Kinetic Theory

Pressure due to an ideal gas is given by
p =
Putting , the density of gas.
p =

The gases carbon monoxide (CO) and nitrogen ( ) are diatomic, so both have equal kinetic energy .

The pressure exerted by the gas,



energy per unit volume, V = 1)
p= E

Step 1: Relation of r.m.s. speed with temperature. For an ideal gas, root mean square speed is: $ T $

Given, that ...(i)
As we know that from Mayer's relation,

where, = universal gas constant
Substitute the value of from the above relation in E (i), we get





Hence, is equal to

The SI unit of the rate of diffusion is square meters per second (m²/s).

The rate of diffusion refers to the speed at which particles of a substance move from an area of higher concentration to an area of lower concentration. This process occurs due to random molecular motion and is influenced by factors such as temperature, pressure, and the medium through which the substance is diffusing.

The unit of diffusion: The SI unit for measuring the rate of diffusion is expressed in square meters per second (m²/s), which indicates the area through which the substance diffuses over a given time period. This unit is commonly used in fields such as physics, chemistry, and biology, where diffusion processes are studied.

Example: If we are studying the diffusion of a gas across a membrane, the rate of diffusion could be measured as the area of the membrane (in m²) through which the gas molecules pass per second (in s).

Key points:

• The rate of diffusion is influenced by factors like temperature, concentration gradient, and the properties of the medium.
• It is important to note that the rate of diffusion can vary depending on whether the substance is a gas, liquid, or solid.

Summary: The SI unit for the rate of diffusion, m²/s, quantifies how much area is diffused per unit of time and is widely applicable in various scientific disciplines studying diffusion processes.

The root mean square (r.m.s.) velocity of gas molecules is determined by the formula: where: - is the universal gas constant, - is the absolute temperature in Kelvin, - is the molar mass of the gas. Step 1: Equating for Hydrogen and Oxygen For hydrogen ( ) and oxygen ( ): Substitute the expression for : Square both sides: Rearrange to solve for : Step 2: Substitute Known Values The temperature of oxygen gas is: Molar masses are: Substitute into the formula for : Simplify: Final Answer:

The given equation for the RMS velocity is: where: - is the root-mean-square velocity, - is the universal gas constant, - is the temperature, and - is the molecular mass. Now, the problem states that the temperature is doubled, i.e., . We are asked to find the new RMS velocity when the temperature is increased to . Substitute the new temperature into the RMS velocity formula: Simplify the expression: Thus, when the temperature is doubled, the new RMS velocity is times the original velocity. However, this result does not match the answer choices provided. Please review the options for more clarification.

Concept: The r.m.s. velocity of a gas molecule is given by: where = absolute temperature and = molar mass. Thus,

Step 1: Equate the r.m.s velocities. Squaring both sides:

Step 2: Substitute molar masses. Temperature of oxygen:

Step 3: Solve for . Thus, the required temperature is