GAS LAWS GL KapadeChemistry Teacher

KVS

Introduction to Gaseous State some points to remember

Gases do not have a fix volume.They assume the volume of the container in

which they are enclosed.The molecules tend have no packing.Gases have very low relative density. The attractive forces between individual

molecules of a gas is negligible.The molecules tend have very high Kinetic &

Thermal Energy.Gases have very high diffusion rate.Gases are “highly compressible”.

THE GAS LAWS

1. Boyle’s law2. Charles law3. Gay Lussac’s law4. Avogadro’s law5. Combined Gas law 6. Dalton’s law7..Kinetic Molecular Theory8. Behavior of Real Gases 9. Liquefaction of Gases

1.BOYLE’S Law (p-V relationship)This law was proposed by Robert Boyle in 1662.

At constant temperature, the pressure of a fixed amount of gas varies inversely with its volume.

Mathematically-

p=k1*1/V , at constant T & n. .......(1.1)where k1 is the proportionality constant.

The value of k1 depends upon the amount of gas, its temperature and the unit in which p and V are expressed.

From equation (1), pV=k1 .……(1.2)

Pressure – volume variation graphs

.

Fig1.1 p versus V graph Fig1.2 p versus V graph at different temperature

Each curve corresponds to a different constant temperature and are known as ISOTHERM.

Fig1.3 p versus 1/V graph.

Finding out pressure (or volume) of a gas at same temperature under different pressure (or volume)

Let V1 be the volume of a given mass of a gas at pressure p1 and at a given temperature T. When the pressure is changed to p2 at the same temperature, the volume changes to V2.

Then according to Boyle’s law,

p1V1=p2V2=constant (k) ……..(1.3)

or p1/p2=V1/V2 ……..(1.4)

The equation (1.4) can used to any of the four quantity if know any three of them.

2.CHARLES’ LAW (V-T RELATIONSHIP) At constant pressure, the volume of a fixed

amount of gas is directly proportional to its “absolute temperature.”

or The volume of a given mass of a gas, at

constant pressure, increases or decreases by 1/273.15 times of he volume of gas at 0°C, for each one degree rise or fall in absolute temperature respectively.

V=k2T ………(2.1)

V/T=k2 ..…….(2.2)

where k2 is the proportionality constant.

Let Vo be the volume of a given mass of a gas at 0°C and Vt is its volume at

any temperature t°C, then the volume, Vt may be written in terms of Charles’ law(at constant pressure) as:

For 1 degree rise in temperature, volume increases=Vo*1/273.15

For t degree rise in temperature, volume increases=Vo*t/273.15

Therefore, volume at t°C, Vt = initial volume + increase in volume

= Vo + Voxt/273.15

= Vo[1+t/273.15] ..……….(2.3)

= Vo[(273.15+t)/273.15]

= VoxT(K)/To {T(K)=t + 273.15} …………(2.4)

where T(K) is the absolute temperature of the gas and To{=273.15(K)} corresponds to 0°C on absolute temperature scale.

Vt = Vo*T/To

Vt/Vo= T/To

Vt/T = Vo/To

Vt/T = k2

**NOTE:- At t=-273.15°C,the volume the gas of tends to become zero, which is physically unviable. This is because all the gases convert into liquid state before reaching this temperature & hence such an absurd answer is received.

VOLUME VERSUS TEMPERATURE GRAPH

Fig2.1 V versus T graph Fig2.2V versus T graph at different pressure

Each line of the volume vs temperature graph is called ISOBAR.

3.GAY LUSSAC’S Law (p-T RELATIONSHIP)This law was put forward by Gay Lussac.

At constant volume, the pressure of a fixed amount of gas is directly proportional to its absolute temperature.

Mathematically- p=k3T ……….(3.1)

p/T=k3 ……….(3.2) p1/t1=p2/t2 ………….(3.3)

Fig3.1 p versus T graph

Each line in the pressure versus absolute temperature graph is called ISOCHORE.

This law was derived by Amedeo Avogadro.

At a constant pressure and temperature, the volume of a gas is directly proportional to its amount.

Mathematically- V=k4n ……….(4.1) where k4 is proportionality constant and n is the number of moles

of the gas.n=6.022*10²³ and is known as the Avogadro’s constant.Thus, V=k4*m/M ………..(4.2)where m is the mass of the gas and M is the molar mass of the gas.

4.Avogadro’s Law (V-n relationship)

5.The COMBINED GAS Law

Combining the three gas laws,

V=k1/p ……… Boyle’s law(1)

V=k2T ……... Charles’ law(2)

V=k4n ……… Avogadro’s law(3)

We arrive at the combined gas law which can be written as:

V=knT/p ……….(5.1)

Or pV=knT=kRT ……….(5.2)

pV=kRT ………..(5.3)

where R is the Universal Gas constant.

pV=nRT is also called the ideal gas equation.

5.1.The nature of the Universal Gas constant ‘R’

The unit of the Universal gas constant can be found out from the ideal gas equation-

pV= nRT R=pV/nTsince p is Force/unit Area R=(F/A * V)/n*T ……….(5.1.1) where F-force, A-area, V-volume, n-moles and T-absolute temperature.

since A=(length)² and V=(length)³. R=F*L/n*T ………..(5.1.2)

since F*L=Work done, R=Work/n*T ………..(5.1.3)Thus, R represents work done per degree per mol or energy per degree

per mol.

5.2.The numerical values of R

I. When pressure is expressed in atmosphere and volume in litres, the value of R is in atmosphere-litre per mole.

At S.T.P., the pressure is 1 atm, volume is 22.4 L and the temperature is 273.15 K.

Therefore,

R=1 atm*22.4 L/(1 mole*273.15 K)

R=0.0827 l atm /(K mol) ……..(5.2.1)

II. When pressure is expressed in Pascal and volume in m³, then R is expressed as Joule per degree per mol.

R=(100000Pa*22.7m³)/(1 mol*273.15K)

=8.314 Nm/(K mol)

R=8.314 J/(K mol) ……..(5.2.2)

III. When the pressure is expressed in bar and volume in dm³, then R is expressed in bar dm³ per mol per K.

R=(1 bar*22.7 dm³)/(1 mol*K)

R=0.0837 bar dm³/(mol*K) ………..(5.2.3)

NOTE:- For pressure-volume calculations, R must be taken in the same units as those for pressure and volume.

6.DALTON’S LAW OF PARTIAL PRESSUREThis law was formulated by John Dalton in 1801. The total pressure exerted by a mixture of

two or more non-reacting gases in a definite volume is equal to the sum of the partial pressure of the individual gas.

Therefore, Dalton’s law can be stated as-

ptotal=p1+p2 ………..(6.1)

the same can be extended to ‘n’ number of gases

ptotal=p1+p2+p3+……+pn .………

(6.2)

Partial pressure:- The partial pressure of each gas means the pressure which that gas would exert if the present alone in the container at the same and constant temperature as that of the mixture

Let us take a mixture of two gases(non-reacting) in a container of volume V at temperature T.

Then , p1=n1RT/V p2=n2RT/V p3=n3RT/Vwhere n1,n2 and n3 are the number of moles of the three gases and

p1, p2 and p3 are the partial pressure of the gases.

According to Dalton's law- ptotal=p1+p2+p3 =n1RT/V+n2RT/V+n3RT/V =(n1+n2+n3)RT/V ………..(6.3)

Dividing equation p1 the whole equation by ptotal, we getP1/ptotal=[n1/(n1+n2+n3)]RTV/RTV =n1/(n1+n2+n3) =n1/n =x1 Where n=n1+n2+n3

Similarly for the other two gases,

p2=x2*ptotal , p3=x3*ptotal

Generally, pi=xi*ptotal ………….(6.4)

7.The KINETIC MOLECULAR Theory of GASESAssumptions of the Kinetic Molecular theory of

Gases-• All the gases are composed of tiny particles called

molecules.• There is negligible intermolecular force of attraction

between the molecules of gases.• The molecules of a gas are always in constant random

motion.• The molecules of a gas do not have any gravitational

force.• The pressure of the gas is due to the collision of gas

molecules on the walls of the container.• The volume of the gas molecules is negligible in

comparison to the total volume of the gas.• The volume of the gas is directly proportional to the

absolute temperature of the gas. (V=kT)

According to the Kinetic Molecular theory of gases:

The Kinetic Gas equation is….

pV=mNurms²/3 ……..(7.1)

where urms is called the root mean square velocity of the molecules.

Average translational kinetic energy of a molecular temperature T is

=murms²/2The total energy of the whole gas containing N molecules is given

by-

Ek=mNurms²/2 …….(7.2)

And the Kinetic Gas equation is

pV=mNurms²/2Therefore,

pV=2mNurms²/(2*3)

pV=2Ek/3 ………(7.3)

Comparing with the ideal gas equation

pV=RT for one mole of gas

Therefore,

RT=2Ek/3 or Ek=3RT/2 ……….(7.4)

Similarly, Kinetic Energy for n moles of the gas

Ek=3nRT/2 ………(7.5) The average kinetic energy per molecule can obtained by both sides

of

the equation by Avogadro number (6.022*10²³) of molecules, NA

The average kinetic energy for one molecule=3RT/2*NA

=3kbT/2 ………(7.6)

where kb=R/NA and is called the Boltzmann constant.

Verification of the gas laws by Kinetic Molecular theory of gases. Charles’ Law K.E.=mu²/2 pV=mNu²/3

pV=mu²/2 (N=1)

therefore, pV=2K.E./3

At constant p,

V=2kt/3 (K.E.=kT)

or V=kT (2k/3=constant)

Boyle’s Law pV=2kT/3

At constant temperature,

pV=k (2kT/3=constant)

or p=k/V , which is the Boyle’s Law.

8.Behavior of Real gases:Deviation from IDEAL GAS Behaviour

The gases which obey the ideal gas equation at all temperatures and pressures is called on ideal gas or perfect gas.

The gases which do not the ideal gas equation at all temperatures and pressures is called a real gas or non-ideal gas.

Causes for deviation from ideal behavior:- Deviation in pressure values They show deviation because molecules tend to interact with each

other. At high pressures molecules of gases are very close to each. Molecular interactions start to operating. At high pressures, molecules do not strike the walls of the container wit full impact because of attraction from other molecules.

This reduces the total pressure of the gas on the walls of the container. Thus, they exert less pressure than an ideal gas.

pideal=preal + an²/V³ ……….(8.1)where n is the no. of moles of the gas and V is the volume of the gas

and a is a measure of the intermolecular attractive forces within the gas and is independent of the temperature and pressure.

Deviation in the value of volume At high pressures, the volume of the individual gas become

significant. Repulsive interactions are short range interactions and are significant when molecules are almost in contact.

The repulsive force causes the molecules to behave as small but impenetrable spheres. As such the volume occupied by the molecules gets reduced. Now they tend to move in volume V-nb where nb is the total volume occupied by molecules themselves.

Vreal= Videal – nb .………..(8.2)Here b is a constant.

Now the ideal gas equation along with the corrections becomes:

[p + an²/V²](V-nb)=nRT ……….(8.3) This equation is known as van der Waals equation.

8.1.Deviation:-Compressibility factor (Z)

The deviation from ideal behavior can also be expressed in terms of the compressibility factor of the gas Z, which is the ratio of product pV and nRT.

Mathematically, Z = pV/nRT …………

(8.1.1)

For ideal gas Z=1 at all temperatures and pressures because pV=nRT.

It can be seen that for gases which

deviate from ideality , value of Z

deviates from unity.

At very low pressure all gases shown in the graph have Z≈1 and behave as ideal gas. At

high pressure all the gases have Z>1. These are more difficult to compress. At

intermediate pressures, most gases have Z<1.

Thus gases show ideal behavior when the volume occupied is so large that the volume of the molecules can be neglected in comparison to it.

Some points to remember:- At low pressures all the gases show ideal behavior.

At high pressures gases have Z>1 and is called negative deviation.

At intermediate pressures gases have Z<1 and is called positive deviation.

Def:- The temperature at which a real gas obeys ideal gas law over an appreciable range of pressure is called Boyle temperature or Boyle point.

The pressure till which a gas shows ideal gas law, is dependent on the nature of the gas and its temperature.

The Boyle point of a gas is depends upon the nature of the gas. Above Boyle point real gases show positive deviation fro ideality and Z values are greater the unity. Below Boyle point real gases first show decrease in Z value with increase in pressure, which reaches a minimum value. On further increase in pressure the value of Z increases continuously.

We can also see further from the following derivation Z=pVreal/n*RT If the gas shows ideal behavior then videal=nRT/pPutting this value in (8.1.2) we have Z=Vreal/VidealFrom (8.1.3) we can see that the ratio of the actual volume of a gas

to the molar volume of it, if it were an ideal gas at that temperature and pressure.

Exception:- We have seen that the behavior of the H2 and He is always increasing.

This is due to the fact that ‘a’ for Hydrogen and He is very small indicating that forces of attraction in these gases are very weak. Therefore, ‘a’/V²is negligible at all pressures so that Z is always greater than unity.

Difference between a Real gas and an Ideal gas:- IDEAL GAS REAL GAS

1. Ideal gas obeys all gas laws under all conditions of temperature and pressure.

1. Real gas obeys gas laws only under low pressure and high temperature.

2. In ideal gas, the volume occupied the molecules is negligible as compared to the volume occupied by the gas.

2. In real gas, the volume occupied by the molecules is significant in comparison to the total volume occupied by the gas.

3. The force of attraction among the molecules of gas are negligible.

3. The force of attraction among the molecules are significant at

all temperatures and pressures.

4. It obeys the ideal gas equation: pV=nRT

4. It obeys the van der Waals equation:

[p + an²/V²](V – nb) = nRT

According to the Kinetic Molecular theory of gases, the forces of attraction between the molecules are negligible. When the temperature is lowered, the K.E. of the molecules decreases. As a result the slow molecules tend to come nearer to one other. At a sufficiently low temperature, some of the molecules cannot resist the force of attraction and they come closer and ultimately the gas changes its state into liquid. They can also be brought closer by increasing the pressure as the volume decreases.

Thus the gases can be liquefied by increasing the pressure or decreasing the temperatures or by both.

Thomas Andrews showed that at high temperatures isotherms look like that of an ideal gas and the gas cannot be liquefied by even at very high pressures. But as the temperature is lowered, the shape of the of the curves changes considerably(Fig9.1). He used CO2 as the gas to show this property of the gases.

From the graph(Fig 9.1) it can be seen that CO2 remains gas up to 73 atm pressure at 30.98°C. This is the highest temperature at liquid CO2 is observed. Above this temperature it remains as a gas.

9.Liquefaction of Gases

Fig9.1 Isotherms of CO2 at different temperatures Fig 9.2 Isotherms of a gas at two different temperatures

Fig9.3 graph showing thermal equilibrium b/w Fig9.4Graph showing the compressibility gas and liquid. factor of a real gas.

Definitions:-1. Critical temperature:- It is the highest temperature at which a gas converts

to a liquid.2. Critical Volume:- The volume of one mole of gas at critical temperature is

called critical volume. 3. Critical pressure:- the pressure at the critical temperature is called critical

pressure.4. Critical constants:- The critical temperature, pressure and volume are

called critical constants.Explaining the graph-

The steep line in the graph(fig 9.1 & 9.2) represents the isotherms of liquid.

a. Increasing pressure:-Further increase in pressure just compresses the liquid CO2 and the curve represents the compressibility of the liquid. Even a slight compression results in rise in pressure indicating low compressibility of the liquid.

Below 30.98°C, the behavior of the gas on compression is quite different. At 21.5°C, CO2 remains as a gas only upto point B. at point B, liquid of a particular volume appears. Further compression does not change the pressure.

b. Condensation :-Liquid and gaseous CO2 co-exist and further application of pressure causes condensation of more gas until the point C is reached. At point c all the gas has been condensed.

c. Phase diagram:- Below 30.98°C, the length of each curve increases. As is clear from

the diagram, the central horizontal line (FG) at 13.1 is larger then that at 21.5°C (BC). Alternatively, we observe that the horizontal portion becomes smaller as the temperature increases. At critical point A, we can say that it represents the gaseous state.

Point C:- It represents the completion of the condensation process.

Point D:- it represents the liquid state. The area under the point D represents the co-existence of liquid

and gaseous CO2 on equilibrium.Points to understand:-• There is always a certain temperature above which a gas

cannot be liquefied, no matter how much pressure is applied.

• The gas can only be liquefied below this temperature.• Every gas has a critical pressure which is dependent on its

temperature.

Critical constants for some substances

Substance Tc (K) Pc (bar) Vc (dm ³/mol)

H₂ 33.2 12.97 0.0650

He 5.3 2.29 0.0577

N₂ 126.0 33.9 0.0900

O₂ 154.3 50.4 0.0744

CO₂ 304.10 73.9 0.0956

H₂O 647.1 220.6 0.0450

NH₃ 405.5 113.0 0.0723

BE THOROUGH WITH THE CHAPTER