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 x being the
velocity along the x direction. 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 x axis being
−9 x . The linear momentum, p, of the particle is given
by the product of its mass and velocity:
p = m9
(db1.1)
so the change in momentum along the x direction,
Δp x , is given by:
Δp x = 2m9 x
(db1.2)
During a time interval Δt, a particle with a velocity 9 x can 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 x Δt are moving towards the
wall and will collide within the time Δt. If the area of the wall is A and 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 = NAm9 x
2
(db1.6)
Total momentum change =
1
2
2
2
N
t A m
NAm
t
x
x
x
9
9
9
Δ
Δ
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
=
( )(
)
Number colliding with wall in Δt =
1
2
N
t A
x
9 Δ
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟ ( )
ϩvx
ϩv
Ϫv
Ϫvx
v x Δt
Figure 1.3 Pressure arises from the
random collisions of gas molecules with
the walls. A molecule traveling with the
momentum m9 x will travel a distance
9 x Δt during the time Δt.
9781405124362_4_001.qxd 4/30/08 19:28 Page 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 x being the
velocity along the x direction. 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 x axis being
−9 x . The linear momentum, p, of the particle is given
by the product of its mass and velocity:
p = m9
(db1.1)
so the change in momentum along the x direction,
Δp x , is given by:
Δp x = 2m9 x
(db1.2)
During a time interval Δt, a particle with a velocity 9 x can 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 x Δt are moving towards the
wall and will collide within the time Δt. If the area of the wall is A and 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 = NAm9 x
2
(db1.6)
Total momentum change =
1
2
2
2
N
t A m
NAm
t
x
x
x
9
9
9
Δ
Δ
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
=
( )(
)
Number colliding with wall in Δt =
1
2
N
t A
x
9 Δ
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟ ( )
ϩvx
ϩv
Ϫv
Ϫvx
v x Δt
Figure 1.3 Pressure arises from the
random collisions of gas molecules with
the walls. A molecule traveling with the
momentum m9 x will travel a distance
9 x Δt during the time Δt.
9781405124362_4_001.qxd 4/30/08 19:28 Page 10