CHAPTER THIRTEEN
KINETIC THEORY
13.1 Introduction
13.2 Molecular nature of matter
13.3 Behaviour of gases
13.4 Kinetic theory of an ideal gas
13.5 Law of equipartition of energy
13.6 Specific heat capacity
13.7 Mean free path Summary Points to ponder
Exercises Additional exercises
13.1 INTRODUCTION
Boyle discovered the law named after him in 1661. Boyle, Newton and several others tried to explain the behaviour of gases by considering that gases are made up of tiny atomic particles. The actual atomic theory got established more than 150 years later. Kinetic theory explains the behaviour of gases based on the idea that the gas consists of rapidly moving atoms or molecules. This is possible as the inter-atomic forces, which are short range forces that are important for solids and liquids, can be neglected for gases. The kinetic theory was developed in the nineteenth century by Maxwell, Boltzmann and others. It has been remarkably successful. It gives a molecular interpretation of pressure and temperature of a gas, and is consistent with gas laws and Avogadro’s hypothesis. It correctly explains specific heat capacities of many gases. It also relates measurable properties of gases such as viscosity, conduction and diffusion with molecular parameters, yielding estimates of molecular sizes and masses. This chapter gives an introduction to kinetic theory.
13.2 MOLECULAR NATURE OF MATTER
Richard Feynman, one of the great physicists of 20th century considers the discovery that “Matter is made up of atoms” to be a very significant one. Humanity may suffer annihilation (due to nuclear catastrophe) or extinction (due to environmental disasters) if we do not act wisely. If that happens, and all of scientific knowledge were to be destroyed then Feynman would like the ‘Atomic Hypothesis’ to be communicated to the next generation of creatures in the universe. Atomic Hypothesis: All things are made of atoms little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another.
Speculation that matter may not be continuous, existed in many places and cultures. Kanada in India and Democritus
Atomic Hypothesis in Ancient India and Greece
Though John Dalton is credited with the introduction of atomic viewpoint in modern science, scholars in ancient India and Greece conjectured long before the existence of atoms and molecules. In the Vaiseshika school of thought in India founded by Kanada (Sixth century B.C.) the atomic picture was developed in considerable detail. Atoms were thought to be eternal, indivisible, infinitesimal and ultimate parts of matter. It was argued that if matter could be subdivided without an end, there would be no difference between a mustard seed and the Meru mountain. The four kinds of atoms (Paramanu — Sanskrit word for the smallest particle) postulated were Bhoomi (Earth), Ap (water), Tejas (fire) and Vayu (air) that have characteristic mass and other attributes, were propounded. Akasa (space) was thought to have no atomic structure and was continuous and inert. Atoms combine to form different molecules (e.g. two atoms combine to form a diatomic molecule dvyanuka, three atoms form a tryanuka or a triatomic molecule), their properties depending upon the nature and ratio of the constituent atoms. The size of the atoms was also estimated, by conjecture or by methods that are not known to us. The estimates vary. In Lalitavistara, a famous biography of the Buddha written mainly in the second century B.C., the estimate is close to the modern estimate of atomic size, of the order of 10–10 m.
In ancient Greece, Democritus (Fourth century B.C.) is best known for his atomic hypothesis. The word ‘atom’ means ‘indivisible’ in Greek. According to him, atoms differ from each other physically, in shape, size and other properties and this resulted in the different properties of the substances formed by their combination. The atoms of water were smooth and round and unable to ‘hook’ on to each other, which is why liquid /water flows easily. The atoms of earth were rough and jagged, so they held together to form hard substances. The atoms of fire were thorny which is why it caused painful burns. These fascinating ideas, despite their ingenuity, could not evolve much further, perhaps because they were intuitive conjectures and speculations not tested and modified by quantitative experiments - the hallmark of modern science.
in Greece had suggested that matter may consist of indivisible constituents. The scientific ‘Atomic Theory’ is usually credited to John Dalton. He proposed the atomic theory to explain the laws of definite and multiple proportions obeyed by elements when they combine into compounds. The first law says that any given compound has, a fixed proportion by mass of its constituents. The second law says that when two elements form more than one compound, for a fixed mass of one element, the masses of the other elements are in ratio of small integers.
To explain the laws Dalton suggested, about 200 years ago, that the smallest constituents of an element are atoms. Atoms of one element are identical but differ from those of other elements. A small number of atoms of each element combine to form a molecule of the compound. Gay Lussac’s law, also given in early 19th century, states: When gases combine chemically to yield another gas, their volumes are in the ratios of small integers. Avogadro’s law (or hypothesis) says: Equal volumes of all gases at equal temperature and pressure have the same number of molecules. Avogadro’s law, when combined with Dalton’s theory explains Gay Lussac’s law. Since the elements are often in the form of molecules, Dalton’s atomic theory can also be referred to as the molecular theory of matter. The theory is now well accepted by scientists. However even at the end of the nineteenth century there were famous scientists who did not believe in atomic theory !
From many observations, in recent times we now know that molecules (made up of one or more atoms) constitute matter. Electron microscopes and scanning tunnelling microscopes enable us to even see them. The size of an atom is about an angstrom (10 -10 m). In solids, which are tightly packed, atoms are spaced about a few angstroms (2 Å) apart. In liquids the separation between atoms is also about the same. In liquids the atoms are not as rigidly fixed as in solids, and can move around. This enables a liquid to flow. In gases the interatomic distances are in tens of angstroms. The average distance a molecule can travel without colliding is called the mean free path. The mean free path, in gases, is of the order of thousands of angstroms. The atoms are much freer in gases and can travel long distances without colliding. If they are not enclosed, gases disperse away. In solids and liquids the closeness makes the interatomic force important. The force has a long range attraction and a short range repulsion. The atoms attract when they are at a few angstroms but repel when they come closer. The static appearance of a gas is misleading. The gas is full of activity and the equilibrium is a dynamic one. In dynamic equilibrium, molecules collide and change their speeds during the collision. Only the average properties are constant.
Atomic theory is not the end of our quest, but the beginning. We now know that atoms are not indivisible or elementary. They consist of a nucleus and electrons. The nucleus itself is made up of protons and neutrons. The protons and neutrons are again made up of quarks. Even quarks may not be the end of the story. There may be string like elementary entities. Nature always has surprises for us, but the search for truth is often enjoyable and the discoveries beautiful. In this chapter, we shall limit ourselves to understanding the behaviour of gases (and a little bit of solids), as a collection of moving molecules in incessant motion.
13.3 BEHAVIOUR OF GASES
Properties of gases are easier to understand than those of solids and liquids. This is mainly because in a gas, molecules are far from each other and their mutual interactions are negligible except when two molecules collide. Gases at low pressures and high temperatures much above that at which they liquefy (or solidify) approximately satisfy a simple relation between their pressure, temperature and volume given by (see Ch. 11)
PV = KT (13.1)
for a given sample of the gas. Here T is the temperature in kelvin or (absolute) scale. K is a constant for the given sample but varies with the volume of the gas. If we now bring in the idea of atoms or molecules then K is proportional to the number of molecules, (say) N in the sample. We can write K = N k . Observation tells us that this k is same for all gases. It is called Boltzmann constant and is denoted by k .
B
PV PV
11 22
As = constant = k(13.2)
NT NTB
11 22
if P, V and T are same, then N is also same for all gases. This is Avogadro’s hypothesis, that the number of molecules per unit volume is same for all gases at a fixed temperature and pressure. The number in 22.4 litres of any gas is 6.02 1023. This is known as Avogadro number and is denoted by NA. The mass of 22.4 litres of any gas is equal to its molecular weight in grams at
S.T.P (standard temperature 273 K and pressure 1 atm). This amount of substance is called a mole (see Chapter 2 for a more precise definition). Avogadro had guessed the equality of numbers in equal volumes of gas at a fixed temperature and pressure from chemical reactions. Kinetic theory justifies this hypothesis.
The perfect gas equation can be written as
PV = μ RT (13.3) where μ is the number of moles and R = NA kB is a universal constant. The temperature T is absolute temperature. Choosing kelvin scale for
John Dalton (1766- 1844)
He was an English chemist. When different types of atoms combine,
they obey certain simple laws. Dalton’s atomic theory explains these
laws in a simple way. He also gave a theory of colour
blindness.
Amedeo Avogadro (1776 – 1856)
He made a brilliant guess that equal volumes of gases have equal number of molecules at the same temperature and pressure. This helped in understanding the combination of different gases in
a very simple way. It is now called Avogadro’s hypothesis (or law). He also suggested that the smallest constituent of gases like hydrogen, oxygen and nitrogen are not atoms but diatomic molecules.
absolute temperature, R = 8.314 J mol–1K–1. Here
MN
(13.4)
M0 NA
where M is the mass of the gas containing N molecules, M0 is the molar mass and NA the Avogadro’s number. Using Eqs. (13.4) and (13.3) can also be written as
PV = kB NT or P = kB nT
i.e., keeping temperature constant, pressure of a given mass of gas varies inversely with volume. This is the famous Boyle’s law. Fig. 13.2 shows comparison between experimental P-V curves and the theoretical curves predicted by Boyle’s law. Once again you see that the agreement is good at high temperatures and low pressures. Next, if you fix P, Eq. (13.1) shows that V ∝ T i.e., for a fixed pressure, the volume of a gas is proportional to its absolute temperature T (Charles’ law). See Fig. 13.3.
P (atm)
Fig.13.1 Real gases approach ideal gas behaviour at low pressures and high temperatures.
where n is the number density, i.e. number of molecules per unit volume. kB is the Boltzmann constant introduced above. Its value in SI units
10–23
is 1.38 J K–1. Another useful form of Eq. (13.3) is
RTP
(13.5)
M0
where ρ is the mass density of the gas.
A gas that satisfies Eq. (13.3) exactly at all pressures and temperatures is defined to be an ideal gas. An ideal gas is a simple theoretical model of a gas. No real gas is truly ideal. Fig. 13.1 shows departures from ideal gas behaviour for a real gas at three different temperatures. Notice that all curves approach the ideal gas behaviour for low pressures and high temperatures.
At low pressures or high temperatures the molecules are far apart and molecular interactions are negligible. Without interactions the gas behaves like an ideal one.
If we fix μ and T in Eq. (13.3), we get
PV = constant (13.6)
Fig.13.2 Experimental P-V curves (solid lines) for steam at three temperatures compared with Boyle’s law (dotted lines). P is in units of 22 atm and V in units of 0.09 litres.
Finally, consider a mixture of non-interacting ideal gases: μ moles of gas 1, μ moles of gas
1 2
2, etc. in a vessel of volume V at temperature T and pressure P. It is then found that the equation of state of the mixture is :
PV = ( μ1 + μ2 +… ) RT (13.7)
RT RT
i.e. P 1 2 ... (13.8)
VV
= P1 + P2 + … (13.9) Clearly P1 = μ1 R T/V is the pressure gas 1 would exert at the same conditions of volume and temperature if no other gases were present. This is called the partial pressure of the gas. Thus, the total pressure of a mixture of ideal gases is the sum of partial pressures. This is Dalton’s law of partial pressures.
density of water molecule may therefore, be regarded as roughly equal to the density of bulk water = 1000 kg m–3. To estimate the volume of a water molecule, we need to know the mass of a single water molecule. We know that 1 mole of water has a mass approximately equal to
•
(2 + 16)g = 18 g = 0.018 kg.
Since 1 mole contains about 6 1023 molecules (Avogadro’s number), the mass of a molecule of water is (0.018)/(6 1023) kg = 3 10–26 kg. Therefore, a rough estimate of the volume of a water molecule is as follows :
Volume of a water molecule = (3 10–26 kg)/ (1000 kg m–3) 10–29
= 3 m3 = (4/3) π (Radius)3
Fig. 13.3 Experimental T-V curves (solid lines) for
Hence, Radius ≈ 2 10-10 m = 2 Å •
CO2 at three pressures compared with
Charles’ law (dotted lines). T is in units of
Example 13.3 What is the average
300 K and V in units of 0.13 litres.
distance between atoms (interatomic
•
distance) in water? Use the data given inWe next consider some examples which give Examples 13.1 and 13.2.
us information about the volume occupied by the molecules and the volume of a single Answer : A given mass of water in vapour statemolecule.
has 1.67103 times the volume of the same mass of water in liquid state (Ex. 13.1). This is alsoExample 13.1 The density of water is 1000 the increase in the amount of volume available
•
kg m–3. The density of water vapour at 100 °C for each molecule of water. When volumeand 1 atm pressure is 0.6 kg m–3. The increases by 103 times the radius increases byvolume of a molecule multiplied by the total V1/3 or 10 times, i.e., 10 2 Å = 20 Å. So thenumber gives ,what is called, molecular average distance is 2 20 = 40 Å. W
volume. Estimate the ratio (or fraction) of
the molecular volume to the total volume
Example 13.4 A vessel contains two non-occupied by the water vapour under the reactive gases : neon (monatomic) andabove conditions of temperature and
•
oxygen (diatomic). The ratio of their partialpressure.
pressures is 3:2. Estimate the ratio of (i) number of molecules and (ii) mass densityAnswer For a given mass of water molecules, of neon and oxygen in the vessel. Atomicthe density is less if volume is large. So the mass of Ne = 20.2 u, molecular mass of O2volume of the vapour is 1000/0.6 = /(6 10 -4 ) = 32.0 u.
times larger. If densities of bulk water and water molecules are same, then the fraction of
Answer Partial pressure of a gas in a mixture is molecular volume to the total volume in liquid
the pressure it would have for the same volume state is 1. As volume in vapour state has
and temperature if it alone occupied the vessel. increased, the fractional volume is less by the
(The total pressure of a mixture of non-reactive same amount, i.e. 610-4. •
gases is the sum of partial pressures due to its constituent gases.) Each gas (assumed ideal)Example 13.2 Estimate the volume of a obeys the gas law. Since V and T are common towater molecule using the data in Example the two gases, we have PV = μ RT and PV =
13.1. 11 2
μRT, i.e. (P/P) = (μ/ μ). Here 1 and 2 refer
2 121 2
Answer In the liquid (or solid) phase, the to neon and oxygen respectively. Since (P1/P2) = molecules of water are quite closely packed. The (3/2) (given), (μ1/ μ2) = 3/2.
(i) By definition μ = (N/N ) and μ= (N/N)
11A2 2A
where N1 and N2 are the number of molecules of 1 and 2, and NA is the Avogadro’s number. Therefore, (N/N) = (μ/ μ) = 3/2.
121 2
(ii) We can also write μ1 = (m1/M1) and μ2 = (m/M) where mand m are the masses of
221 2
1 and 2; and M1 and M2 are their molecular masses. (Both m1 and M1; as well as m2 and M2 should be expressed in the same units). If ρ1 and ρ2 are the mass densities of 1 and 2 respectively, we have
m /Vm M
11 111
m /Vm M
22 222
320.2
0.947 W232.0
13.4 KINETIC THEORY OF AN IDEAL GAS
Kinetic theory of gases is based on the molecular picture of matter. A given amount of gas is a collection of a large number of molecules (typically of the order of Avogadro’s number) that are in incessant random motion. At ordinary pressure and temperature, the average distance between molecules is a factor of 10 or more than the typical size of a molecule (2 Å). Thus the interaction between the molecules is negligible and we can assume that they move freely in straight lines according to Newton’s first law. However, occasionally, they come close to each other, experience intermolecular forces and their velocities change. These interactions are called collisions. The molecules collide incessantly against each other or with the walls and change their velocities. The collisions are considered to be elastic. We can derive an expression for the pressure of a gas based on the kinetic theory.
We begin with the idea that molecules of a gas are in incessant random motion, colliding against one another and with the walls of the container. All collisions between molecules among themselves or between molecules and the walls are elastic. This implies that total kinetic energy is conserved. The total momentum is conserved as usual.
13.4.1 Pressure of an Ideal Gas
Consider a gas enclosed in a cube of side l. Take the axes to be parallel to the sides of the cube, as shown in Fig. 13.4. A molecule with velocity
Fig. 13.4 Elastic collision of a gas molecule with the wall of the container.
(vx, vy, vz ) hits the planar wall parallel to yzplane of area A (= l2). Since the collision is elastic, the molecule rebounds with the same velocity; its y and z components of velocity do not change in the collision but the x-component reverses sign. That is, the velocity after collision is (-vx, vy, vz ) . The change in momentum of the molecule is : –mvx – (mvx) = – 2mvx . By the principle of conservation of momentum, the momentum imparted to the wall in the collision = 2mv .
x
To calculate the force (and pressure) on the wall, we need to calculate momentum imparted to the wall per unit time. In a small time interval Δt, a molecule with x-component of velocity v will hit the wall if it is within the distance v Δxt
x
from the wall. That is, all molecules within the volume Avx Δt only can hit the wall in time Δt. But, on the average, half of these are moving towards the wall and the other half away from the wall. Thus the number of molecules with velocity (vx, vy, vz ) hitting the wall in time Δt is DA v Δt n where n is the number of molecules
x
per unit volume. The total momentum transferred to the wall by these molecules in time Δt is :
Q = (2mvx) (D n A v x Δt ) (13.10)
The force on the wall is the rate of momentum transfer Q/Δt and pressure is force per unit area :
P = Q /(A Δt) = n m vx 2 (3.11)
Actually, all molecules in a gas do not have the same velocity; there is a distribution in velocities. The above equation therefore, stands for pressure due to the group of molecules with speed vx in the x-direction and n stands for the number density of that group of molecules. The
total pressure is obtained by summing over the contribution due to all groups:
P = n m vx 2 (13.12)
wherev2 is the average of v 2 . Now the gas
x x
is isotropic, i.e. there is no preferred direction of velocity of the molecules in the vessel. Therefore, by symmetry,
22 2
v = v = v
xy z
222 2
= (1/3) [v + v + v ] = (1/3) v (13.13)
x yz
where v is the speed and v2 denotes the mean of the squared speed. Thus
P = (1/3) n m v2 (13.14)
Some remarks on this derivation. First, though we choose the container to be a cube, the shape of the vessel really is immaterial. For a vessel of arbitrary shape, we can always choose a small infinitesimal (planar) area and carry through the steps above. Notice that both A and Δt do not appear in the final result. By Pascal’s law, given in Ch. 10, pressure in one portion of the gas in equilibrium is the same as anywhere else. Second, we have ignored any collisions in the derivation. Though this assumption is difficult to justify rigorously, we can qualitatively see that it will not lead to erroneous results. The number of molecules hitting the wall in time Δt was found to be D n Av Δt. Now the collisions
x
are random and the gas is in a steady state. Thus, if a molecule with velocity (vx, vy, vz ) acquires a different velocity due to collision with some molecule, there will always be some other molecule with a different initial velocity which after a collision acquires the velocity (vx, vy, vz ). If this were not so, the distribution of velocities would not remain steady. In any case we are
finding v2 . Thus, on the whole, molecular
x
collisions (if they are not too frequent and the time spent in a collision is negligible compared to time between collisions) will not affect the calculation above.
13.4.2 Kinetic Interpretation of Temperature
Equation (13.14) can be written as PV = (1/3) nV m v2 (13.15a)
Founders of Kinetic Theory of Gases
James Clerk Maxwell (1831 – 1879), born in Edinburgh, Scotland, was among the greatest physicists of the nineteenth century. He derived the thermal velocity distribution of molecules in a gas and was among the first to obtain reliable estimates of molecular parameters from measurable quantities like viscosity, etc. Maxwell’s greatest achievement was the unification of the laws of electricity and magnetism (discovered by Coulomb, Oersted, Ampere and Faraday) into a consistent set of equations now called Maxwell’s equations. From these he arrived at the most important conclusion that light is an electromagnetic wave. Interestingly, Maxwell did not agree with the idea (strongly suggested by the Faraday’s laws of electrolysis) that electricity was particulate in nature.
Ludwig Boltzmann
(1844 – 1906) born in Vienna, Austria, worked on the kinetic theory of gases independently of Maxwell. A firm advocate of atomism, that is basic to kinetic theory, Boltzmann provided a statistical interpretation of the Second Law of thermodynamics and the concept of entropy. He is regarded as one of the founders of classical statistical mechanics. The proportionality constant connecting energy and temperature in kinetic theory is known as Boltzmann’s constant in his honour.
PV = (2/3) N x D m v2 (13.15b) where N (= nV) is the number of molecules in the sample.
The quantity in the bracket is the average translational kinetic energy of the molecules in the gas. Since the internal energy E of an ideal gas is purely kinetic*,
E = N (1/2) m v2 (13.16)
Equation (13.15) then gives : PV = (2/3) E (13.17) We are now ready for a kinetic interpretation
of temperature. Combining Eq. (13.17) with the
M 2 28 –26
mN 4.65 10
kg.
NA 6.02 1026
v2 = 3 kB T / m = (516)2 m2s-2
The square root of v2 is known as root mean square (rms) speed and is denoted by v rms,
( We can also write v2 as < v2 >.)
v = 516 m s-1
rms
The speed is of the order of the speed of sound in air. It follows from Eq. (13.19) that at the same temperature, lighter molecules have greater rms speed.
ideal gas Eq. (13.3), we get
W
Example 13.5 A flask contains argon and
E = (3/2) kB NT (13.18) chlorine in the ratio of 2:1 by mass. The
or E/ N = D m v2 = (3/2) kBT (13.19) i.e., the average kinetic energy of a molecule is proportional to the absolute temperature of the
gas; it is independent of pressure, volume or the nature of the ideal gas. This is a fundamental result relating temperature, a macroscopic measurable parameter of a gas (a thermodynamic variable as it is called) to a molecular quantity, namely the average kinetic energy of a molecule. The two domains are connected by the Boltzmann constant. We note in passing that Eq. (13.18) tells us that internal energy of an ideal gas depends only on temperature, not on pressure or volume. With this interpretation of temperature, kinetic theory of an ideal gas is completely consistent with the ideal gas equation and the various gas laws based on it.
For a mixture of non-reactive ideal gases, the total pressure gets contribution from each gas in the mixture. Equation (13.14) becomes
P = (1/3) [nm2 + nm2 +… ] (13.20)
11 v12 2 v2
In equilibrium, the average kinetic energy of the molecules of different gases will be equal. That is,
D m2 = D m2 = (3/2) k T
1 v12 v2 B
so that
P = (n1 + n2 +… ) kB T (13.21)
which is Dalton’s law of partial pressures.
From Eq. (13.19), we can get an idea of the typical speed of molecules in a gas. At a temperature T = 300 K, the mean square speed of a molecule in nitrogen gas is :
temperature of the mixture is 27 °C. Obtain the ratio of (i) average kinetic energy per molecule, and (ii) root mean square speed v rms of the molecules of the two gases. Atomic mass of argon = 39.9 u; Molecular mass of chlorine = 70.9 u.
Answer The important point to remember is that the average kinetic energy (per molecule) of any (ideal) gas (be it monatomic like argon, diatomic like chlorine or polyatomic) is always equal to (3/2) kBT. It depends only on temperature, and is independent of the nature of the gas.
(i)
Since argon and chlorine both have the same temperature in the flask, the ratio of average kinetic energy (per molecule) of the two gases is 1:1.
(ii)
Now D m v rms2 = average kinetic energy per molecule = (3/2) ) kBT where m is the mass of a molecule of the gas. Therefore,
2
vrms m M
Ar Cl Cl 70.9
2 = =1.77
v mM
rms Cl Ar Ar 39.9
where M denotes the molecular mass of the gas. (For argon, a molecule is just an atom of argon.) Taking square root of both sides,
v
rms Ar
= 1.33
v
rms Cl
You should note that the composition of the mixture by mass is quite irrelevant to the above
* E denotes the translational part of the internal energy U that may include energies due to other degrees of freedom also. See section 13.5.
Maxwell Distribution Function
In a given mass of gas, the velocities of all molecules are not the same, even when bulk parameters like pressure, volume and temperature are fixed. Collisions change the direction and the speed of molecules. However in a state of equilibrium, the distribution of speeds is constant or fixed.
Distributions are very important and useful when dealing with systems containing large number of objects. As an example consider the ages of different persons in a city. It is not feasible to deal with the age of each individual. We can divide the people into groups: children up to age 20 years, adults between ages of 20 and 60, old people above 60. If we want more detailed information we can choose smaller intervals, 0-1, 1-2,..., 99-100 of age groups. When the size of the interval becomes smaller, say half year, the number of persons in the interval will also reduce, roughly half the original number in the one year interval. The number of persons dN(x) in the age interval x and x+dx is proportional to dx or dN(x) = nx dx. We have used nx to denote the number of persons at the value of x.
Maxwell distribution of molecular speeds
In a similar way the molecular speed distribution gives the number of molecules between the speeds v and v+ dv. dN(v) = 4p N a3e–bv2 v2 dv = nv dv. This is called Maxwell distribution. The plot of nv against v is shown in the figure. The fraction of the molecules with speeds v and v+dv is equal to the area of the strip shown. The average of any quantity like v2 is defined by the integral