Page:Radio-activity.djvu/354

 The solution of this equation is of the form

Q = n(ae^{-λ_{1}t} + be^{-λ_{2}t})     (3).

By substitution it is found that a = λ_{1}/(λ_{2} - λ_{1}).

Since Q = 0 when t = 0, b = -λ_{1}(λ_{2} - λ_{1}).

Thus     Q = (nλ_{1}/(λ_{1} - λ_{2}))(e^{-λ_{2}t} - e^{-λ_{1}t})      (4).

Substituting this value of Q in (2), it can readily be shown that

R = n(ae^{-λ_{1}t} + be^{-λ_{2}t} + ce^{-λ_{3}t})     (5),

where

a = λ_{1}λ_{2}/((λ_{1} - λ_{2})(λ_{1} - λ_{3})), b = -λ_{1}λ_{2}/((λ_{1} - λ_{2})(λ_{2} - λ_{3})),

c = λ_{1}λ_{2}/((λ_{1} - λ_{3})(λ_{2} - λ_{3})).

Relative number of atoms of matter A.B.C. present at any time (Case 1).

Fig. 72.

The variation of the values of P, Q, R with the time t, after removal of the source, is shown graphically in Fig. 72, curves A, B, and C respectively. In order to draw the curves for the practical case which will be considered later corresponding to the first three