18.08.2015   NUCLEAR CHEMICAL ENGINEERING

Derivation of the Bateman Equation (2.17)

Consider the general radioactive decay chain

ATj —— ►ЛГ,—— >N3——- ►——————— ►———- >N,——- ► • • •

with Ni atoms of the first member at time zero and none of the other members present at that time. The differential equations are

(2.140s)

^ = x. Af, — 2N2

(2.140b)

(2.140/)

The boundary conditions at t = 0 are

a?

її

—©

(2.141s)

о

II

II

s?

II

II

a?

и

£

(2.141b)

The system of differential equations (2.140) may be transformed to a system of linear equations by taking the Laplace transform and using Eq. (2.138) for the Laplace transform of the first derivatives:

—N° + sNi = ~iNi

(2.142s)

sN2 = 1Nl — 2N2

(2.142b)

■a?

i

T

T

II

(2.1420

where N is the transform of N. These equations may be solved successively for the Ns:

r<

II

■as

(2.143s)

r, _ x.^1 _ Х, л?

2 2 + s (Xj + s)(X2 + s)

(2.143b)

_ П W?

Y _ Xf-rty-t _ *=1

1 i + S і

П (X* + s)

Jt= і

(2.1430

Nt may be found by taking the inverse transform of Eq. (2.143s):

Nt =N? e~Kt (2.144)

|”I 0* + — s)

image226 image227 Подпись: (2.145)

To find the inverse transform of Eqs. (2.143b) to (2.143/) it is necessary to express the denominator as a sum of partial fractions. For Eq. (2.143/) this would be

To find a specific coefficient Aj, multiply each side of Eq. (2.145) by (X;- + s):

Подпись: Ak (Xfc + S) Подпись: (2.146)І

—— ————— — Aj + (X,- + s) £

її (x* + *) кФІ

k*i

and let s approach —X;. When s = — X,-,

Подпись:(2.147)

(2.148)

Because the inverse transform of 1 /(X/ + s) is e V,

i — X — f

Ni = ZV? X,X2 • • • X,_, 2 —’—————————- O’ > 1) (2.149)

/=i ri(x*-w

k=

k*i

which is the Bateman equation (2.17). The product term ї’кФІ (X* — X,) has no meaning when the / species is the initial member of the chain, so Eq. (2.149)necessarily applies only to the daughter species, i. e., />1.