QUESTION NO 4

(a) How are the Boyle’s and Charle’s laws derived from Kinetic gas equation?

The kinetic theory of gases provides a microscopic explanation of the macroscopic behavior of gases described by Boyle’s and Charles’s laws. Here’s how these laws can be derived from the kinetic theory of gases:

Boyle’s Law (Pressure-Volume Relationship at Constant Temperature)

Boyle’s Law states: For a fixed amount of gas at a constant temperature, the volume of the gas is inversely proportional to its pressure.

Mathematically, ( PV = \text{constant} ).

Derivation using the Kinetic Theory of Gases:

1. Kinetic Energy and Pressure Relationship:
   According to the kinetic theory of gases, the pressure ( P ) exerted by a gas is given by:
   [
   P = \frac{1}{3} \frac{N}{V} m \overline{v^2}
   ]
   where ( N ) is the number of molecules, ( V ) is the volume, ( m ) is the mass of one molecule, and ( \overline{v^2} ) is the mean square velocity of the molecules.
2. Constant Temperature Implication:
   At a constant temperature, the average kinetic energy of the gas molecules remains constant. This implies that ( \overline{v^2} ) remains constant because the temperature ( T ) is directly proportional to the average kinetic energy.
3. Inversely Proportional Relationship:
   Since ( m \overline{v^2} ) is constant at a constant temperature, we can write:
   [
   P \propto \frac{1}{V}
   ]
   or
   [
   PV = \text{constant}
   ]
   This is Boyle’s Law.

Charles’s Law (Volume-Temperature Relationship at Constant Pressure)

Charles’s Law states: For a fixed amount of gas at constant pressure, the volume of the gas is directly proportional to its temperature.

Mathematically, ( V \propto T ) or ( \frac{V}{T} = \text{constant} ).

Derivation using the Kinetic Theory of Gases:

1. Pressure Expression:
   From the kinetic theory, the pressure exerted by a gas is:
   [
   P = \frac{1}{3} \frac{N}{V} m \overline{v^2}
   ]
2. Relation Between Temperature and Kinetic Energy:
   The temperature ( T ) of the gas is directly proportional to the average kinetic energy of the molecules. This can be expressed as:
   [
   \frac{1}{2} m \overline{v^2} = \frac{3}{2} k_B T
   ]
   where ( k_B ) is the Boltzmann constant.
3. Expressing Mean Square Velocity:
   Rearranging the above equation gives:
   [
   \overline{v^2} \propto T
   ]
4. Substitute into Pressure Equation:
   Substitute ( \overline{v^2} ) back into the pressure equation:
   [
   P = \frac{1}{3} \frac{N}{V} m (k_B T)
   ]
   Since ( P ) is constant, we get:
   [
   V \propto T
   ]
   or
   [
   \frac{V}{T} = \text{constant}
   ]
   This is Charles’s Law.

In summary, Boyle’s and Charles’s laws can be derived from the kinetic theory of gases by considering the relationships between pressure, volume, and temperature at the molecular level.

(b) At what temperature, r.m.s. velocity of C*O_{2} becomes equal to that of O_{2} at 27 deg * C deg

To determine the temperature at which the root mean square (r.m.s.) velocity of CO₂ becomes equal to that of O₂ at 27°C, we can use the formula for r.m.s. velocity:

[ v_{\text{rms}} = \sqrt{\frac{3kT}{m}} ]

where ( v_{\text{rms}} ) is the root mean square velocity, ( k ) is the Boltzmann constant, ( T ) is the absolute temperature in Kelvin, and ( m ) is the mass of one molecule of the gas.

Let’s denote:

• ( T_{\text{O}_2} = 27^\circ\text{C} = 300 \text{K} ) (since ( 27 + 273 = 300 \text{K} ))
• ( m_{\text{O}_2} ) as the mass of an O₂ molecule
• ( T_{\text{CO}_2} ) as the temperature of CO₂ in Kelvin that we need to find
• ( m_{\text{CO}_2} ) as the mass of a CO₂ molecule

The r.m.s. velocities for O₂ and CO₂ are given by:

[ v_{\text{rms, O}2} = \sqrt{\frac{3kT{\text{O}2}}{m{\text{O}_2}}} ]

[ v_{\text{rms, CO}2} = \sqrt{\frac{3kT{\text{CO}2}}{m{\text{CO}_2}}} ]

We want these velocities to be equal, so:

[ \sqrt{\frac{3kT_{\text{O}2}}{m{\text{O}2}}} = \sqrt{\frac{3kT{\text{CO}2}}{m{\text{CO}_2}}} ]

Squaring both sides and canceling common factors:

[ \frac{T_{\text{O}2}}{m{\text{O}2}} = \frac{T{\text{CO}2}}{m{\text{CO}_2}} ]

Rearranging for ( T_{\text{CO}_2} ):

[ T_{\text{CO}2} = T{\text{O}2} \times \frac{m{\text{CO}2}}{m{\text{O}_2}} ]

To find the masses of the molecules:

• Molar mass of O₂ = 32 g/mol
• Molar mass of CO₂ = 44 g/mol

Since the mass ( m ) of a molecule is directly proportional to the molar mass, we use these values to find:

[ \frac{m_{\text{CO}2}}{m{\text{O}_2}} = \frac{44}{32} = \frac{11}{8} ]

So:

[ T_{\text{CO}_2} = 300 \text{K} \times \frac{11}{8} ]

[ T_{\text{CO}_2} = 412.5 \text{K} ]

Converting this temperature back to Celsius:

[ T_{\text{CO}_2} – 273 = 412.5 – 273 ]

[ T_{\text{CO}_2} \approx 139.5^\circ\text{C} ]

Therefore, the temperature at which the r.m.s. velocity of CO₂ is equal to that of O₂ at 27°C is approximately ( 139.5^\circ\text{C} ).

(c) What are the postulates of Kinetic theory of gases?

The Kinetic Theory of Gases is a model that explains the behavior of gases based on the motion of their molecules. Here are the key postulates:

1. Gas Particles in Constant Motion: Gas molecules are in constant, random motion. They move in straight lines until they collide with other molecules or the walls of the container.
2. Elastic Collisions: Collisions between gas molecules and between molecules and the walls of the container are perfectly elastic. This means that there is no loss of kinetic energy in the collisions.
3. Negligible Volume of Molecules: The volume of the individual gas molecules is negligible compared to the volume of the container. This implies that the size of the molecules is very small relative to the distance between them.
4. No Intermolecular Forces: There are no attractive or repulsive forces between the gas molecules except during collisions. The interactions between the molecules are negligible.
5. Average Kinetic Energy and Temperature: The average kinetic energy of the gas molecules is directly proportional to the absolute temperature (Kelvin) of the gas. This relationship is given by the equation: [ \text{Average Kinetic Energy} = \frac{3}{2} k T ] where ( k ) is the Boltzmann constant and ( T ) is the absolute temperature.
6. Random Motion: The motion of the gas molecules is completely random, and their speeds follow a statistical distribution (the Maxwell-Boltzmann distribution).

These postulates help explain various gas laws and phenomena, such as the relationship between pressure, volume, and temperature in gases.