Introduction to Charle's Law

Application of Charle's law

a) Charle's law is used for making hot air balloon. From Charle's law, when the temperature increases, the gas becomes lighter and easily rises up by displacing atmospheric gas downwardly. These hot air balloons are used as transport and in various experimental observation.

Relation between temperature and pressure of a gas

It staters that " at constant volume, the pressure exerted by the gas is directly proportional to its absolute temperature'.

Mathematically, P∝ T at constant volume

or, P = kT

or, \( \frac {P}{T}\) = k

\( \frac {P_1}{P_2}\) =\( \frac {T_1}{T_2}\)

This law is known as Gay - Lussace's law

Avogadro's law

The volume of the gas not only depends on the pressure and temperature but also in its amount.

It states that ' at constant temperature and pressure, the volume of a gas is directly proportional to the total amount of the gas '.

mathematically,

V∝n at constant T and P

or, V = n . constant

or, \( \frac {V}{n}\) = constant

The constant is same for all gases. So, Avogadro's law can also be defined as " Under the similar condition of temperature and pressure, the equal volume of gas contains the same number of molecules (no. of mole)". At NTP, 1 mole of any gas contains 8.023 x 10²³ number of molecules and occupies 22.4 liters. This number is called Avogadro's number and this volume is called molar volume.

Combined gas equation

From Boyle's law

V∝ \( \frac {1}{P}\) at constant temperature

From Charle's law,

V∝ T at constant pressure

From above equations,

V∝\( \frac{T}{P}\) , under similar condition of temperature and pressure

PV∝ T

or, PV = kT

or, \( \frac {PV}{T}\) = k -(i)

Hence,

\( \frac{P_1V_1}{T_1}\) = \( \frac{P_2V_2}{T_2}\) ------(ii)

Ideal gas equation

The mathematical equation which is obeyed by gases at ideal condition is known as the ideal gas equation.

Derivation

Boyle’s law: V ∝ \( \frac {1}{P}\) (at constant temperature) --------(i)

Charles law: V ∝ T (at constant pressure) -------------(ii)

Avogadro’s law: Equal volume of all the gases contains an equal number of molecules/ moles under the condition of same temperature and pressure.

V ∝ n (at constant temperature and pressure) -----------(iii)

Combining equation (i), (ii) and

V ∝ \( \frac{nT}{P}\)

PV∝ nT

or, PV = nRT -(i)

where proportionaly constant 'R' is called gas constant and the equation (i) is called ideal gas equation.

Universal gas constant R:

The gas constant 'R' present in the ideal gas equation is called the universal gas constant. It is because the value of R is same for all types of ideal gas.

Ideal gas equation: PV = nRT

Or, R = \( \frac {PV}{nT}\)

= \( \frac{Force}{Area}\) x \( \frac{Volume}{No\:of\:mole\:× Temperature\:in\:Kelvin}\)

Volume = Area x Length

Therefore,

R == \( \frac{Force\:Length}{No\:of\:mole\:× Temperature\:in\:Kelvin}\)

Hence, the gas constant 'R' is defined as the amount of work done required for 1 mole of a gas to rise its temperature by 1K

Value of R in liter x atm

Pressure = 1atm

Volume = 22.4 lit

Number of mole (n) = 1

Temperature (T) = 273 K

PV = nRT

Or, R = \( \frac{PV}{nT}\)

or, R = \( \frac { 1× 22.4} { 273× 1}\)

Or, R = 0.0821l atm K^-1 mol^-1.

Value of R in energy unit ( Joule and Calories)

Pressure = 1.01 x 10⁵ Pa

Volume = 22.4 lit = 22.4 x 10^-3 m³

Number of mole (n) = 1

Temperature (T) = 273 K

PV = nRT

Or, R = \( \frac{PV}{nT}\)

or, R = \( \frac { 1.01 ×10^5 × 22.4 × 10^{-3}} { 273× 1}\)

Or, R = 8.31JK^-1 mol^-1.

Again, 1 calorie = 4.2 Joules

Then, R = \( \frac{8.31}{4.2}\) cal K^-1 mol^-1. = 1,987 cal K^-1 mol^-1

Relation between pressure, density and molar mass from ideal gas equation

From ideal gas equation, PV = nRT -------(i)

where, P = Pressure

V = Volume

n = Number of moles

R = Universal Gas constant

T= Temperature

Since, Volume(V) = \(\frac{Mass}{Density}\) nad Number of mole = \( \frac{Given \:mass}{Molar\:Mass}\)

P.\( \frac {m}{d}\) = \( \frac{m}{M}\) RT

or, \( \frac{P}{d}\) = \( \frac{RT}{M}\)

or, PM = dRT

where M = molar mass of the gas

Relation between density, pressure and temperature from combined gas equation

From combined gas equation,

\( \frac{P_1V_1}{T_1}\) = \( \frac{P_2V_2}{T_2}\)

Since volume (V) = \( \frac{Mass(M)}{Density(d)}\)

\( \frac{P_1m_1}{d_1T_1}\) =\( \frac{P_2m_2}{d_2T_2}\)

For any particular gas, m₁ = m₂

or, \( \frac{T_1d_1}{P_1}\) = \( \frac{T_2d_2}{P_2}\)

Dalton’s law of partial pressure

It was introduced by John Dalton in 1807. It is the law to calculate the pressure of the gaseous mixture from the partial pressure of the component gases.

Dalton’s law of partial pressure states that “the total pressure of the mixture of gases is equal to the sum of the partial pressure of the component gases at constant temperature and pressure.

Let P_T be the total pressure and P₁, P₂ and P₃ be the partial pressure of the gases A, B, and C respectively.

Then,

P_T = P₁ + P₂ + P₃

Mathematical deduction of Dalton's law

Let n₁, n₂, n₃be the number of moles of three non-reacting gases enclosed in a vessel at pressure P and volume V.

Then, from ideal gas equation

PV = nRT

or, PV =(n₁+n₂+n₃) RT
or, P = \(\frac{n_1}{V}\) RT +\(\frac{n_2}{V}\) RT +\(\frac{n_3}{V}\) RT

or, P= P₁ + P₂ + P₃ [At constant temperature]

Dalton's law in terms of mole fraction

P= P₁ + P₂ + P₃ [At constant temperature]

P₁ =\(\frac{n_1}{V}\) RT ----(i)

P₂ =\(\frac{n_2}{V}\) RT ----(ii)

P₃ =\(\frac{n_1}{V}\) RT ----(i)

From ideal gas equation

P =\(\frac{n}{V}\) RT ----(iv)

Dividing (iv) by (i)

\( \frac{P_1}{P}\) = \( \frac{\frac{n_1}{V}RT}{\frac{n}{V}RT}\)

or,\( \frac{P_1}{P}\) =\( \frac{n_1}{n}\)

Hence, P₁ =\( \frac{n_1}{n}\) P --(v)

P₁ = X₁.P where, X₁ = mole fraction of gas

P₂ = X₂.P

P₃ = X₃.P

In general,

P_i = X_iP --------(vi)

The mole fraction of the gas can be defined as the ratio of a number of mole of the individual component of gas to the total number of moles.

Applications of Dalton's law 's laws

a) Dalton's law of partial pressure is used to calculate the pressure exerted by pure (dry) gas which is collected over water. The pressure exerted by water is called water vapour pressure. It is calculated as

P_dry = P_moist - P_{water vapor}

The pressure exerted by water vapor at a particular temperature is called aqueous tension and is denoted by f. It is known for any temperature

P_dry = P_moist - Aqueous tnesion

Graham's law of diffusion

The property of a gas by virtue of which they intermix with each other is known as diffusion of gas.

The diffusion of gas is explained by Graham's law of diffusion. It states "at constant temperature and pressure, the rate of diffusion of different gases are inversely proportional to square root of their density"

Mathematically,

r∝ \( \frac{1}{\sqrt d}\)

Let r₁ and r₂ be the rate of diffusion of two gases having density d₁ and d₂respectively. Then from Graham's law,

r₁∝ \( \frac{1}{\sqrt d_1}\) --(i)

r₂∝ \( \frac{1}{\sqrt d_2}\) --(ii)

KINETIC THEORY OF GASES

Postulates

i) The molecules of a gas do not exert any attractive force with each other, that means the intermolecular force of attraction is almost negligible.

ii) The pressure exerted by the gas molecule is due to the collision of gas molecules on the wall of the container in which it is contained.

iii) The average kinetic energy of a gas molecule is directly proportional to the absolute temperature i.e. molecules have the same level of energy at fixed temperature.

iv) Different molecules move with different motion/speed. Further, the speed of molecules goes on increasing due to the collision. In spite of this fact, the distribution of molecular speed remains unchanged at fixed temperature This distribution of molecular speed is known as Maxwell - Boltzmann distribution

Qualitative explanation of Boyle's law and Charle's law

i) Boyle's law:From kinetic theory of gas, the pressure exerted by a gas is due to the collision of gas molecules on the wall of the container. For a fixed mass of gas, the average kinetic energy is also fixed which also fixes the average velocities.

For a fixe mass of gas, a total number of molecules will also be fixed. When the volume is decreased, a total number of molecules per unit volume will increase. This increased number of molecules also increases the collision of gas molecules increases and ultimately increases the pressure of the gas. On the other hand, if the volume is increased, the number of molecules per unit volume decreases. The molecules are far apart from each other which causes striking of molecules on the wall to decrease. Hence, the pressure deceases on increasing volume at the constant temperature which is Boyle's law.

ii) Charle's law

If a fixed mass of gas is heated in a container, or a fixed volume, its pressure increases. If the volume is not fixed ad pressure is kept constant, then, on heating the container, the gas expands and expansion continues till the pressure decreases to its original value. Hence, the result of the increase in temperature is increased in volume of the container at constant pressure which is Charle's law.

Ideal gas and Real Gas

A gas which obeys ideal gas equation PV = nRT at all conditions of temperature and pressure is called ideally, has. But, all known gas are found to obey ideal behavior at only low pressure and high temperature. The real gas shows the marked deviation from ideal behavior at high pressure and low temperature.

Deviation of real gas from ideal behaviour

The real gas deviates from ideal behaviour which can be explained by the following equation,

PV = nRT (for ideal gas)

and PV = z(nRT) (For real gas)

where 'z' measures the degree of deviation of real gas from ideal behavior.

When,

i) z = 1, PV = nRT , the real gas shows ideal behaviour under all conditions of temperature and pressure

ii) When z < 1, real gas shows negative deviation i.e. the gases are more compressible than the ideal gas.

iii) When z>1, the real gas shows positive deviation i.e. the gases are less compressible than ideal gas

The factor 'z' is called compressibly factor which measures the degree of deviation from ideal behaviour.

For a real gas, the effect of deviation can be explained in terms of temperature and pressure

i) Effect of pressure on deviation

The effect of pressure can be explained by ' z VS P' graph of different gases at a fixed temperature

ii) Effect of temperature on deviation

The effect of temperature can be explained by plotting 'z vs P' graph of particular gas at the different temperature.

The temperature at which real gas shows ideal behaviour over the wide range of pressure is called 'Boyle's temperature'.

i) When T < T_B , z < 1 at lower pressure and greater than 1 at higher pressure.

ii) When T > T_B, z > 1 at all pressure

Hydrogen and Helium gas always show positive deviation from ideal behaviour because the Boyle's temperature for these gases is quite low ( -165⁰C for hydrogen and 240⁰C for helium)

Cause of deviation

At low pressure and high temperature, gas shows ideal behaviour but at the reverse condition, (high pressure and low temperature) , real gas deviate from ideal behaviour . This fact may be attributed by following two faulty assumptions made in kinetic theory of gas:

i) The total volume occupied by a gas molecule is negligible as compared to the total volume of the gas

ii) There is no intermolecular force of attraction between gas molecules

Above two assumptions are valid at low pressure and high temperature because, at this conditions, gas molecules are far apart from one another. But at low temperature and high pressure, gas molecules come closer to another and its volume cannot be neglected. Similarly, when gas molecules come closer, they feel the same kind of operative force

To account this fact, Van Der Waal modified ideal gas equation as

( P + \( \frac{an^2}{V^2}) \) ( V - nb) = nRT

where 'an' and 'b' are Van Der Waal's constants

and this equation is called van Der Waal's equation