A sample contains a mixture of ^ 110 Ag and ^ 108 Ag isotopes, each having an initial activity of 8.0 × 10^ 8 disintegrations per second. ^ 108 Ag is known to have a larger half-life than ^ 110 Ag . If one plots (A_ 110

Concepts Of Physics MCQ Edition [Volume 2]

Concepts Of Physics MCQ Edition [Volume 2] · Physics · The Nucleus

A sample contains a mixture of ^ 110 Ag and ^ 108 Ag isotopes, each...

A sample contains a mixture of ^{110}\text{Ag}

and ^{108}\text{Ag}

isotopes, each having an initial activity of

8.0 \times 10^{8}

disintegrations per second. ^{108}\text{Ag}

is known to have a larger half-life than ^{110}\text{Ag}

. If one plots

\ln(A_{110}/A_{0,110})

versus time for ^{110}\text{Ag}

for the first

50 \text{ s}

, what is the expected shape of the plot?<br><table style="border: 1px solid black; border-collapse: collapse; width: auto;"><tr><td style="border: 1px solid black; padding: 10px;">Time (s)</td><td style="border: 1px solid black; padding: 10px;">Activity

(

10^{8}

disintegrations

\texts^{-1}

)</td><td style="border: 1px solid black; padding: 10px;">Time (s)</td><td style="border: 1px solid black; padding: 10px;">Activity

(

10^{8}

disintegrations

\texts^{-1}

)</td></tr><tr><td style="border: 1px solid black; padding: 10px;">

</td><td style="border: 1px solid black; padding: 10px;">

11.799

</td><td style="border: 1px solid black; padding: 10px;">

200

</td><td style="border: 1px solid black; padding: 10px;">

3.0828

</td></tr><tr><td style="border: 1px solid black; padding: 10px;">

</td><td style="border: 1px solid black; padding: 10px;">

9.1680

</td><td style="border: 1px solid black; padding: 10px;">

300

</td><td style="border: 1px solid black; padding: 10px;">

1.8899

</td></tr><tr><td style="border: 1px solid black; padding: 10px;">

</td><td style="border: 1px solid black; padding: 10px;">

7.4492

</td><td style="border: 1px solid black; padding: 10px;">

400

</td><td style="border: 1px solid black; padding: 10px;">

1.1671

</td></tr><tr><td style="border: 1px solid black; padding: 10px;">

</td><td style="border: 1px solid black; padding: 10px;">

6.2684

</td><td style="border: 1px solid black; padding: 10px;">

500

</td><td style="border: 1px solid black; padding: 10px;">

0.7212

</td></tr><tr><td style="border: 1px solid black; padding: 10px;">

100

</td><td style="border: 1px solid black; padding: 10px;">

5.4115

B. A straight line with a negative slope.

The activity of a single radioactive isotope follows the exponential decay law:

A_{110} = A_{0,110} e^{-\lambda_{110} t}

where

A_{110} is the activity at time

,

A_{0,110}

is the initial activity, and

\lambda_{110}

is the decay constant of ^{110}\text{Ag}

.

Dividing by

A_{0,110}

and taking the natural logarithm on both sides, we get:

\ln\left(\dfrac{A_{110}}{A_{0,110}}\right) = -\lambda_{110} t

This equation represents a linear relationship of the form

y = mx

, where the y-axis is

\ln \left(\dfrac{A_{110}}{A_{0,110}}\right) and the x-axis is time

.<br><br>The slope of this line is

m = -\lambda_{110}

. Since the decay constant

\lambda_{110}

is a positive quantity, the slope is negative.<br><br>Thus, the expected shape of the plot is a straight line with a negative slope. The provided table of total activity is not required to determine the theoretical shape of the decay curve for a single isotope.