#### Expansion of Gases, Physics tutorial

Expansion of Gases:

Gases as well expand on heating. Their volume expansion is much greater than that of liquids. When a gas is heated at constant pressure, its volume rises and when a gas is heated at constant volume, its pressure rises due to expansion. Likewise, if we have to study the variation of pressure by means of temperature, its volume should be kept constant.

Kinetic Molecular Theory of Gas and its Assumptions:

We utilize the kinetic molecular theory of gases to describe the effect of temperature on the volume and pressure of a gas. The theory makes the given assumptions.

1) A gas is assumed to be made up of a large number of molecules acting similar to elastic spheres.

2) The molecules are in a state of constant motion knocking against one other and the walls of the containing utensil.

3) The particles apply attractive forces on one other.

4) The particles have kinetic energy due to their motion.

5) The volume of molecules of a gas is negligible as compared by the volume of the container.

Boyle's Law:

Boyle's law defines that the pressure of a fixed mass of gas is inversely proportional to the volume, given that the temperature is remain constant.

In symbolic form, Boyle's law is written as:

P = 1/V

PV = constant

Here, P = pressure and V = Volume of a fixed mass of gas.

This law can be deduced as an equation which associates the initial volume (V1) and the initial pressure (P1) to the final volume (V2) and the final pressure (P2). At constant temperature, Boyle's Law Formula is symbolically represented as:

P1V1 = constant

P2V2 = constant

By comparing both the above equations:

P1V1 = P2V2

Use of Kinetic Theory to describe Boyle's Law:

Now, let us suppose a given mass of gas at constant temperature. At constant temperature, the average velocity of the gas molecules is constant and the number of collisions they make per unit area of the wall of the container is also constant. Therefore, the gas applies a certain constant pressure, P, on the walls of the vessel.

When we twice the original volume of the gas, the gas molecules will be spread out and it will take longer time to bombard the walls and therefore, fewer impact per second. As an outcome, the pressure of the gas will be halved. Therefore, at constant temperature, volume of a gas rises as pressure reduces.

On other hand, when we half the original volume of the gas, the gas molecules will be more closely pack and then it will take less time to bombard the walls of the container. The gas pressure rises (or double) as more impact per second are made. Therefore, at constant temperature, reduce in volume of a gas leads to rise in pressure.

This signifies that the pressure of a given mass of gas differs inversely by its volume at constant temperature in accordance by the Boyle's law.

Charles' Law:

In this we concern in the connection between the changes in volume with temperature change of a fixed mass of gas kept at constant pressure introduced by Charles in the year 1787 and independently by Gay-Lussac in the year 1802.

Statement: Charles' law defines that the volume of a fixed mass of gas rises by 1/273 of its volume at 0oC per degree Celsius rise in temperature given its pressure remains constant.

We can as well define Charles law by employing Kelvin temperature scale. You are aware that a temperature toC on the Celsius scale is associated to an absolute or Kelvin temperature T by the equation:

T (K) = 273 + toC

By employing the Kelvin scale thus, Charles' law can be defined as the volume of a fixed mass of gas is directly proportional to its absolute temperature when its pressure remains constant.

In symbolic form, we deduce Charles' law as:

V ∝ T (if pressure is kept constant)

V = KT

V/T = K

V1/T1 = V2/T2

Here, V = volume; T = Kelvin temperature; K = a mathematical constant; V1 = volume of gas at temperature T1; V2 = volume of gas at temperature T2 of gas at constant pressure.

Cubic Expansivity of a Gas:

You can as well employ Charles law experiment to get the value of the cubic expansivity of a gas at constant pressure. This is stated as:

Increase in volume per unit volume at 0oC per degree Celsius increase in temperature.

That is, when γ = cubic expansivity of a gas, Vo = volume at temperature 0oC, Vt = volume at temperature t oC

Then

γ = (Vt - Vo)/(Vo x t)

Use of Kinetic Molecular Theory to Explain Charles Law:

Let us assumed that a fixed mass of a gas confirmed in a container at constant pressure. Now, when we heat this gas, the molecules attain more K.E., move faster and collide more frequently with the walls of the vessel.

As an outcome, rises the pressure they apply. For us to maintain a constant pressure, we then raise the volume of the container to let the similar mass of gas to travel a longer distance before hitting the walls of the container. Therefore a raise in temperature leads to a raise in the gas volume. This is in agreement with Charles's law.

Pressure Law or Gay - Lussac's Law:

In this we are concern with the connection between the pressure changes with temperature of a fixed mass of gas maintained at constant volume. This connection was proposed through Gay-Lussac in the year 1802 and these are two alternatives of stating the relation (law).

Statement of Pressure Law:

We can now define the law in the two alternative manners therefore:

=> The law defines that the pressure of a fixed mass of gas at constant volume increases by 1/273 of its pressure at 0oC for each and every degree Celsius (or Kelvin) rise in temperature.

We write the above in the symbolic form as:

Pt = Po [1 + (t/273)]

Here, Po = pressure at temperature 0oC, Pt = Pressure at temperature toC.

=> The pressure of a fixed mass of gas at constant volume is proportional to the absolute temperature of the gas.

In symbolic form, we can state the law as:

P ∝ T

P/T = constant

That is, P1/T1 = P2/T2

Use of Kinetic Molecular Theory to Explain Pressure Law:

We are familiar that heat is a form of energy and that if a gas is heated the molecules gain kinetic energy and move about faster. As an outcome, the momentum changes generated through the molecules at the walls per impact is greater. This as well leads to rise in the number of impacts of each molecule per second due to the increase in speed.

Therefore, the pressure rises with rise in temperature at content volume. This is the statement of Gay-Lussac's law or Pressure law.

General Gas Law or Equation:

By using the gas laws we are familiar that the volume of a gas depends on both its temperature and pressure. We can sum up the relationship among these three variables: volume, temperature and pressure to get the general gas law.

=> From the Boyle's law,

PV = constant when T is constant

=> From Charles' law,

V/T = constant when P is constant

=> From the Gay - Lussac's or pressure law,

P/T = constant at contact volume

We can join any two of these three equations to get the general gas equation.

That is, PV/T = K

Here, K is a constant for a fixed mass of gas.

(P1V1)/T1 = (P2V2)/T2

Here, P1V1 are the gas pressure and volume at temperature T1 and P2V2 are the gas pressure and volume at temperature T2.

Ideal Gas Equation:

The ideal gas equation associate usually pressure, volume and absolute temperature of a fixed mass of gas in the general gas equation.

(PV)/T = constant

Though, for an ideal gas, at standard temperature and pressure, for 1 mole of gas we encompass:

PV = RT

Here, R is a constant termed as the molar gas constant and its value is 8.31J mol-1K-1

For n moles, though, we encompass

PV = nRT

Conversion to Standard Temperature and Pressure (S.T.P):

We have observed that the volumes of gases changes amazingly with changes in temperature and pressure. Due to this reason, it is not simple to compare the outcome of experiments between country and towns across the world having temperate and tropical weathers. This examination made scientist to choose 0oC or 273K and 760 mm Hg or 1.01 x 105 Nm-2 to be the standard temperature and pressure (STP) at which the gas volumes are given. This makes it simple and possible for us to compare the volumes of various masses of a gas.

Intermolecular Energy and Forces:

The physical properties of boiling point, melting point, vapor pressure, viscosity, evaporation, surface tension and solubility are associated to the strength of attractive forces among molecules. These attractive forces are termed as Intermolecular Forces. The amount of 'stick togetherness' is significant in the interpretation of the different properties.

There are mainly four kinds of intermolecular forces.

a) Ionic Forces:

The forces holding ions altogether in ionic solids are the electrostatic forces. Opposite charges attract one other. These are the strongest intermolecular forces. Ionic forces hold numerous ions in a crystal lattice structure.

b) Dipole Forces:

Polar covalent molecules are at times illustrated as 'dipoles', signifies that the molecule consists of two 'poles'. One end (pole) of the molecule consists of a partial positive charge whereas the other end consists of a partial negative charge. The molecules will orientate themselves in such a way that the opposite charges attract principle operates efficiently.

c) Hydrogen Bonding:

The hydrogen bond is in reality a special case of dipole forces. A hydrogen bond is the attractive force among the hydrogen joined to an electronegative atom of one molecule and an electronegative atom of dissimilar molecule.

d) Induced Dipole Forces:

Forces between fundamentally non-polar molecules are the weakest of all intermolecular forces. 'Temporary dipoles' are made by the shifting of electron clouds in molecules. Such temporary dipoles repel or attract the electron clouds of close by non-polar molecules.

Deductions from the Ideal Gas Equation:

1) Avogadro's Law defines that equivalent volumes of all ideal gases (that is, at similar temperature and pressure) have the same number of molecules.

V/n = constant => n1/V1 = n2/V2

2) Boyle's Law defines that equivalent pressure is inversely proportional to volume (that is, if temperature is constant).

P x V = constant => P1 x V1 = P2 x V2

3) Charles's Law defines that the volume is proportional to temperature (that is, if pressure is constant). Keep in mind that the temperature should be evaluated in Kelvin.

V/T = constant => V1/T1 = V2/T2

4) Gay-Lussac's Law defines that the pressure is proportional to temperature (that is, if volume is constant).

P/T = constant => P1/T1 = P2/T2

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