Molecular Theory of gases - Physics II _ For HSC Part 2 / XII / Class 12 (Science Group) - Section C - ERQs (Long Answered Questions)

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Unit 15: Molecular Theory of gases
Physics II -
For HSC Part 2 / XII / Class 12 (Science Group)

SECTION C
ERQs (Long Answered Questions)

Q.1: What is temperature? Explain the scales of temperature in detail.
Ans: TEMPERATURE:
It is the measurement of hotness or coldness in a body. It determines the direction of flow of heat.
Definition:
"Temperature is a measure of the average translational kinetic energy of the molecules of body."
Unit:
The SI unit of temperature is Kelvin.

Measurement Of Temperature:
Temperature is measured by thermometer. It can be determine by measuring the physical properties, as properties changes with the temperature.

Scales of Temperature OR Types Of Thermometer:
There are three scales of temperature which are commonly used these days.
i. Centigrade or Celsius scale
ii. Fahrenheit Scale
iii. Kelvin or Absolute Scale

i) Centigrade or Celsius Scale (Anders Celsius):
In the Celsius scale the freezing point of water or Melting point of ice is marked as 0 °C and boiling point of water is marked as 100 °C. The interval between these two points is divided into hundred equal parts. Each part thus represents one degree centigrade or Celsius and is represented by °C.

ii) Fahrenheit Scale (Daniel Fahrenheit):
In Fahrenheit scale the freezing point of water is marked 32 °F and boiling point of water is marked as 212°F respectively and the interval between the two fixed points (freezing & boiling points) is divided into 180 equal parts. Each part is called one degree Fahrenheit and is represented by 1°F.

(iii) Kelvin Scale (James Lord Kelvin):
In this scale, the melting point (freezing point) of ice is marked as 273 K and the boiling point of water is marked as 373 K. The interval between two fixed points (freezing & boiling points) is divided into 100 equal parts. The temperature is given in units called Kelvin instead of degrees and is represented by K. The lowest temperature is 0 K known as absolute zero.

Empirical Formulae:
In order to derive empirical formulae among Centigrade, Fahrenheit and Kelvin scales, let the three thermometer placed in a bath tub and the mercury in each thermometer rises to the same level. Thus we arrive at the relation.

Relation Between Different Temperature Scales:
Celsius (Centigrade) Scale, Kelvin Scale & Fahrenheit Scale:

• Centigrade temperature to Kelvin temperature by simply adding 273 to the centigrade temperature i.e.
  T_K = T_°C + 273
  
  
• Kelvin to Celsius
  T_°C = T_K - 273
  
  
• Fahrenheit scale To Celsius Scale:
  T_°C = (⁵/₉ x T °F) - 32
  
  
• Celsius To Fahrenheit scale:
  T_°F = (⁹/₅ x T °C) + 32
  
  
• Scale Division:
  * The scale divisions on the Kelvin scale are equal in size to the divisions on Celsius scale. i.e. 1K= 1 °C
  * 1 °C = 1.8 °F
  * Fahrenheit and Celsius scales coincide at - 40 °C OR - 40 °F
  * Fahrenheit and Kelvin scales coincide at 574.25° F or 574.25K

Q.2: Define and explain Boyle's law, Charles's and Avogadro's law.
Ans: (a) BOYLE's LAW:
In 1662 Robert Boyle proposed gases law about the relationship between volume and pressure of an ideal at constant temperature and mass is known as Boyle's law.

Definition (Statement):
Boyle's law states that:

"Volume V of any given mass of a gas is inversely proportional to the pressure P, provided the temperature T of the gas remains constant."

Explanation:
Mathematically:
If V is the volume of the gas and P is its pressure, then Boyle's law can be written as:

V ∝ ¹ /_P (at constant temperature)
V = ^K/_P
OR
PV = K

Where K is a constant and depends upon sample of gas.
For change in state from initial to final, Boyle's law can be written as:

P₁ V₁ = P₂ V₂ = Constant

Real gases obey Boyle's law at low pressure.

Graphically:
We can also represent Boyle's law on a graph, as:

• The graph plotted between P and V at constant temperature is a curve called hyperbola showing the inverse relation between them for two different states.
• While the second graph of P plotted against ¹ /_v is a straight line passing through the origin, showing direct proportionality.

The relation between pressure and volume & pressure and inverse of volume respectively

Relation with the mass (m) of a gas:
The inverse relation of volume and pressure of a gas remain stable until the mass of the gas remains constant. If, however, the mass of gas varies, then the product of PV will be directly proportional to the mass of a gas:

PV ∝ m
PV = Constant x m
OR
PV / m = Constant

For different state of gases

P₁ V₁/ m₁ = P₂ V₂/m₂ = Constant

(b) CHARLE'S LAW:
In 1787 French scientist J. Charles proposed his law to explain the relationship between volume and absolute temperature of a given mass of a gas, at the pressure constant is known as Charle's Law.

Definition (Statement): This law states that:

"The volume V of any given mass of a gas is directly proportional to the absolute temperature T at constant pressure P".

Explanation:
Mathematically:
Charles law can be written as:

V ∝ T (at constant pressure)
V = KT
OR
V / T = K

Where K is constant and depends upon sample of a gas.
If V₁ and V₂ are the volumes of the gas at temperature T₁ and T₂ respectively then for two different states, For change in state from initial to final, Charles's law is represented as:

V₁/T₁ = V₂/T₂ = constant

Real gases obey Charle's law at high temperature.

Graphically:

• From the graph at 0°C the gas still has a volumeV₀.
• The graph between volume and temperature is a straight line, which shows that volume increase with the increase in temperature.
• If the this line is extrapolated (projected) backward, it meets the temperature axis at -237 °C. indicating volume of a gas is zero.
• Thus (-273°C) is the lowest possible temperature that a gas can achieve and is called Absolute Zero or 0 Kelvin.

Relation between temperature and volume

Relation with the mass (m) of a gas:
The direct relation of volume and temperature of a gas remain stable until the mass of the gas remains constant. If, however, the mass of gas varies, then the ratio will be directly proportional to the mass of a gas:

V/T ∝ m
V/T = Constant x m
OR
V / mT = Constant

For different state of gases:

V₁/ m₁T₁ = V₂/m₂T₂ = Constant

(c) AVOGARDRO'S LAW:
In 1811, Italian scientist Amedeo Avogadro suggested his hypothesis regarding the relationship between volume and number of molecules of a gas. This hypothesis now called Avogadro's law.

Definition (Statement):
Avogadro's law states that:

"equal volume of all gases contains the same number of molecules at the same temperature and pressure".

Explanation:
Mathematically:
Thus, the volume of a gas is directly proportional to number of moles of the gas at constant temperature and pressure. In symbol, we can write as:

V ∝ n (at constant temperature and pressure)
V = Kn
OR
V/n = K

Where K is constant and depends upon a sample of a gas, V is volume and n is the number of moles of a gas.
It can be written as:

V₁/n₁ = V₂/n₂ = Constant

When V₁ and V₂ are volumes of gas and n1₁ and n₂ are amount of gas.

Thus 1 dm³ (or cm³, m3³) of oxygen contains the same number of molecules as 1 dm³ or 1 cm³, 1m³ etc of hydrogen or of any other gas, provided the volumes are measured under the same conditions of temperature and pressure.

Example:
Blowing up a balloon is an example of Avogadro's law, because as we blow more molecules of air into the balloon it expands.

Q.3: Derive general gas law by making use of gas laws.
Ans: GENERAL GAS LAW:

An interrelation among the physical quantities e g. pressure (P), volume (V), temperature (T) and mass (m) of a given sample of gas which determine the state of a gas is termed as "equation of state" of gas or General gas law.

In order to derive general gas law, we make use of Boyle's law, Charles's law and Avogadro's law.
According to Boyle's law:

V ∝ 1/P (when n number of moles and temperature T are kept constant)

According to Charles's law:

V ∝ T (when n and pressure P are kept constant)

According to Avogadro's law:

V ∝ n (when T and P are kept constant)

Lets Consider that none of the variable are to be kept constant, then by the combination of all the above three relationships, we can write as:

Where R is constant of proportionally and is called General gas constant or universal gas constant and does not depend on the quantity of gas in the sample. It is denoted by R.
Above equation is written as:

PV = nRT

S.I unit:
If P is measured in Nm^-2 V in m³ and T in Kelvin then the volume of universal gas constant is R = 8.314 J mol^-1 k^-1

Q.4: Describe the molecular movement causes the pressure exerted by gas, derive pressure equation?
Ans: PRESSURE OF GAS:
The pressure exerted by a gas is merely the momentum transferred to the walls of the container per second per unit area due to the continuous collisions of molecules of the gas.

PRESSURE EQUATION:
To Calculate The Pressure Of An Ideal Gas From Kinetic Theory:

• Let us consider a cubical container having side length L whose walls are perfectly elastic contains N number of molecules each of mass m.
• A molecule which has a velocity V₁ can be resolved into three rectangular components V₁x, V₁y and V₁z parallel to three co-ordinates axes x, y and z.
• A molecule which collides with the face ABCDA of the cubical container, it will rebound elastically in opposite direction, such that component of the velocity V₁ is reversed, and other components remain unaffected.

Now Consider a single molecule of mass m moving with velocity V₁ parallel to x-axis. It moves back and forth, colliding at reguarl intervals with the ends of the box (cubical container) and thereby contributing to the pressure of the gas.
The molecule from the left face of the cube moves with the velocity V₁x and collide with the right face of the cube and rebounds with the velocity -V₁x
Therefore,
Initial momentum of molecule (before collision) = P_i = mV₁x
Final momentum of molecule (after collision) = P_f = m (-V₁x) = - mV₁x
and

Change in momentum = △P = P_i — P_f
= mV₁x — (—mV₁x) = mV₁x + mV₁x
= 2mV₁x .........(1)

After recoil the molecule travels to opposite face and collides with it, rebounds and travels back to the face ABCDA after covering a distance 2L. i.e.

S = 2L
But S = vt
2L = v₁x x t
t = 2L / v₁x
The time t between two successive collisions with face ABCDA = △t
so t = △t
Or △t = 2L / v₁x .........(2)

Now we can find the force exerted by one molecule on face ABCDA, using Newton's 2^nd law of motion. This says that the rate of change of momentum of the molecule is equal to force applied by the wall. According to Newton's 3^rd law of motion, force F₁ exerted the molecule on face ABCDA is equal but opposite.

Similarly, the forces due to all other molecules can be determined. Thus, the total x-directed F due to N number of molecules of the gas moving with velocities V₁, V₂, V₃.........V_n is:

F = F₁ + F₂ + F₃ + ........... + F_n

Q.5: Interpret mathematically that temperature is a measure of average translational K.E of the molecules of a gas. OR
Prove that the average translatory kinetic energyof gas molecule is directly proportional to the Absolute temperature of the gas. OR
Prove that T ∝ (K.E)_ave OR
On the basis of KMT of gases show that ¹/₂ mV² = ³/₂ KT
Ans: THE RELATION BETWEEN KINETIC ENERGY OF MOLECULE AND ABDOLUTE TEMPERATURE:
Consider a gas in a cylinder having density "𝝆", Pressure "P", Volume "V", moles "n" and total number of molecules "N" for such a gas using kinetic theory of gases can be written as:

Note: Average Transitional Kinetic Energy is also called Kinetic Energy per molecule of the gas i.e.:

K.E per molecule = K.E_avg = ³/₂ kT

While the Kinetic Energy of per mole of a gas is:

K.E per mole = ³/₂ RT

More Long Answered Questions

Q.6: Write down the relation of different scales in detail?
Ans: Relations Between Different Temperature Scales:
Relation between Celsius and Fahrenheit Scales:
Let us consider that both, a Celsius and a Fahrenheit, thermometers are dipped in the same water bath placed at room temperature. The final and initial points in Celsius thermometer are labelled as A and B, and those are in the Fahrenheit thermometer as A' and B' respectively, whereas the mercury levels in both are labelled as C and C'.
The general relation between two scales of temperature can be written as:

Relation between Kelvin and Fahrenheit Scales:

Relation between Celsius and Kelvin Scales:

• T °C = TK — 273
• TK = T °C + 273

Q.7: Define and explain triple point of a water?
Ans: The Triple Point of Water:
Definition:

The specific temperature and pressure, where all three states or phases (vapour or gas, liquid and solid) of water are coexist in thermodynamic equilibrium is called Triple Point of Water. This means that at the triple point, ice, liquid water, and water vapor can exist together without any phase changing to another.

Value and Conditions:

• The particular temperature of triple point of water at which pure water, pure ice, and pure water vapour can coexist in a stable equilibrium is precisely 273.16 K (0.01°C) and 32.02°F 
• While the pressure is 4.58 mm of mercury and 611.73 Pascal (N/m²) and is used to calibrate thermometer.
•  These conditions are used to define the Kelvin temperature scale.
• The temperature at which Triple point of water has been set by international agreement to be T₃ = 273.16 K. In T3₃ the subscript 3 means "Triple Point" this agreement also sets the size of the Kelvin as ¹/_273.16 of the difference between the Triple point temperature of water and absolute zero.

Graph OR Phase Diagrams:

• In a graph or phase diagram, the triple point is represented as the point (at the intersection) where the lines separating the solid, liquid, and gas phases meet.
• This graph visually demonstrates the relationship between temperature and pressure for the different phases of water.
• In each of the three regions of the diagram, the substance is in a single state (or phase).
• The dark lines that act as the boundary between those regions represent the conditions under which the two phases are in equilibrium.

Triple Point Cell:

• A triple point cell is a device used to determine the triple point of a substance like water.
• Inside the cell conditions are maintained, so that solid ice, liquid water and water vapour co-exist in thermal equilibrium.
• The bulb of the constant volume gas thermometer is inserted into the well of the cell to measure the temperature at this fixed point.

Q.8: What is kinetic molecular theory of a gas? State the basic postulates of Kinetic theory of gas? Also write down example of KMT
Ans: KINETIC MOLECULAR THEORY:
The Kinetic Theory of Gases (KMT) is a microscopic model that explains the behavior of ideal gases by considering their molecules. It provides a link between the macroscopic properties of gases, such as pressure and temperature, and the microscopic characteristics of molecules, such as their speed and collisions. This theory helps us understand the underlying reasons for the observed properties of gases and forms the basis for many fundamental gas laws.

BASIC POSTULATES OF KINETIC MOLECULAR THEORY:
The basic postulates of Kinetic Theory of gases are as under :

1. MOLECULES:
   A gas contains a very large number of particles called molecules. Depending on the gases each molecule consists of an atom or a group of atoms.
2. VOLUME OF A GAS:
   A finite volume of a gas consists of very large number of molecules. This assumption is justified by experiments. At standard conditions there are 3 x 10²⁵ molecules per cubic meter.
3. SIZE OF MOLECULES:
   The size of the molecules is much smaller than separation between molecules; it is about 3 x 10^-10 m.
4. COLLISION:
   The molecules move in all directions with various speeds collide elastically with one another and with the walls of the container.
5. FORCES BETWEEN MOLECULES:
   The molecules exert no forces on one another except during collisions. In the absence of the external forces, they move freely in straight lines.
6. LAWS OF MECHANICS:
   Laws of mechanics are assumed to be applicable to the motion of molecules.

EXAMPLE:
Popcorn is a fun way to learn about the kinetic molecular theory of gases, the phase change of water from a liquid to a gas. When heated, the water inside turns to steam, making the kernel pop and expand up to 50 times to its size.