# IB Physics Unit 3 Modelling a gas Notes

### 3.2 Modelling a gas

Essential Idea:
The properties of ideal gases allow scientists to make predictions of the behaviour of real gases

Understandings:

• Pressure
• Equation of state for an ideal gas
• Kinetic model of an ideal gas
• Mole, molar mass and the Avogadro constant
• Differences between real and ideal gases

Applications and Skills:

• Solving problems using the equation of state for an ideal gas and gas laws
• Sketching and interpreting changes of state of an ideal gas on pressure– volume, pressure–temperature and volume–temperature diagrams
• Investigating at least one gas law experimentally

Data booklet reference:

### BEHAVIOUR OF GASES

#### IDEAL GAS

In an ideal gas, we assume that molecules are point masses and there is no mutual attraction between them. The ideal gas obeys following laws :

BOYLE’S LAW
According to Boyle’s law for a given mass of ideal gas, the pressure of a ideal gas is inversely proportional to the volume at constant temperature
i.e.,

$$P\propto \frac{1}{V}$$ When T is constant

CHARLE’S LAW
For a given mass, the volume of a ideal gas is proportional to temperature at a constant pressure
i.e.,

GAY-LUSSAC’S LAW
For a given mass of ideal gas, the pressure is proportional to temperature at constant volume
i.e.,

According to Avogadro’s law, the number of molecules of all gases are same at same temperature, pressure and volume
i.e., for same P, V and T.
The value of Avogadro number is 6.02 × 1023 molecules.

GRAHAM’S LAW
• At constant temperature and pressure, the rms speed of diffusion of two gases is inversely proportional to the square root of the relative density
i.e., vrms
• According to Graham’s law, the rate of diffusion of a gas is inversely proportional to the square root of its density, provided pressure and temperature are constant

DALTON’S LAW
The pressure exerted by a gaseous mixture is equal to sum of partial pressure of each component gases present in the mixture,
i.e.,  P = P1 + P2 + P3+……………… Pn
A relation connecting macroscopic properties  P, V and T of a gas describing the state of the system is called equation of state.
The equation of state for an ideal gas of n mole is
where R is universal gas constant whose value is
R = 8.31 J/mol K and R = NAk, where k is Boltzmann’s constant (NA is Avogadro number). n is the number of moles of a gas and
where m is the mass of a gas, N is the number of molecules and M is the molecular weight of a gas.

#### EQUATION OF REAL GAS

The real gas follows Vander Wall’s law. According to this: ;
here a and b are Vander Waal’s constant

#### CRITICAL TEMPERATURE, VOLUME AND PRESSURE

• The temperature at or below which a gas can be liquefied by applying pressure alone is called critical temperature TC. It is given by
• The volume of gas at a critical temperature TC is called critical volume VC, where VC = 3b
• The pressure of gas at a critical temperature TC is called critical pressure PC, where

#### GAS EQUATION

• The gases found in nature are real gases.
• The real gas do not obey ideal gas equation but they obey Vander Waal’s gas equation
• a’ depends upon the intermolecular force and the nature of gas.
• b’ depends upon the size of the gas molecules and represents the volume occupied by the molecules of the gas.
• The molecules of real gas have potential energy as well as kinetic energy.
• The real gas can be liquefied and solidified.
• The real gases like CO2, NH3, SO2 etc. obey Vander Wall’s equn at high pressure and low temperature.

### KINETIC THEORY OF AN IDEAL GAS

Any sample of a gas is made of molecules. A molecule is the smallest unit having all the chemical properties of the sample. The observed behaviour of a gas results from the detailed behaviour of its large number of molecules. The kinetic theory of gases attempts to develop a model of the molecular behaviour which should result in the observed behaviour of an ideal gas.
The basic assumptions of kinetic theory are :
• A gas consist of particles called molecules which move randomly in all directions.
• The volume of molecule is very small in comparison to the volume occupied by gas i.e., the size of molecule is infinitesimally small.
• The collision between two molecules or between a molecule and wall are perfectly elastic and collision time (duration of collision) is very small.
• The molecules exert no force on each other or on the walls of containers except during collision.
• The total number of molecules are large and they obey Newtonian mechanics.
….(i)
where is called mean square velocity and is called root mean square velocity
i.e,  ….(ii)
and   ….(iii)
where m = mass of one molecule and n = number of molecules
….(iv)
or
or   ….(v)
where E is translational kinetic energy per unit volume  of the gas. It is clear that pressure of ideal gas is equal to 2/3rd of translational kinetic energy per unit volume.

Kinetic interpretation of temperature : From eqn. (iv), we get
For 1 mole of a gas at temperature T :
PV = RT  so
(is Boltzmann constant)
or       ….(vi)
or
or          ….(vii)
or

It is clear from eq.(vi) that at a given temperature, the average translational kinetic energy of any gas molecules are equal i.e., it depends only on temperature.
From eqn. (vii) It is clear that
• , where M is molecular mass of the gas.

The kinetic molecular theory can be used to explain each of the experimentally determined gas laws.

### DEGREE OF FREEDOM

The degree of freedom of a particle is the number of independent modes of exchanging energy or the number of independent motion, which the particle can undergo.
For monatomic gas such as helium, argon, neon etc. the molecules can have three independent motion i.e., it has
3 degree of freedom, all translational.
For a diatomic gas molecules such as H2, O2, N2, etc. it has two independent rotational motion besides of three independent translational motion, so it has 5 degree of freedom.
In polyatomic gas molecules such as CO2, it can rotate about any of three coordinate axes. It has six degree (three translational + three rotational) of freedom. At high temperature the molecule can vibrate also and degree of freedom due to vibration also arises, but we neglect it.

### MEAN FREE PATH

The distance covered by the molecules between two successive collisions is called the free path.
The average distance covered by the molecules between two successive collisions is called the mean free path.
i.e.,
where, n = number of molecules per unit volume
d = diameter of each molecule
KB = Boltzmann’s constant
T = temperature
P = pressure
Mean free path depends on the diameter of molecule (d) and the number of molecules per unit volume n.
At N.T.P., λ for air molecules is 0.01 µm.

### LAW OF EQUIPARTITION OF ENERGY

If we dealing with a large number of particles in thermal equilibrium to which we can apply Newtonian mechanics, the energy associated with each degree of freedom has the same average value (i.e., ), and this average value depends on temperature.
From the kinetic theory of monatomic ideal gas, we have
….(i)
or
Energy of molecule =  (number of degree of freedom) ×  ….(ii)
So it is clear from equation (i) and (ii) that monatomic gas has three degree of freedom and energy associated per degree of freedom is (where k is Boltzmann’s constant)
Since we know that internal energy of an ideal gas depends only on temperature and it is purely kinetic energy. Let us consider an 1 mole ideal gas, which has N molecules and f degree of freedom, then total internal energy
U = (total number of molecules)
× (degree of freedom of one molecule)
× (energy associated with each degree of freedom)
=(N) (f) × ()
U = Nf ….(iii)
Differentiating eq.(iii) w.r.t  T, at constant volume, we get,
….(iv)
Now molar, specific heat at constant volume is defined as
….(v)
so   ….(vi)
where R is universal gas constant.
Now by Mayer’s formula,
…. (vii)
So ratio of specific heat is      …(viii)
where f is degree of freedom of one molecule.
(a) For monatomic gas,  f = 3, ,
so
(b) For diatomic gas,  f = 5,  ,
(c) For polyatomic gas,  f = 6, ,

KEEP IN MEMORY
1. Real gases behave like perfect gas at high temperature and low pressure.
In real gas, we assume that the molecules have finite size and intermolecular attraction acts between them.
1. Real gases deviate most from the perfect gas at high pressure and low temperature.
2. Gaseous state of matter below critical temperature is called vapours. Below critical temperature gas is vapour and above critical temperature vapour is gas.
3. Random motion of the constituents of the system involving exchange of energy due to mutual collisions is called thermal motion.
4. Total kinetic energy or internal energy or total energy does not depend on the direction of flow of heat. It is determined by the temperature alone.
5. The internal energy of a perfect gas consists only of kinetic energy of the molecules. But in case of the real gas it consists of both the kinetic energy and potential energy of inter molecular configuration.

### DISTRIBUTION OF MOLECULAR SPEEDS

The speed of all molecules in a gas is not same but speeds of individual molecules vary over a wide range of magnitude.
Maxwell derived the molecular distribution law (by which we can find distribution of molecules in different speeds) for sample of a gas containing N molecules, which is
…(1)
where N(v)dv is the number of molecules in the gas sample having speeds between v and v + dv.
where T = absolute temperature of the gas
m = mass of molecule
k = Boltzmann’s constant
The total number of molecules N in the gas can be find out by integrating equation (1) from 0 to ∞ i.e.,
….(2)
Figure shows Maxwell distribution law for molecules at two different temperature T1 and T2(T2 > T1).
The number of molecules between v1 and v2 equals the area under the curve between the vertical lines at v1 and v2 and the total number of molecules as given by equation (2) is equal to area under the distribution curve.
The distribution curve is not symmetrical about most probable speed, vP,(vP is the speed, which is possessed in a gas by a large number of molecules) because the lowest speed must be zero, whereas there is no limit to the upper speed a molecule can attain. It is clear from fig.1 that
vrms(root mean square) > (average speed of molecules) > (most probable speed) vP

### AVERAGE, ROOT MEAN SQUARE AND MOST PROBABLE SPEED

#### AVERAGE SPEED

To find the average speed , we multiply the number of particles in each speed interval by speed v characteristic of that interval. We sum these products over all speed intervals and divide by total number of particles
i.e.,  (where summation is replaced by integration because N is large)
….(1)

#### ROOT MEAN SQUARE SPEED

In this case we multiply the number of particles in each speed interval by v2 characteristic of that interval; sum of these products over all speed interval and divide by N
i.e.,  ….(2)
Root mean square speed is defined as

#### MOST PROBABLE SPEED

It is the speed at which N(v) has its maximum value (or possessed by large number of molecules), so
….(3)
It is clear from eqn. (1), (2) and (3)
The root mean square velocity of a particle in thermal system is given by
where R  universal gas constant
T  temperature of gas
m  mass of the gas
M  molecular weight of gas or mass of one mole of a gas.
The average speed of the gas molecules is given by
where M is the mass of one mole and m is the mass of one particle.
Most probable speed is that with which the maximum number of molecules move. It is given by
The most probable speed, the average speed as well as root mean square speed increases with temperature.

KEEP IN MEMORY
1. Brownian motion, provides a direct evidence for the existence of molecules and their motion. The zig-zag motion of gas molecules is Brownian motion.
2. Average speed
1. Root mean square speed,
1. Most probable speed
1. Vrms : : Vmp = 1.73 : 1.60 : 1.41
Vrms > > Vmp.

### Ideal Gases

#### 3.2.1 State the macroscopic gas laws relating pressure, volume and temperature.

Boyle’s Law – For a gas at a constant temperature the volume and the pressure are inversely proportional:

(1)

\begin{align} p \propto {1 \over V} \end{align}
 Plotting pressure vs. volume produces the following graph: Plotting pressure vs. inverse volume produces a straight line graph:

If a thermodynamic system changes, but its temperature remains constant then the following the initial pressure and volume are related to the final pressure and volume by the following equation:

(2)

$$P_1 V_1 = P_2 V_2$$

Charles’ Law – If the mass and pressure of a gas are held constant, the volume of the gas is directly proportional to its absolute temperature.

(3)

\begin{align} V \propto T \end{align}

If a thermodynamic system changes, but its mass and pressure are held constant the initial volume and temperature are related to the final volume and temperature by the following equation:

(4)

\begin{align} {V_1 \over T_1} = {V_2 \over T_2} \end{align}

Gay-Lussac’s Law ¬– If the volume of a sample of gas remains constant, the absolute pressure of the gas is directly proportional to its absolute temperature.

(5)

\begin{align} P \propto T \end{align}

If a thermodynamic system changes, but the volume are held constant the initial pressure and temperature are related to the final pressure and temperature by the following equation:

(6)

\begin{align} {P_1 \over T_1} = {P_2 \over T_2} \end{align}

All of the above mentioned laws required special circumstances that are not often satisfied. A more general relationship between the volume, pressure and temperature can be found:

(7)

\begin{align} {P_1 V_1 \over T_1} = {P_2 V_2 \over T_2} \end{align}

This still does not take into account a change in mass. If the temperature and pressure of a gas is held constant and more gas is added (mass increases) the volume must increase. If the volume and temperature of a gas are held constant while more gas is added the pressure must increase. If we combine these experimental observations we can write an even more general relation:

(8)

\begin{align} {P_1 V_1 \over m_1 T_1} = {P_2 V_2 \over m_2 T_2} \end{align}

#### 3.2.2 Define the terms mole and molar mass

In general we use relatively small amount of gas or just a few numbers of atoms or molecules. It is possible to determine the mass of individual atoms or molecules, but the numbers are messy, a gram is an incredibly large mass compared to the mass of an atom. So we define a new mass unit, called an atomic mass unit or amu.

An atomic mass unit is defined as 1/12th the mass of a carbon-12 atom. Basically the mass of a proton or neutron, or a proton or neutron have mass 1 amu. Note: Neutrons and protons do not have exactly the same mass, but its close enough. Thus one carbon-12 atom has mass 12 amu.

Helium-4 has an mass 4 amu or approximately and fluorine-19 has a mass of 19 amu or approximately $1.99 \times 10^{-23} g$. Which numbers would you rather use?

It would be a rare find to find a scale that measures in amu. So how do we relate amu to grams? Here we define the concept of a mole. A mole (mol) is the number of particles in a sample such that the mass in grams of the sample is equal to the mass in amu of a single particle. So 1 mol of carbon-12 atoms has a mass of 12 g, 1 mol of helium-4 has a mass of 4 g.

From this we can define the concept of molar mass, this is simply the grams per mole. Carbon-12’s molar mass is $12g \cdot mol^-1$.

#### 3.2.3 Define the Avogadro constant

One day some body got bored and wondered exactly how many particles are in a mole, this guy’s name was Avogadro (written in symbol form NA). I don’t think he actually made a good measurement of his number, but hey he got a symbol named after him, better than I could do.

(9)

\begin{align} N_A = {Number of Particles \over Number of Moles} \end{align}

This number is called Avogadro’s number of Avogadro’s constant. The accepted value is:

(10)

\begin{align} N_A = 6.023 \times 10^23 \end{align}

#### 3.2.4 State that the equation of state of an ideal gas is

The number of moles of gas to the mass of gas is related by the molecular mass of the gas. Therefore we can write the mass as:

(11)

$$m = nM$$

Where m is the mass, n is the number of moles and M is the molecular mass. If we substitute this expression in to the general gas law:

#### 3.2.1 State the macroscopic gas laws relating pressure, volume and temperature.

Boyle’s Law – For a gas at a constant temperature the volume and the pressure are inversely proportional:

(12)

\begin{align} p \propto {1 \over V} \end{align}
 Plotting pressure vs. volume produces the following graph: Plotting pressure vs. inverse volume produces a straight line graph:

If a thermodynamic system changes, but its temperature remains constant then the following the initial pressure and volume are related to the final pressure and volume by the following equation:

(13)

$$P_1 V_1 = P_2 V_2$$

Charles’ Law – If the mass and pressure of a gas are held constant, the volume of the gas is directly proportional to its absolute temperature.

(14)

\begin{align} V \propto T \end{align}

If a thermodynamic system changes, but its mass and pressure are held constant the initial volume and temperature are related to the final volume and temperature by the following equation:

(15)

\begin{align} {V_1 \over T_1} = {V_2 \over T_2} \end{align}

Gay-Lussac’s Law ¬– If the volume of a sample of gas remains constant, the absolute pressure of the gas is directly proportional to its absolute temperature.

(16)

\begin{align} P \propto T \end{align}

If a thermodynamic system changes, but the volume are held constant the initial pressure and temperature are related to the final pressure and temperature by the following equation:

(17)

\begin{align} {P_1 \over T_1} = {P_2 \over T_2} \end{align}

All of the above mentioned laws required special circumstances that are not often satisfied. A more general relationship between the volume, pressure and temperature can be found:

(18)

\begin{align} {P_1 V_1 \over T_1} = {P_2 V_2 \over T_2} \end{align}

This still does not take into account a change in mass. If the temperature and pressure of a gas is held constant and more gas is added (mass increases) the volume must increase. If the volume and temperature of a gas are held constant while more gas is added the pressure must increase. If we combine these experimental observations we can write an even more general relation:

(19)

\begin{align} {P_1 V_1 \over n_1 M T_1} = {P_2 V_2 \over n_2 M T_2} \end{align}

Assuming the molecular mass is constant:

(20)

\begin{align} {P_1 V_1 \over n_1 T_1} = {P_2 V_2 \over n_2 T_2} \end{align}

This can be written yet again as:

(21)

\begin{align} {P_1 V_1 \over n_1 T_1} =R \end{align}

Or as the ideal gas law:

(22)

$$pV = nRT$$

An ideal gas being a gas that obeys the ideal gas law at all pressures, volumes and temperatures. No real gas is ideal, but most gases obey the ideal gas law for low pressure and high temperatures (above their freezing point).

R is called the universal gas constant, the value of R is dependent on the choice of units:

(23)

\begin{align} R = 8.314 J \cdot mol^-1 \cdot K^-1 \end{align}

The universal gas constant is given in the IB formula booklet.

#### 3.2.5 Describe the concept of absolute zero and the Kelvin scale

If the volume of a gas is measured at constant pressure but at different temperatures a linear relationship is found. If the line is extended to the left it will eventually intercept the temperature axis, if the line went farther than this it would represent a negative volume which does not make physical sense. If lines are plotted for several different gases it can be found that they have different slopes, but intercept the temperature axis at the same point. This temperature is in theory a minimum temperature, or absolute zero. Absolute zero does not mean there is no internal energy, but only that internal energy is at a minimum.

It should be noticed also that the equations presented above would provide nonsensical answers for negative temperatures, i.e. negative volume or negative pressure. This is simply a mathematical issue. Due to this, a new scale the Kelvin scale was developed where 0 K is absolute zero. One degree Kelvin is the same change in temperature as one degree Celsius. 0 K is defined as -273.16°C, for the purposes of the IB -273°C is sufficient.

#### 3.2.8 Explain the macroscopic behavior of an ideal gas in terms of molecular model

The assumptions or postulates of the moving particle theory are extended for an ideal gas to include:

• Gases consist of tiny particles called atoms (monatomic gases such as neon and argon) or molecules.
• The total number of molecules in any sample of gas is extremely large.
• The molecules are in constant random motion.
• The range of the intermolecular forces is small compared to the average separation of the molecules.
• The size of the particles is relatively small compared with the distance between them.
• Collisions of short duration occur between molecules and the walls of the container and the collisions are perfectly elastic.
• No forces act between particles except when they collide, and hence particles move in straight lines.
• Between collisions the molecules, obey Newton’s Laws of motion

The view of an ideal gas is one of molecules moving in random straight line paths at constant speeds until they collide with the sides of the container or with one another. Their paths over time are therefore zig-zags. Because the gas molecules can move freely and are relatively far apart, they occupy the total volume of a container.

The large number of particles ensures that the number of particles moving in all directions is constant at any time.

The pressure that the molecules exert is due to their collisions with the sides of the container. As the temperature of a gas is increased, the average kinetic energy per molecule increases. The increase in velocity of the molecules leads to a greater rate of collisions, and each collision involves greater impulse. Hence the pressure of the gas increases as the collisions with the sides of the container increase. When a force is applied to a piston in a cylinder containing a volume of gas, the molecules take up a smaller volume and hence collisions are more frequent leading to an increase in pressure.

Because the collisions are perfectly elastic there is no loss in kinetic energy as a result of the collisions. Temperature is a measure of the average kinetic energy per molecule.