Radio-activity/Chapter 9

by Ernest Rutherford

 

CHAPTER IX.

THEORY OF SUCCESSIVE CHANGES.

193. Introduction. We have seen in previous chapters that the radio-activity of the radio-elements is always accompanied by the production of a series of new substances with some distinctive physical and chemical properties. For example, thorium produces from itself an intensely radio-active substance, Th X, which can be separated from the thorium in consequence of its solubility in ammonia. In addition, thorium gives rise to a gaseous product, the thorium emanation, and also to another substance which is deposited on the surface of bodies in the neighbourhood of the thorium, where its presence is indicated by the phenomenon known as "excited activity."

A close examination of the origin of these products shows that they are not produced simultaneously, but arise in consequence of a succession of changes originating in the radio-element. Thorium first of all gives rise to the product Th X. The Th X produces from itself the thorium emanation, and this in turn is transformed into a non-volatile substance. A similar series of changes is observed in radium, with the exception that there is no product in radium corresponding to the Th X in the case of thorium. Radium first of all produces an emanation, which, like thorium, is transformed into a non-volatile substance. In uranium only one product, Ur X, has been observed, for uranium does not give off an emanation and in consequence does not produce excited activity on bodies.

As a typical example of the evidence, from which it is deduced that one substance is the parent of another, we will consider the connection of the two products Th X and the thorium emanation. It has been shown (section 154) that after the separation of Th X from a thorium solution, by precipitation with ammonia, the precipitated thorium hydroxide has lost to a large extent its power of emanating. This cannot be ascribed to a prevention of escape of the emanation produced in it, for very little emanation is observed when a current of air is drawn through the hydroxide in a state of solution, when most of the emanation present would be carried off. On the other hand, the solution containing the Th X gives off a large quantity of emanation, showing that the power of giving off an emanation belongs to the product Th X. Now it is found that the quantity of emanation given off by the separated Th X decreases according to an exponential law with the time, falling to half value in four days. The rate of production of emanation thus falls off according to the same law and at the same rate as the activity of the Th X measured in the ordinary manner by the α rays. Now this is exactly the result to be expected if the Th X is the parent of the emanation, for the activity of Th X at any time is proportional to its rate of change, i.e., to the rate of production of the secondary type of matter by the emanation in consequence of a change in it. Since the rate of change of the emanation (half transformed in 1 minute) is very rapid compared with the rate of change of Th X, the amount of emanation present will be practically proportional to the activity of the Th X at any instant, i.e., to the amount of unchanged Th X present. The observed fact that the hydroxide regains its power of emanating in the course of time is due to the production of fresh Th X by the thorium, which in turn produces the emanation.

In a similar way, excited activity is produced on bodies over which the emanation is passed, and in amount proportional to the activity of the emanation, i.e., to the amount of the emanation present. This shows that the active deposit, which gives rise to the phenomenon of excited activity, is itself a product of the emanation. The evidence thus seems to be conclusive that Th X is the parent of the emanation and that the emanation is the parent of the deposited matter.

194. Chemical and Physical properties of the active products. Each of these radio-active products is marked by some distinctive chemical and physical properties which differentiate it from the preceding and succeeding products. For example, Th X behaves as a solid. It is soluble in ammonia, while thorium is not. The thorium emanation behaves as a chemically inert gas and condenses at a temperature of -120° C. The active deposit from the emanation behaves as a solid and is readily soluble in sulphuric and hydrochloric acids and is only slightly soluble in ammonia.

The striking dissimilarity which exists in many cases between the chemical and the physical properties of the parent matter and the product to which it gives rise is very well illustrated by the case of radium and the radium emanation. Radium is an element so closely allied in chemical properties to barium that, apart from a slight difference in the solubility of the chlorides and bromides, it is difficult to distinguish chemically between them. It has a definite spectrum of bright lines similar in many respects to the spectra of the alkaline earths. Like barium, it is non-volatile at ordinary temperature. On the other hand, the emanation which is continually produced from radium is a radio-active and chemically inert gas, which is condensed at a temperature of -150° C. Both in its spectrum and in the absence of definite chemical properties, it resembles the argon-helium group of inert gases, but differs from these gases in certain marked features.

The emanation must be considered to be an unstable gas which breaks down into a non-volatile type of matter, the disintegration being accompanied by the expulsion of heavy atoms of matter (α particles) projected with great velocity. This rate of breaking up is not affected by temperature over the considerable range which has been examined. After a month's interval, the volume of the emanation has shrunk to a small portion of its initial value. But the most striking property of the emanation, which, as we shall see later (chapter XII), is a direct consequence of its radio-activity, is the enormous amount of energy emitted from it. The emanation in breaking up through its successive stages emits about 3 million times as much energy as is given out by the explosion of an equal volume of hydrogen and oxygen, mixed in the proper proportions to form water; and yet, in this latter chemical reaction more heat is emitted than in any other known chemical change. We have seen that the two emanations and the products Ur X, Th X lose their activity with the time according to a simple exponential law, and at a rate that is independent—as far as observation has gone—of the chemical and physical agents at our disposal. The time taken for each of these products to fall to half its value is thus a definite physical constant which serves to distinguish it from all other products.

On the other hand, the variation of the excited activity produced by these emanations does not even approximately obey such a law. The rate of decay depends not only on the time of exposure to the respective emanations, but also, in the case of radium, on the type of radiation which is used as a means of comparative measurement. It will be shown, in succeeding chapters, that the complexity of the decay is due to the fact that the matter in the active deposits undergoes several successive transformations, and that the peculiarities of the curves of decay, obtained under different conditions, can be explained completely on the assumption that two changes occur in the active deposit from both thorium and actinium and six in the active deposit from radium.

195. Nomenclature. The nomenclature to be applied to the numerous radio-active products is a question of great importance and also one of considerable difficulty. Since there are at least seven distinct substances produced from radium, and probably five from thorium and actinium, it is neither advisable nor convenient to give each a special name such as is applied to the parent elements. At the same time, it is becoming more and more necessary that each product should be labelled in such a way as to indicate its place in the succession of changes. This difficulty is especially felt in discussing the numerous changes in the active deposits from the different emanations. Many of the names attached to the products were given at the time of their discovery, before their position in the scheme of changes was understood. In this way the names Ur X, Th X were applied to the active residues obtained by chemical treatment of uranium and thorium. Since, in all probability, these substances are the first products of the two elements, it may be advisable to retain these names, which certainly have the advantage of brevity. The name "emanation" was originally given to the radio-active gas from thorium, and has since been applied to the similar gaseous products of radium and actinium.

Finding the name "radium emanation" somewhat long and clumsy, Sir William Ramsay^[1] has recently suggested "ex-radio" as an equivalent. This name is certainly brief and is also suggestive of its origin; but at least six other ex-radios, whose parentage is as certain as that of the emanation, remain unnamed. A difficulty arises in applying the corresponding names ex-thorio, ex-actinio to the other gaseous products, for, unlike radium, the emanations of thorium and actinium are probably the second, not the first, disintegration product of the radio-elements in question. Another name thus has to be applied to the first product in these cases. It may be advisable to give a special name to the emanation, since it has been the product most investigated and was the first to be isolated chemically; but, on the other hand, the name "radium emanation" is historically interesting, and suggests a type of volatile or gaseous matter. Since the term "excited" or "induced" activity refers only to the radiations from the active body, a name is required for the radiating matter itself. The writer in the first edition of this book suggested the name "emanation X."^[2] This title was given from analogy to the names Ur X and Th X, to indicate that the active matter was product of the emanation. The name, however, is not very suitable, and, in addition, can only be applied to the initial product deposited, and not to the further products of its decomposition. It is very convenient in discussing mathematically the theory of successive changes to suppose that the deposited matter called A is changed into B, B into C, C into D, and so on. I have therefore discarded the name emanation X, and have used the terms radium A, radium B, and so on, to signify the successive products of the decomposition of the emanation of radium. A similar nomenclature is applied to thorium and actinium. This system of notation is elastic and simple, and I have found it of great convenience in the discussion of successive products. In speaking generally of the active matter, which causes excited activity, without regard to its constituents, I have used the term "active deposit." The scheme of nomenclature employed in this book is clearly shown below:—

Radium Thorium Uranium Actinium
  [v] [v] [v] [v]
Radium emanation Th X Ur X Actinium X
  [v] [v] [v] [v]
Radium A } Thorium emanation Final product Actinium emanation
  [v] } [v] [v]
Radium B } Thorium A } Actinium A }
  [v] } Active [v] } [v] }
Radium C } deposit Thorium B } Active Actinium B } Active
  [v] } [v] } deposit [v] } deposit
Radium D } Thorium C } Actinium C }
  [v] } (final product) } (final product) }
  &c.

Each product on this scheme is the parent of the product below it. Since only two products have been observed in the active deposit of thorium and actinium, thorium C and actinium C respectively refer to their final inactive products. It will be shown in the next chapter that, as in the case of thorium, an intermediate product exists between actinium and its emanation. From analogy to the products Th X and Ur X, this substance is termed "actinium X."

196. Theory of Successive Changes. Before considering the evidence from which these changes are deduced, the general theory of successive changes of radio-active matter will be considered. It is supposed that the matter A changes into B, B into C, C into D, and so on.

Each of these changes is supposed to take place according to the same law as a monomolecular change in chemistry, i.e., the number N of particles unchanged after a time t is given by N = N_{0}e^{-λt}, where N_{0} is the initial number and λ the constant of the change.

Since dN/dt = -λN, the rate of change at any time is always proportional to the amount of matter unchanged. It has previously been pointed out that this law of decay of the activity of the radio-active products is an expression of the fact that the change is of the same type as a monomolecular chemical change. Suppose that P, Q, R represent the number of particles of the matter A, B, and C respectively at any time t. Let λ_{1}, λ_{2}, λ_{3} be the constants of change of the matter A, B, and C respectively. Each atom of the matter A is supposed to give rise to one atom of the matter B, one atom of B to one of C, and so on. The expelled "rays" or particles are non-radio-active, and so do not enter into the theory. It is not difficult to deduce mathematically the number of atoms of P, Q, R, . . . of the matter A, B, C, . . . existing at any time t after this matter is set aside, if the initial values of P, Q, R, . . . are given. In practice, however, it is generally only necessary to employ three special cases of the theory which correspond, for example, to the changes in the active deposit, produced on a wire exposed to a constant amount of radium emanation and then removed, (1) when the time of exposure is extremely short compared with the period of the changes, (2) when the time of exposure is so long that the amount of each of the products has reached a steady limiting value, and (3) for any time of exposure. There is also another case of importance which is practically a converse of Case 3, viz. when the matter A is supplied at a constant rate from a primary source and the amounts of A, B, C are required at any subsequent time. The solution of this can, however, be deduced immediately from Case 3 without analysis. 197. Case 1. Suppose that the matter initially considered is all of one kind A. It is required to find the number of particles P, Q, R of the matter A, B, C respectively present after any time t.

Then P = ne^{-λ_{1}t}, if n is the number of particles of A initially present. Now dQ, the increase of the number of particles of the matter B per unit time, is the number supplied by the change in the matter A, less the number due to the change of B into C, thus

               dQ/dt = λ_{1}P - λ_{2}Q (1).
Similarly dR/dt = λ_{2}Q - λ_{3}R (2).

Substituting in (1) the value of P in terms of n,

dQ/dt = λ_{1}ne^{-λ_{1}t} - λ_{2}Q.

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 changes in radium A, the values of λ_{1}, λ_{2}, λ_{3} were taken as 3·85 × 10^{-3}, 5·38 × 10^{-4}, 4·13 × 10^{-4} respectively, i.e., the times required for each successive type of matter to be half transformed are about 3, 21, and 28 minutes respectively. The ordinates of the curves represent the relative number of atoms of the matter A, B, and C existing at any time, and the value of n, the original number of atoms of the matter A deposited, is taken as 100. The amount of matter B is initially zero, and in this particular case, passes through a maximum about 10 minutes later, and then diminishes with the time. In a similar way, the amount of C passes through a maximum about 37 minutes after removal. After an interval of several hours the amount of both B and C diminishes very approximately according to an exponential law with the time, falling to half value after intervals of 21 and 28 minutes respectively. 198. Case 2. A primary source supplies the matter A at a constant rate and the process has continued so long that the amount of the products A, B, C, . . . has reached a steady limiting value. The primary source is then suddenly removed. It is required to find the amounts of A, B, C, . . . remaining at any subsequent time t.

In this case, the number n_{0} of particles of A, deposited per second from the source, is equal to the number of particles of A which change into B per second, and of B into C, and so on. This requires the relation

n_{0} = λ_{1}P_{0} = λ_{2}Q_{0} = λ_{3}R_{0} (6),

where P_{0}, Q_{0}, R_{0} are the maximum numbers of particles of the matter A, B, and C when a steady state is reached.

The values of P, Q, R at any time t after removal of the source are given by equations of the same form as (3) and (5) for a short exposure. Remembering the condition that initially

P = P_{0} = n_{0}/λ_{1},
Q = Q_{0} = n_{0}/λ_{2},
R = R_{0} = n_{0}/λ_{3},

it can readily be shown that

P = (n_{0}/λ_{1})e^{-λ_{1}t} (7),
Q = (n_{0}/(λ_{1} - λ_{2}))((λ_{1}/λ_{2})e^{-λ_{2}t} - e^{-λ_{1}t}) (8),
R = n_{0}(ae^{-λ_{1}t} + be^{-λ_{2}t} + ce^{-λ_{3}t}) (9),

where

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

Relative number of atoms of matter A.B.C. present at any instant (Case 2).

Fig. 73.

The relative numbers of atoms of P, Q, R existing at any time are shown graphically in Fig. 73, curves A, B, C respectively. The number of atoms R_{0} is taken as 100 for comparison, and the values of λ_{1}, λ_{2}, λ_{3} are taken corresponding to the 3, 21, and 28-minute changes in the active deposit of radium. A comparison with Fig. 72 for a short exposure brings out very clearly the variation in the relative amounts of P, Q, R in the two cases. Initially the amount of R decreases very slowly. This is a result of the fact that the supply of C due to the breaking up of B at first, nearly compensates for the breaking up of C. The values of Q and R after several hours decrease exponentially, falling to half value in 28 minutes. 199. Case 3. Suppose that a primary source has supplied the matter A at a constant rate for any time T and is then suddenly removed. Required the amounts of A, B, C at any subsequent time.

Suppose that n_{0} particles of the matter A are deposited each second. After a time of exposure T, the number of particles P_{T} of the matter A present is given by

P_{T} = n_{0}[integral]_{0}^T e^{-λ_{1}t}dt = (n_{0}/λ_{1})(1 - e^{-λ_{1}T}.

At any time t, after removal of the source, the number of particles P of the matter A is given by

P = P_{T}e^{-λ_{1}t} = (n_{0}/λ_{1})(1 - e^{-λ_{1}T})e^{-λ_{1}t}.

Consider the number of particles n_{0}dt of the matter A produced during the interval dt. At any later time t, the number of particles dQ of the matter B, which result from the change in A, is given (see equation 4) by

dQ = (n_{0}λ_{1})/(λ_{1} - λ_{2})(e^{-λ_{2}t} - e^{-λ_{1}t})dt = n_{0}f(t)dt (10).

After a time of exposure T, the number of particles Q_{T} of the matter B present is readily seen to be given by

Q_{T} = n_{0}[f(T)dt + f(T - dt)dt + . . . + f(0)dt]
          = n_{0}[integral]_{0}^T f(t)dt.

If the body is removed from the emanation after an exposure T, at any later time t the number of particles of B is in the same way given by

Q = n_{0}[integral]_{t}^{T + t} f(t)dt.

It will be noted that the method of deduction of Q_{T} and Q is

independent of the particular form of the function f(t). Substituting the particular value of f(t) given in equation (10) and integrating, it can readily be deduced that

            Q/Q_{T} = (ae^{-λ_{2}t} - be^{-λ_{1}t})/(a - b) (11), where a = (1 - e^{-λ_{2}T})/λ_{2}, b = (1 - e^{-λ_{1}T})/λ_{1}.

In a similar way, the number of particles R of the matter C present at any time can be deduced by substitution of the value of f(t) in equation (5). These equations are, however, too complicated in form for simple application to experiment, and will not be considered here. 200. Case 4. The matter A is supplied at a constant rate from a primary source. Required to find the number of particles of A, B, C at any subsequent time t, when initially A, B, C are absent.

The solution can be simply obtained in the following way. Suppose that the conditions of Case 2 are fulfilled. The products A, B, C are in radio-active equilibrium and let P_{0}, Q_{0}, R_{0} be the number of particles of each present. Suppose the source is removed. The values of P, Q, R at any subsequent time are given by equations (7), (8) and (9) respectively. Now suppose the source, which has been removed, still continues to supply A at the same constant rate and let P_{1}, Q_{1}, R_{1} be the number of particles of A, B, C again present with the source at any subsequent time. Now we have seen, that the rate of change of any individual product, considered by itself, is independent of conditions and is the same whether the matter is mixed with the parent substance or removed from it. Since the values of P_{0}, Q_{0}, R_{0} represent a steady state where the rate of supply of each kind of matter is equal to its rate of change, the sum of the number of particles A, B, C present at any time with the source, and in the matter from which it was removed, must at all times be equal to P_{0}, Q_{0}, R_{0}, . . ., that is

P_{1} + P = P_{0},
Q_{1} + Q = Q_{0},
R_{1} + R = R_{0}.

This must obviously be the case, for otherwise there would be a

destruction or creation of matter by the mere process of separation of the source from its products; but, by hypothesis, neither the rate of supply from the source, nor the law of change of the products, has been in any way altered by removal.

Substituting the values of P, Q, R from equations (7), (8), and (9), we obtain

P_{1}/P_{0} = 1 - e^{-λ_{1}t},
Q_{1}/Q_{0} = 1 - (λ_{1}e^{-λ_{2}t} - λ_{2}e^{-λ_{1}t}) / (λ_{1} - λ_{2}),
R_{1}/R_{0} = 1 - λ_{3}(ae^{-λ_{1}t} + be^{-λ_{2}t} + ce^{-λ_{3}t}),

where a, b, and c have the values given after equation (9). The curves representing the increase of P, Q, R, are thus, in all cases, complementary to the curves shown in Fig. 73. The sum of the ordinates of the two curves of rise and decay at any time is equal to 100. We have already seen examples of this in the case of the decay and recovery curves of Ur X and Th X.

201. Activity of a mixture of products. In the previous calculations we have seen how the number of particles of each of the successive products varies with the time under different conditions. It is now necessary to consider how this number is connected with the activity of the mixture of products.

If N is the number of particles of a product, the number of particles breaking up per second is λN, where λ is the constant of change. If each particle of each product, in breaking up, emits one α particle, we see that the number of α particles expelled per second from the mixture of products at any time is equal to λ_{1}P + λ_{2}Q + λ_{3}R + . . ., where P, Q, R, . . . are the numbers of particles of the successive products A, B, C, . . . . Substituting the values of P, Q, R already found from any one of the four cases previously considered, the variation of the number of α particles expelled per second with the time can be determined.

The ideal method of measuring the activity of any mixture of radio-active products would be to determine the number of α or β particles expelled from it per second. In practice, however, this is inconvenient and also very difficult experimentally.

Certain practical difficulties arise in endeavouring to compare the activity of one product with another. We shall see later that, in many cases, all of the successive products do not emit α rays. Some give out β and γ rays alone, while there are several "rayless" products, that is, products which do not emit either α, β, or γ rays. In the case of radium, for example, radium A gives out only α rays, radium B no rays at all, while radium C gives out α, β, and γ rays.

In practice, the relative activity of any individual product at any time is usually determined by relative measurements of the saturation ionization current produced between the electrodes of a suitable testing vessel.

Let us consider, for example, the case of a product which gives out only α rays. The passage of the α particles through the gas produces a large number of ions in its path. Since the α particles from any individual product are projected with the same average velocity under all conditions, the relative amount of the ionization produced per second in the testing vessel serves as an accurate means of determining the variation of its activity. No two products, however, emit α particles with the same average velocity. We have seen that the rays from some products are more readily stopped in the gas than others. Thus the relative saturation current, due to two different products in a testing vessel, does not serve as an accurate method of comparing the relative number of α particles expelled per second. The ratio of the currents will in general depend upon the distance between the plates of the testing vessel, and, unless the relative ionization due to the average α particle from the two products is known from other data, the comparison of the currents can, at best, be only an approximate guide to the relative number of α particles escaping into the gas.

202. Some examples will now be considered to show how the factors, above considered, influence the character of the curves of activity obtained under different experimental conditions. For the purpose of illustration, we shall consider the variation after removal of the excited activity on a body exposed for different times to a constant supply of the radium emanation. The active deposit on removal consists in general of a mixture of the products radium A, B, and C. The nature of the rays from each product, the time for each product to be transformed, and the value of λ are tabulated below for convenience:—

  Product Rays T. λ (sec^{-1})
Radium A α rays 3 min. 3·85 × 10^{-3}
Radium B no rays 21 min. 5·38 × 10^{-4}
Radium C α, β, γ rays 28 min. 4·13 × 10^{-4}

Since only the product C gives rise to β and γ rays, the activity measured by either of these types of rays will be proportional to the amount of C present at any time, i.e. to the value of R at any time. For a long exposure, the variation of activity with time measured by the β and γ rays will thus be represented by the upper curve CC of Fig. 73, where the ordinates represent activity. This curve will be seen to be very similar in shape to the experimental curve for a long exposure which is given in Fig. 68.

Since radium B does not give out rays, the number of α particles expelled from the active deposit per second is proportional to λ_{1}P + λ_{3}R. The activity measured by the α rays, using the electrical method, is thus proportional at any time to λ_{1}P + Kλ_{3}R, where K is a constant which represents the ratio of the number of ions, produced in the testing vessel, by an α particle from C compared with that from an α particle emitted by A.

It will be seen later that, for this particular case, K is nearly unity. Taking K = 1, the activity at any time after removal is proportional to λ_{1}P + λ_{3}R.

Case 1. We shall first consider the activity curve for a short exposure to the radium emanation. The relative values of P, Q, and R at any time corresponding to this case are graphically shown in Fig. 74. The activity measured by the α rays at any time will be the sum of the activities due to A and C separately.

Let curve AA (Fig. 74) represent the activity due to A. This decreases exponentially, falling to half value in 3 minutes. In order to show the small activity due to C clearly in the Figure, the activity due to A is plotted after an interval of 6 minutes, when the activity has been reduced to 25 per cent. of its maximum value. The activity due to C is proportional to λ_{3}R, and in order to represent the activity due to C to the same scale as A, it is necessary to reduce the scale of the ordinates of curve CC in Fig. 72 in the ratio λ_{3}/λ_{1}.

Fig. 74.

The activity due to C is thus represented by the curve CCC, Fig. 74. The total activity is thus represented by a curve A + C whose ordinates are the sum of the ordinates of A and C.

This theoretical activity curve is seen to be very similar in its general features to the experimental curve shown in Fig. 66, where the activity from a very short exposure is measured by the α rays.

Case 2. The activity curve for a long exposure to the emanation will now be considered. The activity after removal of A and C is proportional to λ_{1}P + λ_{3}R, where the values of P and R are graphically shown in Fig. 75 by the curves AA, CC. Initially after removal, λ_{1}P_{0} = λ_{3}R_{0}, since A and C are in radio-active equilibrium, and the same number of particles of each product break up per second. The activity due to A alone is shown in curve AA, Fig. 75. The activity decreases exponentially, falling to half value in 3 minutes. The activity due to C at any time is pro-

• portional to R, and is initially equal to that of A. The activity

curve due to C is thus represented by the curve CC, which is the same curve as the upper curve CC of Fig. 73. The activity of A and C together is represented by the upper curve A + C (Fig. 75), where the ordinates are equal to the sum of the ordinates of the curves A and C. This theoretical curve is seen to be very similar in shape to the experimental curve (Fig. 67) showing the decay of activity of the active deposit from a long exposure measured by the α rays.

Fig. 75.

203. Effect of a rayless change on the activity curves. Certain important cases occur in the analysis of radio-active changes, when one of the products does not give rise to rays and so cannot be detected directly. The presence of this rayless change can, however, be readily observed by the variations which occur in the activity of the succeeding product.

Let us consider, for example, the case where the inactive matter A, initially all of one kind, changes into the matter B which gives out rays. The inactive matter A is supposed to be transformed according to the same law as the radio-active products. Let λ_{1}, λ_{2} be the constants of the change of A and B respectively. If n is the number of particles of A, initially present, we see from the equation (4), section 197, that the number of particles of the matter B present at any time is given by

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

Differentiating and equating to zero, it is seen that the value of Q passes through a maximum at a time T given by the equation

λ_{2}e^{-λ_{2}T} = λ_{1}e^{-λ_{1}T}.

For the sake of illustration, we shall consider the variation of the activity of the active deposit of thorium, due to a very short exposure to the emanation. Thorium A gives out no rays, and thorium B gives out α, β, and γ rays, while thorium C is inactive.

The matter A is half transformed in 11 hours, and B is half transformed in 55 minutes. The value of λ_{1} = 1·75 x 10^{-5}(sec.)^{-1} and λ_{2} = 2·08 x 10^{-4}(sec.)^{-1}.

The activity of the mixture of products A + B is due to B alone, and will, in consequence, be always proportional to the amount of B present, that is, to the value of Q.

Fig. 76.

The variation of activity with time is shown graphically in Fig. 76. The activity rises from zero to a maximum in 220 minutes and then decays, finally decreasing, according to an exponential law, with the time, falling to half value in 11 hours. This theoretical curve is seen to agree closely in shape with the experimental curve (Fig. 65), which shows the variation of the activity of the active deposit of thorium, produced by a short exposure in presence of the emanation.

There are several points of interest in connection with an activity curve of this character. The activity, some hours after removal, decays according to an exponential law, not at the rate of the product B, from which the activity rises, but at the same rate as the first rayless transformation. This will also be the case if the rayless product has a slower rate of change than the succeeding active product. Given an activity curve of the character of Fig. 76, we can deduce from it that the first change is not accompanied by rays and also the period of the two changes in question. We are, however, unable to determine from the curve which of the periods of change refers to the rayless product. It is seen that the activity curve is unaltered if the values of λ_{1}, λ_{2}, that is, if the periods of the products are interchanged, for the equation is symmetrical in λ_{1}, λ_{2}. For example, in the case of the active deposit of thorium, without further data it is impossible to decide whether the period of the first change has a value of 55 minutes or 11 hours. In such cases the question can only be settled by using some physical or chemical means in order to separate the product A from B, and then testing the rate of decay of their activity separately. In practice, this can often be effected by electrolysis or by utilizing the difference in volatility of the two products. If now a product is separated from the mixture of A and B which loses its activity according to an exponential law, falling to half value in 55 minutes (and such is experimentally observed), we can at once conclude that the active product B has the period of 55 minutes.

The characteristic features of the activity curve shown in Fig. 76 becomes less marked with increase of the time of exposure of a body to the emanation, that is, when more and more of B is mixed with A at the time of removal. For a long time of exposure, when the products A and B are in radio-active equilibrium, the activity after removal is proportional to Q, where

Q = (n_{0}/(λ_{1} - λ_{2}))((λ_{1}/λ_{2})e^{-λ_{2}t} - e^{-λ_{1}t),

(see equation 8, section 198). The value of Q, in this case, does

not increase after removal, but at once commences to diminish. The activity, in consequence, decreases from the moment of removal, but more slowly than would be given by an exponential law. The activity finally decays exponentially, as in the previous case, falling to half value in 11 hours.

In the previous case we have discussed the activity curve obtained when both the active and inactive product have comparatively rapid rates of transformation. In certain cases which arise in the analysis of the changes in actinium and radium, the rayless product has a rate of change extremely slow compared with that of the active product. This corresponds to the case where the active matter B is supplied from A at a constant rate. The activity curve will thus be identical in form with the recovery curves of Th X and Ur X, that is, the activity I at any time t will be represented by the equation I_{t}/I_{0} = 1 - e^{-λ_{2}t}, where I_{0} is the maximum value of the activity and λ_{2} the constant of change of B.

204. In this chapter we have considered the variation with time, under different conditions, of the number of atoms of the successive products, when the period and number of the changes are given. It has been seen that the activity curves to be expected under various conditions can be readily deduced from the simple theory. In practice, however, the investigator has been faced with the much more difficult inverse problem of deducing the period, number, and character of the products, by analysis of the activity curves obtained under various conditions.

In the case of radium, where at least seven distinct changes occur, the problem has been one of considerable difficulty, and a solution has only been possible by devising special physical and chemical methods of isolation of some of the products.

We shall see later that two rayless changes occur in radium and actinium and one in thorium. It is at first sight a very striking fact that the presence of a substance which does not emit rays can be detected, and its properties investigated. This is only possible when the rayless product is transformed into another substance which emits rays; for the variation of the activity of the latter may be such as to determine not only the period but also the physical and chemical properties of the parent product. In the two following chapters the application of the theory of successive changes will be shown to account satisfactorily for the complicated processes occurring in the radio-elements.

1. Ramsay, Proc. Roy. Soc. p. 470, June, 1904; C. R. 138, June 6, 1904.
2. Phil. Mag. February, 1904.