No. of collisions =(1)/(4) n =v Now, (v^')/(v)=(n^')/(n) (⟨v^'⟩)/(⟨v>)=(1)/(η) √((T)/(T)) (Whe

step 1 Let X be the number of collision here which is equal to 1 by 4 and average V which is equal to V

No. of collisions =(1)/(4) n<v>=v Now, (v^')/(v)=(n^')/(n)...

No. of collisions $=\frac{1}{4} n<v>=v$ Now, $\frac{v^{\prime}}{v}=\frac{n^{\prime}}{n} \frac{\left\langle v^{\prime}\right\rangle}{\langle v>}=\frac{1}{\eta} \sqrt{\frac{T}{T}}$ (When the gas is expanded $\eta$ times, $n$ decreases by a factor $\eta$ ). Also $T(\eta V)^{2 / i}=T V^{2 / i}$ or $T=\eta^{2 / i} T$ so, $\frac{v^{\prime}}{v}=\frac{1}{\eta} \eta^{-1 / i}=\eta^{\frac{-i-1}{i}}$ i.e. collisions decrease by a factor $\eta^{\frac{i+1}{i}}, i=5$ here.

No. of collisions $=\frac{1}{4} n<v>=v$
Now, $\frac{v^{\prime}}{v}=\frac{n^{\prime}}{n} \frac{\left\langle v^{\prime}\right\rangle}{\langle v>}=\frac{1}{\eta} \sqrt{\frac{T}{T}}$
(When the gas is expanded $\eta$ times, $n$ decreases by a factor $\eta$ ). Also $T(\eta V)^{2 / i}=T V^{2 / i}$ or $T=\eta^{2 / i} T$ so, $\frac{v^{\prime}}{v}=\frac{1}{\eta} \eta^{-1 / i}=\eta^{\frac{-i-1}{i}}$
i.e. collisions decrease by a factor $\eta^{\frac{i+1}{i}}, i=5$ here.

Key Concepts

Collision Frequency

This concept refers to the number of collisions that gas particles undergo in a given time interval. It is derived using kinetic theory and depends on the number density of the particles and their average speed. In simplified models, collision frequency is proportional to the product of the particle density and their mean velocity, providing an estimate of how often interactions occur in a gas.

Density and Volume Relationship

When a gas expands, its density decreases. The number density of particles is inversely proportional to the volume of the container. Therefore, if a gas expands by a factor of ? in volume, the number density decreases by the same factor ?. This relationship is fundamental in predicting how macroscopic properties of a gas, such as collision frequency, change during expansion.

Adiabatic Process

An adiabatic process is one in which no heat is exchanged with the surroundings, and any changes in the gas properties, such as temperature and volume, are governed solely by internal energy changes. In such processes, the gas follows a specific relation between temperature and volume, often expressed through an exponent that depends on the degrees of freedom of the gas particles. This relation is crucial for understanding how temperature scales when the gas is expanded or compressed.

Degrees of Freedom in Gases

The degrees of freedom (represented by i) indicate the number of independent ways in which a gas molecule can store energy. This parameter influences the specific heat capacities and the relationship between temperature and volume during adiabatic processes. For instance, in diatomic gases, a typical value of i might be 5, and this value is used to derive scaling laws that affect the collision frequency and temperature during expansion or compression.

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