Physics for Scientists and Engineers
Physics for Scientists and Engineers
6th Edition
ISBN: 9781429281843
Author: Tipler
Publisher: MAC HIGHER
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Chapter 40, Problem 73P

a.

To determine

To Justify:The equation dNDdt=λPNPλDND

a.

Expert Solution
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Explanation of Solution

Introduction:

Often, the daughter nucleus of a radioactive parent nucleus is itself radioactive. The number of daughter nuclei ND are given by the solution to the differential equation dNDdt=λPNPλDND

The rate of decay of D nuclei subtracted from the rate of change of ND is the rate of generation of D nuclei. The generation rate is equal to the decay rate if P nuclei, which equals λPNP . The decay rate of D nuclei is λDND .

Conclusion:

The rate of change of ND is the rate of generation of D nuclei minus the rate of decay of D nuclei. The generation rate is equal to the decay rate if P nuclei, which equals λPNP . The decay rate of D nuclei is λDND .

b.

To determine

To Show: The equation ND(t)=NPoλPλDλP(eλpteλDt) where NPo is the number of parent nuclei present at t=0 when there are no daughter nuclei.

b.

Expert Solution
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Answer to Problem 73P

  ND(t)=NPoλPλDλP(eλpteλDt) is the solution of the equation dNDdt=λPNPλDND

Explanation of Solution

Given:

Parent nucleus decay constant λP

Daughter nucleus decay constant λD

  ND(t)=NPoλPλDλP(eλpteλDt)

Formula Used:

  ND(t)=NPoλPλDλP(eλpteλDt)

Calculations:

Differentiate ND(t)=NPoλPλDλP(eλpteλDt) with respect to t

  ddt[ND(t)]=NPoλPλDλPddt(eλpteλDt)=NPoλPλDλP(λpeλpt+λDeλDt)

Substituting in dNDdt=λPNPλDND

  NPoλPλDλP(λpeλpt+λDeλDt)=NPoλPeλptλD[NPoλPλDλP(λpeλptλDeλDt)]

Multiply both sides by λDλPλDλP and simplify

  NPoλD(λpeλpt+λDeλDt)=λDλPλDNPoeλptNPo(eλpteλDt)

  =NPoeλptNPoλPλDeλptNPoeλpt+NPoeλDt

  =NPoλPλDeλpt+NPoeλDt

  =NPoλD[λpeλpt+λDeλDt]

Which is an identity and confirms that ND(t)=NPoλPλDλP(eλpteλDt) is the solution of the equation dNDdt=λPNPλDND

Conclusion:

  ND(t)=NPoλPλDλP(eλpteλDt) is the solution of the equation dNDdt=λPNPλDND

c.

To determine

Show that the expression for ND in part b gives ND(t)>0 whether λP>λD or λD>λP .

c.

Expert Solution
Check Mark

Explanation of Solution

Introduction:

Frequently, the daughter nucleus of a radioactive parent nucleus is itself radioactive. The number of daughter nuclei ND are given by the solution to the differential equation dNDdt=λPNPλDND

  ND(t)=NPoλPλDλP(eλpteλDt) is the solution of the equation dNDdt=λPNPλDND

If λP>λD , the denominator and (eλpteλDt) are both negative for t>0 . If λP<λD the denominator and (eλpteλDt) are both positive for t>0 .

d.

To determine

Make a plot of NP(t) and ND(t) as a function of time when τD=3τP .

d.

Expert Solution
Check Mark

Explanation of Solution

Introduction:

The number of daughter nuclei ND are given by the solution to the differential equation dNDdt=λPNPλDND . ND(t)=NPoλPλDλP(eλpteλDt) is the solution of the equation dNDdt=λPNPλDND .

  Physics for Scientists and Engineers, Chapter 40, Problem 73P

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