Monday, May 6, 2019
Notes to a video lecture on http://www.unizor.com
and Volume of Ideal Gas
Temperature is an observable macro property of an object. It's related to a particular instrument we use to measure this property.
Let's examine the mechanism of this measurement using a classic mercury-based or alcohol-based thermometer.
Our first step in measurement is to make a physical contact between a thermometer and an object of measurement (for example, a human body or air in a room). When accomplished, we expect that the measurement of a thermometer would correspond to a state of average kinetic energy of molecules of an object. The reason it happens (and it takes some time to happen) is that on a micro level molecules at the surface of an object are colliding with molecules at the surface of a thermometer, and exchange the kinetic energy, eventually equalizing it. The molecules close to a surface, in turn, collide with surface molecules and also eventually equalize their average kinetic energies. This process continues until the average level of kinetic energy in all parts of an object and in a thermometer equalize.
What is important in this case is that the total amount of kinetic energy of all molecules of an object and a thermometer remains the same. So, if an object has more intense movement of molecules and a thermometer's molecules are moving slower, the kinetic energy is transferred from an object to a thermometer. If the molecules of a thermometer are, on average, faster, then the exchange of kinetic energy will be from a thermometer to an object.
In any case, the average kinetic energy of molecules of both an object and a thermometer equalize.
An important consideration is that the contact between an object and a thermometer changes the average level of kinetic energy in both. The process of measuring, therefore, is not completely neutral towards an object. However, what happens in most cases is that the number of molecules inside an object we measure is usually significantly greater that the number of molecules in a thermometer. As a result, equalizing the average kinetic energy of all molecules does not significantly change the level of average kinetic energy of molecules of an object, and the level of average kinetic energy of the molecules of a thermometer is a good representation of this characteristic of an object.
As explained in the Heat and Energy lecture of this course, the temperature in mercury or alcohol thermometers is an observable expansion of the volume of liquid inside a thermometer. We also indicated in that lecture that this thermo-expansion is proportional to an average of squares of velocities of molecules, that is proportional to average kinetic energy of the molecules of a thermometer, which, in turn, is equalized with average kinetic energy of molecules of an object.
Thus, by observing the expansion of liquid in a thermometer we measure the average kinetic energy of molecules of an object, which allows us to write the following equation:
T ≅ AVE(Ekin) = Eave
where T is an observable level of liquid inside a thermometer in some units (that is, temperature) and
AVE(Ekin) = Eave is average kinetic energy of molecules of an object.
This looks more natural in a form
AVE(Ekin) = Eave ≅ T
since average kinetic energy of molecules Eave (micro characteristic) cannot be easily observed, while the temperature T (macro characteristic) can.
So, we have established that the temperature (in some units, starting from absolute zero) and average kinetic energy of molecules are proportional. The coefficient of proportionality between the temperature and average kinetic energy of molecules remains unknown and is different for different substances.
Obviously, the measure of liquid in a thermometer should be calibrated and, for this equation to be true, we have to assign the zero temperature to a state of an object when all its molecules are at rest, which happens when there is no source of energy around, like in open space far from stars.
The convenient scale is the Kelvin scale with zero temperature on this scale corresponding to this state of molecules at complete rest and a unit of measurement of the temperature is a degree with the distance between the temperature of melting ice and boiling water assigned as 100 degrees on this scale.
From the previous lecture about kinetics of ideal gas we know the relationship between the pressure of ideal gas on the walls of a reservoir, the volumeof a reservoir and average kinetic energy of the molecules
p = (2/3)N·Eave /V
p is the gas pressure against the walls of a reservoir,
N is the number of gas molecules in a reservoir,
V volume of a reservoir,
Eave is average kinetic energy of molecules,
Alternatively, it can be written as
Eave = (3/2)·p·V/N
Comparing this with a derived above relationship between temperature (counting from the absolute zero) and average kinetic energy of molecules, we have established a relationship
Eave = (3/2)·p·V/N ≅ T
p·V/N = k·T
where k is an unknown coefficient of proportionality.
The only thing that prevents us from determining average kinetic energy of molecules by its temperature is an unknown coefficient of proportionality k.
Now we will concentrate attention on gases, as an object of measuring temperature and average kinetic energy of molecules.
Different gases have different molecules and different molecular mass. Using certain theory and using chemical and physical experiments, we can compare the masses of different molecules and even measure this mass in certain "units of mass" called atomic mass units.
For such a unit of molecular mass scientists used 1/12 of a mass of a single atom of carbon. So, in this system of units hydrogen molecule of 2 hydrogen atoms H2 has approximate mass of 2, oxygen molecule of two oxygen atoms O2 - approximately 32, carbon dioxide molecule CO2 had a measure of, approximately, 44 atomic mass units, etc.
Using this measurement, we can always establish experiments with the same number of molecules of different gases. For example, if the mass of certain amount of oxygen (atomic mass of molecules O2 is 32) is 16 times greater than the mass of certain amount of hydrogen (atomic mass of molecules H2 is 2), we can assume that the number of molecules in both cases is the same.
It has been experimentally established that, if the same number of different gas molecules are placed in reservoirs of the same volume and hold them at the same temperature, the pressure on the walls in both cases will be the same. Alternatively, if the pressure is the same, the temperature will be the same too.
In other words, the coefficient kin formula
p·V/N = k·T
does not depend on the type of gas we deal with, it's a universal constant called the Boltzman's constant, which is equal to
kB = 1.381·10−23 (J/°K)
This was the reason to introduce a concept of ideal gas. All gases are, approximately, ideal to a certain degree of precision. This is related to the fact that molecules of the gas are flying with high speeds and on large distances from each other, much larger than their geometric sizes.
Now we can write the equation between the temperature T (in degrees °K from absolute zero), average kinetic energy of molecules (in units of SI joules J), pressure p (in units of SI newton/m²), volume V (in m³) and number of molecules in a reservoir for ideal gas:
Eave = (3/2)·p·V/N = (3/2)·kB·T
Consequently, if we are dealing with certain fixed amount of gas (N molecules) then
p·V/T = kB·N = const
That means that changing the pressure, volume and temperature of the same amount of gas preserves the expression p·V/T
which is called the Combined Ideal Gas Law.
For example, if the absolute temperature remains the same, but volume taken by certain amount of gas increases (decreases) by some factor, the pressure will decrease (increase) by the same factor, that is pressure and volume are inversely proportional to each other (Boyle-Mariotte's Law).
If the pressure remains the same, but the volume taken by certain amount of gas increases (decreases) by some factor, the absolute temperature will increase (decrease) by the same factor, that is volume and absolute temperature are proportional to each other (Charles' Law).
If the volume taken by certain amount of gas remains the same, but the absolute temperature increases (decreases) by some factor, the pressure will increase (decrease) by the same factor, that is absolute temperature and pressure are proportional to each other (Gay-Lussac's Law).
Out of curiosity, let's use the formula
Eave = (3/2)·kB·T
to calculate how fast the molecules of oxygen are flying in the room at some normal temperature.
Assume the pressure at the ground level is about 100,000N/m² and the temperature in the room is about 20°C=293°K. Then the average kinetic energy of a molecule of oxygen is
Eave = (3/2)·1.381·10−23·293 =
= 6.06·10−21 J
Mass of a molecule of oxygen O2 is m=5.31·10−26 kg
From the formula for kinetic energy E=m·v²/2 we derive the average of squares of velocities of oxygen molecules as
AVE(v²) = 2·Eave /m =
= 2·6.06·10−21/5.31·10−26 =
Therefore, the average speed of oxygen molecule will be equal to a square root of this number:
AVE(v) = 478 m/sec
Pretty fast moving! Take into consideration, however, that real oxygen molecules, as molecules of any real gas, are chaotically colliding with other and change the direction all the time.
Wednesday, May 1, 2019
Notes to a video lecture on http://www.unizor.com
Kinetics of Ideal Gas
In this lecture we will discuss a concept of pressure of gas, enclosed in some reservoir, against the walls of this reservoir.
Usually, speaking about pressure, we have in mind an object of certain weight on a horizontal table, in which case the constant force of its weight exerts a pressure on a table equal to this weight divided by an area of the table occupied by an object or, as it's sometimes presented, the weight per unit of area.
As an introduction to kinetics of gases, let's consider a reservoir shaped as a cube with sides aligned parallel to coordinate planes and with the length of each edge L.
Assume that there is a single molecule of gas of mass mflying between the opposite walls perpendicularly to them, elastically reflecting off these walls.
Keep in mind that, according to the Third Newton's Law, the force exerted by the wall towards the molecule, which causes the reflection of a molecule from the wall, is equal to the force exerted by a molecule towards the wall, which causes pressure.
The area of each of these opposite walls equals to L², which will be used to calculate the pressure.
In this case of a single gas molecule, flying between the opposite walls of a reservoir, the force is variable. It exists during the time when a molecule hits the wall and then the force becomes zero until the next time this molecule hits the same wall.
A proper definition of pressurein this case is based on a concept of the average force during certain amount of time.
Firstly, let's evaluate the average force during a single period of oscillation of a molecule between walls from a moment of time it's near one wall to a next moment it's at this position after flying to the opposite wall and returning back.
As we know, a change of a momentum of an object equals to an impulse of the force that caused this change
m·vend − m·vbeg = F·τ
m is a mass of an object
vbeg is the velocity of an object in the beginning of the action of the force
vend is the velocity of an object at the end of the action of the force
F is the force, acting on an object
τ is the time duration of the force, acting on an object.
Keep in mind that velocity and force are vectors in the above equation.
In our case we consider one period of oscillation of a molecule as the time period of average force acted on it.
So, τ is the time between two consecutive events when a molecule is at the wall opposite to the one it collides with:
τ = 2L/|v|.
where |v| is now an absolute speed of a molecule (scalar).
Since we are considering elastic reflection of a molecule in the opposite direction to its initial trajectory,
vend = −vbeg
and the equation above can be written in scalar form
2m·|v| = |F|·τ = |F|·2L/|v|
from which we determine the absolute average force
|F| = m·|v|²/L
Mass m of a molecule is, obviously, a constant during this process.
The average pressure of a molecule on a wall is the average force divided by the area of a wall:
p = m·|v|²/L³ = m·|v|²/V =
= 2Ekin /V
where V is the volume of a reservoir and Ekin is a kinetic energy of a molecule.
Our first step towards a comprehensive theory is to consider a case of many molecules flying parallel to each other in the same reservoir as above, but with, generally speaking, different speeds. Obviously, the pressure against the wall will be greater because each molecule contributes its own force against the wall during a collision.
If vi is a speed of the ithmolecule, the combined pressure will be
p = Σpi = Σm·vi²/V =
= 2Etot /V
where Etot is a total kinetic energy of all molecules moving parallel to each other.
Considering the number of molecules N remains the same, it's more convenient to express this formula in terms of an average of squares of molecular speed and average kinetic energy of molecules:
v²ave = (1/N)Σvi²
Eave = (1/N)ΣEi
Using this average of squares of molecular speed, the pressure is
p = N·m·v²ave /V = 2N·Eave /V
which brings us to a principle of proportionality between pressure of the gas against the walls of a reservoir and an average of squares of the molecular speed or to an average of kinetic energy of the molecules.
Granted, we showed this only for a case of all molecules moving between two opposite walls parallel each other and perpendicularly to the walls they collide with.
Let's make another step towards comprehensive theory and consider the chaotic movement of all gas molecules.
To quantitatively approach molecular movement, we will use a model of an ideal gas.
This model assumes that molecules of ideal gas are point-mass objects completely chaotically moving in all directions with equal probabilities and elastically colliding at random times among themselves or with all the walls of a reservoir that contains this gas.
We further assume that no interacting forces (like gravity or electromagnetic forces) exist between them. So, only kinetic energy of these molecules plays the role in evaluation of the characteristics of the ideal gas.
Every vector of velocity of any molecule can be represented as a sum of three vectors along the XYZ-coordinates
v = vx+vy+vz
So, every movement of a molecule can be represented as simultaneous movement in three different directions. Depending on which wall of the reservoir is hit by a molecule, its pressureagainst the wall is determined only by one component - the one that is perpendicular to the surface of a wall.
If molecules inside the reservoir are moving completely chaotically, they are hitting all walls with approximately the same frequency. So, for each of the six walls of a cube we can use the same logic as above for one side, except for the walls parallel to YZ coordinate plane the pressures px will be proportional to the average of vx², for the walls parallel to XZ coordinate plane the pressures py will be proportional to the average of vy² and for the walls parallel to XY coordinate plane the pressures pz will be proportional to the average of vz².
Considering completely chaotic character of the molecular motion in the ideal gas, the three pressures px, py and pzmust be the same. Similarly, averages of vx², vy² and vz² must be also the same.
Since v = vx+vy+vz, according to three-dimensional equivalent of Pythagorean Theorem,
v² = vx² + vz² + vz²
and, similarly, for averages
AVE(v²) = v²ave =
From the equality of averages for the ideal gas we come to the following equalities:
v²ave = 3·AVE(vx²)
AVE(vx²) = (1/3)·v²ave
px = N·m·AVE(vx²)/V =
= (1/3)N·m·v²ave /V =
= (2/3)N·Eave /V
py = N·m·AVE(vy²)/V =
= (1/3)N·m·v²ave /V =
= (2/3)N·Eave /V
pz = N·m·AVE(vz²)/V =
= (1/3)N·m·v²ave /V =
= (2/3)N·Eave /V
And, since pressures on all walls are the same in the ideal gas,
p = (1/3)N·m·v²ave /V =
= (2/3)N·Eave /V = (2/3)Etot /V
The expression of a pressure in terms of average kinetic energy and volume is more general than in terms of mass, average of squares of molecular speed and volume, as it encompasses a case with molecules of different masses.