This section we will discuss about the fundamental theory of gases as we focus on kinetic theory of gases (perfect or ideal gas), real gases, intermolecular forces, and kinetic theory. Besides that in kinetic theory of gases we will see perfect gases' state functions in term of energy and as well for real gases. In this section we will also see briefly about diffusion and effusion.
To begin with, most of us acknowledge that gases are the simplest state of matter that we know. Besides that, it also has low density and the molecules move independently. The independent movement of gases is caused by the separation each molecules that around 10 times their size on average. Therefore, to understand the behaviour of gases, we need to see the state functions of gases. The state functions are not necessarily solid, liquid, or even gas; but it also tell us about the energy.
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| Robert Boyle |
The first state function of gases is the relationship between pressure and volume in constant temperature. This relationship was found by Robert Boyle, some say it was found firstly by Henry Power, in 1660 and he stated that the pressure is inversely proportional to the volume. Therefore, to increase the pressure of gases, we just need to compress the gases or make smaller volume.
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| Jacques Alexandre César Charles |
Besides that, Jacques Charles in 1787 stated the relationship between volume is proportional to the temperature of the gases with the condition constant pressure and constant amount of gas molecules. Hence, as the gases are heated, the kinetic energy of the gases increases, so the gas molecules expand.
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William Thomson
1st Baron Kelvin |
Moreover, in 1848 William Thomson, 1st Baron Kelvin, extrapolated the graphs between volume and temperature and he found they had point of interaction. From this concept, it appears the temperature scale in Kelvin (K) which is 0 K is equal with -273.15 °C.
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Lorenzo Romano Amedeo Carlo Avogadro di Quaregna e di Cerreto
Count of Quaregna and Cerreto |
In 1811, Amedeo Avogadro, Count of Quaregna and Cerreto, found there is a direct proportionality between the volume and the number of gas molecules in constant temperature and pressure. Hence, from this relationship, increasing the number of gas molecules which mean bigger volume is required (which is quite obvious). Below is an excerpt from Avogadro about this relationship.
"Volumi eguali di sostanze gassose, a eguale temperatura a pressione, rappresentano lo stesso numero di molecole, in modo che le densita dei diversi gas sono la misura delle masse delle loro molecolo e i rapporti dei volumi nelle combinazioni fra gas altro non sono che i rapporti fra i numeri di molecole che si combinano per formare molecole composte."
Amedeo Avogadro, 14 July 1811
From Avogadro's relationship about volume and the number of gas molecule in a system, John Dalton developed a term about partial pressures. Partial pressures occur in a mixture of gasses as the individual component of the mixture contribute to the total pressure. This relationship can be applied to all gasses and it does not depend on the gas. The ideal gas equation does not care about chemistry of the gas. From the explanation above, we can write the total pressure of gas mixture as
Then, by using definition of mole fraction, x,
Therefore, we can rewrite the equation as:
From three relationships that mentioned earlier, we can plot the perfect gas behaviour as the state functions of pressure, volume, and temperature to give a curve as shown below.
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A region of the p,V,T surface of a fixed amount of perfect gas.
The points forming the surface represent the only states of the gas that can exist |
As we discussed briefly about behaviour of gases, we can discuss about kinetic theory of gases (KTG). As mentioned, the gas molecules move independently and as the gas molecules move around, the gas molecules have kinetic energy. In kinetic theory of gases there are 3 main assumptions that are used. Firstly, the gas consist of particles of mass, m, in constant ceaseless random motion (except at 0 K - quantum mechanics dictates that even at absolute 0 motion still exists. This is zero point motion and is a consequence of the uncertainty principle). Secondly, the particles have no interactions except brief, infrequent, elastic collision. In this case, elastic means that kinetic energy is conserved in the collision. Lastly, the size of particle is negligible compared to the distance over which it moves between collisions. In short, KTG states that gas molecules move, have no interactions, and have no size.
In early section, we mentioned about states of matter briefly. However, states can be defined by state functions which includes n (amount of particles, moles), V (volume, m3), T (temperature, K), and p (pressure, Pa or Nm-2). These states are related each other in equation of state for a system (solid, liquid, or gas). From 3 relationships that we saw earlier (Boyle, Charles, and Avogadro), we can defined equation of state for perfect (ideal) gas which is
Where R is a gas constant (8.31451 J mol-1 K-1) and the "standard state" for a gas is defined as Standard Ambient Temperature and Pressure (SATP). SATP is defined as temperature of 298.15 K (25°C), at pressure (p') 1 bar (100 kPa); so at SATP volume molar (Vm) for perfect gas is 0.02479 m3 mol-1.
After we discussed about the state functions of gas molecules, we can see the gasses in molecular picture. Firstly, gases in a vessel exert a pressure as the gas molecules in continual motion which mean the gas molecules possess kinetic energy. After that, the gas molecules collide with the walls of vessel impart brief force. Moreover, it is not only one or two collisions but it can reach billions of collisions (1 mol of gas = 6.022 14 x 1023 particles) per second which implies constant force and constant force per unit area is pressure. This phenomenon can be treated quantitatively with kinetic theory of gases.
In kinetic theory of gases we need to remember that we assume the size of molecules is negligible and the molecules do not interact except during collisions. Hence, the ideal gas equation can be rewrite as:
Where M is the molar mass (kg mol-1) and c is the root mean square (r.m.s.) of speed of the gas which is:
Then, assuming that KTG gives right expression of pV, so for the perfect gas:
At 298 K, c for a typical gas (N2) is around 500 m s-1, but there are 3 point which should be remembered. Firstly, gas molecules have distribution of speeds; secondly individual speeds constantly changing; and lastly, the typical collision frequency is around 10-9 s.
In perfect gas, all energy is kinetic energy and average of kinetic energy of a gas molecules:
Then, we can substitute
c from the previous equation, so the total energy of 1 mole of gas molecules is:
and for
n moles of a gas is:
As additional notes, temperature is a direct measure of kinetic energy and
U (internal energy) of perfect gas depends only on
T.
Besides that, for 1 molecule, the average of kinetic energy is:
This equation the kinetic energy is thermal energy and it is based on the equipartition principle which means the kinetic energy per degree of freedom.
In the equation of r.m.s. of speed, there is one of the three important notes which is gas molecules have distribution of speeds. The gas molecules have constant redistribution of molecular speeds and it is the fraction of molecules with speeds between
s and
s + d
s. James Clerk Maxwell formulated the distribution of speeds as shown below in 1866.
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| James Clerk Maxwell |
The implication of distribution of speeds is the gas molecules possess distribution of energies (kinetic energy) as well. Ludwig Boltzmann stated that probabilities,
p and
p', of finding a system in states of energy, separated by difference of energies is given by:
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| Ludwig Eduard Boltzmann |
From Maxwell and Boltzmann equations we can plot Maxwell-Boltzmann distribution which states
as T changes the area (number of molecules) stays constant. The Maxwell-Boltzmann distribution plot is shown below.
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| The distribution of molecular speeds with temperature and molar mass |
Furthermore, there are some proofs of kinetic theory of gas which is diffusion and effusion. Diffusion and effusion is basically the same as the random movement of gas molecules and both processes depends on
c. However, diffusion is a process of movement of gas molecules in a vessel, and effusion the movement of gas molecules through a hole or gap. Thomas Graham formulated the rate of effusion is proportional to the mass of molecules to the power -1/2. Hence, the Graham's law can be formulated as
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| Thomas Graham |
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| The variations of compression factors at 273 K |
Until this far, we only discuss about the perfect gas which is assumed as no volume and no long-range interactions. Real gases show deviations from the perfect gas law because molecules interact with one another. A point to keep in mind is that repulsive forces between molecules assist expansion and attractive forces assist compression. Repulsive forces are significant only when molecules are almost in contact: they are short-range interactions, even on a scale measured in molecular diameters. Because they are short-range interactions, repulsions can be expected to be important only when the average separation of the molecules is small. This is the case at high pressure, when many molecules occupy a small volume. On the other hand, attractive intermolecular forces have a relatively long range and are effective over several molecular diameters. They are important when the molecules are fairly close together but not necessarily touching. Attractive forces are ineffective when the molecules are far apart. Intermolecular forces are also important when the temperature is so low that the molecules travel with such low mean speeds that they can be captured by one another.
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| Johannes Diderik van der Waals |
At low pressures, when the sample occupies a large volume, the molecules are so far apart for most of the time that the intermolecular forces play no significant role,and the gas behaves virtually perfectly. At moderate pressures, when the average separation of the molecules is only a few molecular diameters, the attractive forces dominate the repulsive forces. In this case, the gas can be expected to be more compressible than a perfect gas because the forces help to draw the molecules together. At high pressures, when the average separation of the molecules is small, the repulsive forces dominate and the gas can be expected to be less compressible because now the forces help to drive the molecules apart.
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| A region of the p,V,T surface of a fixed amount of perfect gas. |
Because of deviations from the perfect gas equation of state, so another constants should be introduced to the equation. Van der Waals introduced 2 new constants, a and b, which is b for the repulsions forces from the gas molecules. The repulsions reduce volume available to molecules and the reduction is proportional to the amount of gas molecules. In short b is a measure of the size of the gas molecules, to be explicit it is the hard sphere volume of the molecule and this shows up in the minimum separation in a collision.
The constant a is due to attractive forces that reduce pressure. The total attractive interaction is proportional to the concentration of the gas molecules and pressure is proportional to the rate of impacts multiplied with average strength, so it becomes proportional to the concentration squared. In short, a is a measure of the strengths of the interactions between the molecules of the gas. Hence, van der Waals equation is shown as below.
Hence, from the van der Waals equation, we have a new surface plot as shown above.
To begin with, most of us acknowledge that gases are the simplest state of matter that we know. Besides that, it also has low density and the molecules move independently. The independent movement of gases is caused by the separation each molecules that around 10 times their size on average. Therefore, to understand the behaviour of gases, we need to see the state functions of gases. The state functions are not necessarily solid, liquid, or even gas; but it also tell us about the energy.
The first state function of gases is the relationship between pressure and volume in constant temperature. This relationship was found by Robert Boyle, some say it was found firstly by Henry Power, in 1660 and he stated that the pressure is inversely proportional to the volume. Therefore, to increase the pressure of gases, we just need to compress the gases or make smaller volume.
Besides that, Jacques Charles in 1787 stated the relationship between volume is proportional to the temperature of the gases with the condition constant pressure and constant amount of gas molecules. Hence, as the gases are heated, the kinetic energy of the gases increases, so the gas molecules expand.
In 1811, Amedeo Avogadro, Count of Quaregna and Cerreto, found there is a direct proportionality between the volume and the number of gas molecules in constant temperature and pressure. Hence, from this relationship, increasing the number of gas molecules which mean bigger volume is required (which is quite obvious). Below is an excerpt from Avogadro about this relationship.
Then, by using definition of mole fraction, x,
Therefore, we can rewrite the equation as:
Where R is a gas constant (8.31451 J mol-1 K-1) and the "standard state" for a gas is defined as Standard Ambient Temperature and Pressure (SATP). SATP is defined as temperature of 298.15 K (25°C), at pressure (p') 1 bar (100 kPa); so at SATP volume molar (Vm) for perfect gas is 0.02479 m3 mol-1.
Where M is the molar mass (kg mol-1) and c is the root mean square (r.m.s.) of speed of the gas which is:
Then, assuming that KTG gives right expression of pV, so for the perfect gas:
At 298 K, c for a typical gas (N2) is around 500 m s-1, but there are 3 point which should be remembered. Firstly, gas molecules have distribution of speeds; secondly individual speeds constantly changing; and lastly, the typical collision frequency is around 10-9 s.
Then, we can substitute c from the previous equation, so the total energy of 1 mole of gas molecules is:
and for n moles of a gas is:
As additional notes, temperature is a direct measure of kinetic energy and U (internal energy) of perfect gas depends only on T.
This equation the kinetic energy is thermal energy and it is based on the equipartition principle which means the kinetic energy per degree of freedom.
In the equation of r.m.s. of speed, there is one of the three important notes which is gas molecules have distribution of speeds. The gas molecules have constant redistribution of molecular speeds and it is the fraction of molecules with speeds between s and s + ds. James Clerk Maxwell formulated the distribution of speeds as shown below in 1866.
From Maxwell and Boltzmann equations we can plot Maxwell-Boltzmann distribution which states as T changes the area (number of molecules) stays constant. The Maxwell-Boltzmann distribution plot is shown below.
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