BioPHYSICAL chemistry

10 CHAPTER 1 BASIC THERMODYNAMIC AND BIOCHEMICAL CONCEPTS

Derivation box 1.1 Relationship between the average velocity and pressure

In kinetic theory, gas molecules are considered to randomly collide with the wall and on

average exert a certain pressure due to the sum of the collisions averaged over time (Figure 1.3).

Consider a single molecule moving in three dimensions with a velocity 9 , with (^9) xbeing the
velocity along the xdirection. After collision, the molecule is assumed to have the same kinetic
energy but to be moving in the opposite direction with the velocity along the xaxis being
− (^9) x. The linear momentum, p, of the particle is given
by the product of its mass and velocity:
p=m 9 (db1.1)
so the change in momentum along the xdirection,
Δpx, is given by:
Δpx= 2 m (^9) x (db1.2)
During a time interval Δt, a particle with a velo-
city (^9) xcan travel a distance given by the product
of the velocity and time:
Distance =velocity ×time =( (^9) x)(Δt) (db1.3)
On average, half of the particles that are within the distance (^9) xΔtare moving towards the
wall and will collide within the time Δt. If the area of the wall is Aand the number of
molecules per unit area is N, then the number that will hit the wall is:
(db1.4)
and the total momentum change is given by the product of the number (eqn db1.4) and the
momentum change of each molecule (eqn db1.2):
(db1.5)
and the rate of momentum change is given by the total momentum change divided by the
time interval, yielding:
Rate of total momentum change =NAm (^92) x (db1.6)
Total momentum change =

1

2

NtAmNAmt (^999) xxxΔΔ 2 2

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟()( )=

Number colliding with wall in Δt=

1

2

NtA (^9) xΔ

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟()

vx

v

v

vx

vxΔt

Figure 1.3Pressure arises from the

random collisions of gas molecules with

the walls. A molecule traveling with the

momentum m (^9) xwill travel a distance
(^9) xΔtduring the time Δt.