situation. However, if they are compressed (to about one molecular diameter or less) then there will be very strong repulsive force, and so the gas pressure will be greater than an Ideal Gas in the same situation.
And there is a Third reason why Real Gases do not act like Ideal Gases as they are compressed (But this force has nothing to do with attraction or repulsion exactly, but rather a basic property of matter – for matter to exist, it must occupy space. This compares to Ideal Gas molecules, as they theoretically do not occupy space as individual molecules).
Real Gas molecules, however, are not points of infinite smallness, they are real matter (That is why compressing a Real Gas becomes like putting 50 marbles in your pocket, they are already so close that they can’t get any closer, they need room in which to exist because they are real, not perfect).
So, Notice that “Intermolecular Forces” do not exist in eq. 7
The first slide above shows a fixed volume filled with a Real Gas ( This one is Ozone). The Gas molecules are condensed into a liquid form (shown in the next slide). As you can see, It shows how that same volume of gas can be represented by that tiny amount of violet liquid (both contain the same amount of Ozone molecules!).
As I mentioned above, the Intermolecular forces “Attractive” and “Repulsive” express themselves as Real variations in Pressure (as Compared to Ideal Gas in the same situation). These Real variations in Pressure can either be manifested as either a Lower or Higher pressure (compared to what eq.7 would predict).
The X axis represents distance between two Real Gas molecules,
The Y axis represents the resulting potential energy in maintaining the distance X, Fig 13.
A) When two molecules are very very close (less than one molecular diameter), it is difficult to keep them together (and so the potential energy is very high),
B) And when two molecules are just a little bit far apart (several molecular diameters) they are just close enough to experience some attractive force from each other (but not yet close enough to be repelled by each other), that is, the “dip” in the graph where attractions are dominant (“dominant” as in more significant than repulsion).
C) The point where the graph crosses the X axis is where Attractive forces exactly equal the repulsive forces (they “cancel each other out” in terms of potential energy or pressure,and just at that instant, Real gases appear to act like Ideal gases).
D) Notice that as the distance between two molecules increases more and more beyond several molecular diameters, the Real Gas looks more and more like an Ideal Gas (because it asymptotically approaches Y=0 on the graph, and Ideal gases have Zero Attractive/Repulsive forces, and therefore Zero intermolecular potential energy).
 Summary 
Recall that eq. 7 represents Ideal Gases but not Real Gases; the Real resulting P (after you plug in for n,T,V) is that it is usually either higher or lower than the P of the Ideal Gas (That deviation, in terms of potential energy between the Real Gas molecules is represented by fig. 113).
When Real Gas molecules are between one and several molecular diameters apart, they are attracted to each other, and the resulting Real Pressure is lower than Ideal Pressure (Ideal Gas in the same situation). Or, Instead of describing the comparison in terms of the final effect on Pressure, we could have also said Keeping Real and Ideal Pressure the same (along with Real and Ideal Temperature, as an input variable) the final resulting Real volume would be smaller than the Ideal volume because less volume is needed as the attracting molecules get closer and accommodate each other easily (lower potential energy).
When Real Gas molecules are less than one molecular diameter apart (High pressure, Low temperature), their electron clouds repel each other (Coulombic repulsion), and the resulting Real Pressure is greater than Ideal Pressure (Ideal gas in the same situation).
Or, putting it in terms of Volume instead of pressure, (therefore keeping pressure as an input variable along with Temperature) we can say More volume is needed (as a Real gas compared to Ideal) to maintain a given pressure (otherwise pressure would rise) as molecules are compressed closer and closer (one molecular diameter or less).
When Real Gas molecules are beyond several molecular diameters, they are virtually no longer attracted to each other (and their behavior asymptotically approaches Ideal behavior. That is, the potential energy between molecules asymptotically approaches zero), the resulting Real Pressure approaches Ideal Pressure as intermolecular distance approaches zero.
COMPRESSIBILITY
The simplest way to express the idea of Compressibility is simply that it is a ratio of Real Volume to Ideal Volume (where n, T and P are all the same input variables and V is the only output variable).
*Real Volume divided by Ideal Volume results in:
A) Greater than 1 if the Real volume is greater than the Ideal volume (and so, Greater than 1 also represents intermolecular repulsion – positive intermolecular potential energy, relative to zero potential energy for Ideal).
B) Less than 1 if the Real volume is less than the Ideal volume (and so, Less than 1 also represents intermolecular attraction – negative intermolecular potential energy, relative to zero potential energy for Ideal).
That equation (Compressibility) would look like this:
Z=Vm/Vm0 (eq. 8)
Vm = molar volume of the Real Gas,
Vm0 = molar volume of the Ideal Gas.
It gives an instant representation of by how many times the Real Gas deviates from the Real. (example; Z=2 describes a particular Real gas requiring twice the volume as an Ideal gas for the same input values for n, T and P).
Figures 113 and 114 respectively show local minima. The local minima for their respective gases represent greatest possible intermolecular attraction.
1Fig14.docx
Fig 114 shows how Z compressibility being graphed according to specific given P pressures.
While, Fig 113 shows changes in potential energy as intermolecular distance is varied.
That is:
For fig 113, X= Distance, Y= Potential Energy.
For fig 114, X= Pressure, y= Relative Volume (Real/Ideal).
Both graphs (Fig 114 and Fig 113) show that as pressure approaches zero (intermolecular distance increasing beyond several molecular diameters) the Real gas behaves more and more Ideally as Z approaches 1. Fig 113 expresses this as an asymptotic approach to zero potential energy between real molecules (just like there is no potential energy between ideal molecules).
Both graphs (Fig 114 and Fig 113) show that as pressure approaches infinity (intermolecular distance approaching zero, having started from one molecular diameter and going down to zero distance) the Real gas potential energy approaches infinity as Z approaches infinity. We can observe this rize in Z value at 600800 atm on fig 114. Fig 113 expresses this as a rise in intermolecular potential energy as distance approaches zero.
Taking a closer look at the graph you have the minima for Real gases like C2H2 and CH4 at about 200 atm. So, at about 200 atm the molecules are experiencing their greatest attraction possible (for the constant temperature we selected).
Interestingly, Hydrogen Gas does not show us a minima!
This may suggest that, between H2, the attractive forces, are always weaker than whatever repulsive forces are present, anywhere all along the pressure range, for the constant T we selected. (That does not mean that there are no attractive forces between H2 molecules, it just means that attractive forces are always weaker than repulsive forces, For this specific constant T).
We can express eq. 8 in a perhaps more versatile form. We can exchange Vm0 for the variables that Vm originally represented to give us a similar equation:
Z= (PVm/RT) (eq. 9).
Or if we just want a formula for Ideal molar volume, Vm0:
Vm0 = RT/P (eq. 10)
Equation 7 unites the variables T, V and P. This allows us to keep any one variable constant while making an input and output relationship out of the two remaining variables.
Well, we have a vocabulary to say “keeping something (T, or V or P) the same”:
Keep T the same = Isothermic (So P can be input while V can be output, or vice versa).
Keep P the same = Isobaric (So T can be input while V can be output, or vice versa).
Keep V the same = Isochoric (So P can be input while T can be output, or vice versa).
Graph 19 shows a 3D graph (which is like a 3D Loaf of bread).
Here’s how you use the graph (but you must keep one of the three variables constant):
1) Choose any particular T as constant = Isothermic (So P can be input while V can be output, or vice versa).
Slice (with a knife) the loaf of bread along any T you want.
The out line of the new surface area that you just created represents how P varies with V
(You are looking at an Isotherm).
2) Choose any particular P as constant = Isobaric (So T can be input while V can be output, or vice versa).
Similarly, you can slice along any particular P you choose.
Notice how the outline of the new surface area represents how T varies with V.
(You are looking at an Isobar).
3) Choose any particular V as constant = Isochoric (So P can be input while T can be output, or vice versa).
Slicing your knife along a particular V creates a representation of how T and P relate to each other as you keep that particular V constant.
(You are looking at an Isochore).
If eq. 7 is arranged to P=RT/Vm to relate pressure and volume (volume=X, Pressure=Y ), then that new equation follows the pattern Y=1/(X),
as in Figure 14:
Temperature T essentially acts as just a multiplying factor to Y=1/(X).
As we choose greater and greater values for T, we see a multiplying effect:
For example, if we notice that Isotherm A gives a value for P that is twice as great as for the P value given by Isotherm B – both at that particular value for volume then that automatically means that the Temperature of isotherm A is twice the Temperature of isotherm B (Of course, this is an Ideal senario).
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