Question

Derive an equation for the pressure due to an ideal gas during kinetic theory of gases?

Answer 1

Let gas filled in a cubical container of sides length 'a '
→velocity of gaseous molecule along x , y , and z -axis are Vx, Vy , and Vz .
→ molecule always moves with constant velocity . e.g acceleration = 0
→ collision between molecule and wall of cubical container is perfectly elastic .

Let edge length of cubical container is a then time taken by molecule to cover displacement between wall ,
t = a/Vx or , a/Vy or a/Vz
time interval between consequetive collision of gaseous molecule with a single is =====
total time taken =2a/Vx

Force applied by molecule on wall
Fx = ∆P/∆t
=(Pf -Pi )/∆t
= -2mVx/2a/Vx
=-mVx²/a

similarly ,
Fy = -mVy²/a
Fz = -mVz²/a

|F| =( |Fx| +|Fy| +|Fy|)/3
= 1/(3a)m{ Vx² + Vy² + Vz²}
m is mass of single molecule
for N number of molecule
F =1/(3a) mN(Vx² + Vz² +Vy²)

Vx² + Vy² + Vz² = C² { RMS velocity }

now,
F =1/(3a)NmC²
F/A =1/{3a.A}NmC²
P =1/3(a³) NmC²
P =1/3N(m/V) C²

here P is the pressure , m is the mass of single , N is the number of molecule and C is RMS velocity .