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Derivation of kinetic gas equation

Consider a certain mass of gas enclosed in a cubic box at a fixed temperature.

Suppose that:

                        the length of each side of the box = \textit{l} cm

                        the total number of gas molecules = N

                        the mass of one molecule               = m

                        the velocity of a molecule               = v

(1) According to kinetic molecular theory the molecule of a gas can move with a velocity (v) in any direction X, Y and Z axes.

v_x, v_y and v_z are perpendicular to each other so:

The magnitude of velocity of all directions x, y and z equlas:

v = \sqrt{v^2_x + v^2_y + v^2_z} v^2 = v^2_x + v^2_y + v^2_z

(2) The time for collision on a face A of the cube:

Consider a molecule moving in X direction between opposite faces A and B. It will strike the
face A with velocity Vx and rebound with velocity – Vx. To hit the same face again, the molecule must
travel l cm to collide with the opposite face B and then again l cm to return to face A.

Time = \frac{Distance}{Velocity} = \frac{2\textit{l}}{v_x}

(3) Total change in momentum on all faces of box by one molecule:

Each impact of the molecule on the face A causes a change of momentum (mass × velocity) :
the momentum before the impact = mvx
the momentum after the impact = m (– vx)
∴ the change of momentum = mvx – (– mvx)
= 2 mvx

But the number of collisions per second on face A due to one molecule = \frac{v_x}{2\textit{l}}

Therefore, the total change of momentum per second on face A caused by one molecule

= 2mv_x x (\frac{v_x}{2\textit{l}}) = \frac{mv^2_x}{\textit{l}}

The change of momentum on both the opposite faces A and B along X-axis (2 sides) would be double i.e., \frac{2mv^2_x}{\textit{l}} similarly, the change of momentum along Y-axis and Z-axis will be \frac{2mv^2_y}{\textit{l}} and \frac{2mv^2_z}{\textit{l}} respectively. Hence, the overall change of momentum per second on all faces of the box will be

(4) Total change of momentum per second due to impacts of all the molecules on all faces of the box:
Suppose there are N molecules in the box each of which is moving with a different velocity v1, v2,
v3 , etc. The total change of momentum due to impacts of all the molecules on all faces of the box

Where u2 is the Mean Square Velocity.

(5) calculate the pressure from change of momentum; Derivation of Kinetic Gas Equation

Since force may be defined as the change in momentum per second, we can write

Force = \frac{2mN u^2}{\textit{l}}

Pressure = \frac{Force}{Area}

Since the total area of each a cube = 6\textit{l}^2

P = \frac{2mN u^2}{\textit{l}}X \frac{1}{6\textit{l} ^2} = \frac{1}{3} \frac{2mN u^2}{\textit{l}^3} 

As \textit{l}^3is the volume of the cube then it is V

P = \frac{1}{3} \frac{mN u^2}{V}

PV = \frac{1}{3}mN u^2

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