EXPONENTIAL FUNCTIONS

The exponential function is the MOST commonly used function to describe natural phenomena.

A$($t$)=$ A$0$ebt

e $=$ that number $(2\mathrm{.}718...)$ and b is a constant that governs how fast A$($t$)$ changes

PART $1$

$1.$ On graph paper, make an Activity versus time graph with the data provided at the end of this

assignment. Sketch by hand. Draw in the data points and then draw a curve to represent the points.

$2.$ Find the constant $$b$$ for your curve.

a$.$ Find a place where your curve overlaps where two graph paper lines are intersecting, NOT a

data point. Circle that place on your curve.

b$.$ Think of the y$-$value of the circled spot as an A$0$ initial value, and think of the x$-$value of your

circled spot as an initial time t$0.$ Write those numerical values next to the circled spot.

c$.$ Find the place on the curve where A$($t$)=$ e$-1$A$0=0\mathrm{.}368$ A$0.$ Note the value of t there.

d$.$ Knowing that A$($t$)=$ A$0$ebt and that A$($t$)=$ e$-1$A$0=0\mathrm{.}368$ A$0$ at the place on the curve from part c$,$

find the numerical value for b$,$ the constant in the exponential equation.

e$.$ Use the method described above to pick $3$ different values of Ao$.$ Determine b for each case.

Average the $3$ values of b you found to find a final value for b$.$

The value for t that results in bt $=-1$ is commonly referred to as the time constant $\backslash$tau when dealing

with exponential functions involving time.

$3.$ In many situations, it is useful to know the time it takes to go from Ao to $(1)/(2)$A$0.$ For radioactive decay,

this is referred to as the half$-$life T$1/2.$

a$.$ To find the half$-$life for this graph, choose $3$ different values of Ao$.$ For each Ao$,$ find the time t to

reach $(1)/(2)$A$0.$ Average the values of t to determine the average half$-$life T$1/2.$

b$.$ The relationship between half$-$life T$1/2$ and b depends on the constant ln$2.$ Verify this relationship.

$4.$ The true initial activity Ao $($the activity when t $=0)$ can be found by using any point read

directly from the curve and the main equation above. Again choose three different points that lie on

the curve. Use the value of A and t for each point, together with baverage $($from $2.$ above$),$ to calculate

Ao $=$ A$($t$=0).$ Average the three values of Ao and compare with the known value of $70.$

PART $2$

$1.$ On graph paper, make an ln$($A$)$ versus time graph with the data provided at the end of this

assignment. Sketch by hand. Draw in the data points and then draw a curve to represent the points.

$2.$ From this graph, determine the slope and y$-$intercept.

$3.$ Considering the following math, find b and A$0$ from the slope and y$-$intercept.

A $=$ Ao ebt

lnA $=$ ln$($Ao ebt$)=$ lnAo $+$ ln$($ebt$)$

lnA $=$ lnAo $+$ bt $=$ bt $+$ lnAo

lnA $=$ bt $+$ lnAo $->$ y $=$ mx $+$ b

$4.$ Compare your b and A$0$ from part $1$ to the b and A$0$ found in part $2.$

PART $3$

$1.$ Use a spreadsheet and graphing software like Excel in order to generate a plot of the data like in part $1$

for A versus t$.$ Add a fit to the curve and include an equation for the fit on the graph. Be careful to use the

proper variables and units: do not use y and x$,$ instead use A$0$ and t$.$ Identify A$0$ and b$.$

$2.$ Use a spreadsheet and graphing software like Excel in order to generate a plot of the data like in part $2$

for lnA versus t$.$ Add a fit to the curve and include an equation for the fit on the graph. Be careful to use

the proper variables and units: do not use y and x$,$ instead use A$0$ and t$.$ Identify A$0$ and b$.$

$3.$ Compare your A$0$ and b values from part $3$ to the ones found in parts $1$ and $2.$

RADIOACTIVE DECAY DATA

Activity $($counts$)$ Time $($seconds$)$

$65\mathrm{.}55$

$52\mathrm{.}610$

$41\mathrm{.}815$

$34\mathrm{.}520$

$26\mathrm{.}225$

$20\mathrm{.}730$

$16\mathrm{.}535$

$14\mathrm{.}440$

$10\mathrm{.}645$

$8\mathrm{.}3850$

$7\mathrm{.}5155$

$5\mathrm{.}9360$

$3\mathrm{.}8765$

$3\mathrm{.}1370$

$3\mathrm{.}3575$