PHYSICS
PHYSICS
5th Edition
ISBN: 2818440038631
Author: GIAMBATTISTA
Publisher: MCG
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Chapter 29, Problem 68P

(a)

To determine

The ratio of the decay rates or activity at t=0

(a)

Expert Solution
Check Mark

Answer to Problem 68P

The ratio of the decay rates or activity at t=0 is 4.0.

Explanation of Solution

Two radioactive sample with equal number of nuclides A and B, (NA=NB=N0)  at t=0 has half-life of 3.0 h and 12.0 h respectively.

Write the formula for half-life

T1/2=τln2                                                                                             (I)

Here, T1/2 is the half-life and τ is the time constant.

Write the formula for the decay constant

λ=1τ                                                                                                    (II)

Here, λ is the decay constant

Write the formula for the activity

R=R02t/T1/2                                                                                             (III)

Here, R is the activity after time t, R0 is the activity at t=0 and t is the time.

Write the formula for the activity

R=λN                                                                                           (III)

Here, N is the number of nuclei.

Conclusion:

Equating (IV) in (III) to solve for R

R=R02t/T1/2=λN

The ratio of activities of A and B is

RARB=RA020RB020=RA0RB0=λAN0λBN0=λAλB

Substitute equation (II)  and (I) in above equation

RARB=τBτA=TB/ln2TA/ln2=TBTA

Substitute 3.0 h for TA and 12.0 h for TB in the above equation to find the ratio of RARB

RARB=12.0 h3.0 h=4.0

Thus, the ratio of the decay rates or activity at t=0 is 4.0.

(b)

To determine

The ratio of the decay rates or activity at t=12.0 h

(b)

Expert Solution
Check Mark

Answer to Problem 68P

The ratio of the decay rates or activity at t=12.0 h is 0.50.

Explanation of Solution

The ratio of RARB at t=12.0 h is

RARB=RA02t/TARB02t/TB

Substitute RA0RB0=4.0, 3.0 h for TA and 12.0 h for TB in the above equation to find the ratio of RARB at t=12.0 h

RARB=4.0×2t(1TB1TA)RARB=4.0×212.0 h×(112.0 h13.0 h)RARB=0.50

Thus, the ratio of the decay rates or activity at t=12.0 h is 0.50.

(c)

To determine

The ratio of the decay rates or activity at t=24.0 h

(c)

Expert Solution
Check Mark

Answer to Problem 68P

The ratio of the decay rates or activity at t=24.0 h is 0.063.

Explanation of Solution

The ratio of RARB at t=24.0 h is

RARB=RA02t/TARB02t/TB

Substitute RA0RB0=4.0, 3.0 h for TA and 12.0 h for TB in the above equation to find the ratio of RARB at t=24.0 h

RARB=4.0×2t(1TB1TA)RARB=4.0×224.0 h×(112.0 h13.0 h)RARB=0.063

Thus, the ratio of the decay rates or activity at t=24.0 h is 0.063.

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Chapter 29 Solutions

PHYSICS

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