What is the half-life of (Na^24) if a research assistant made 160 mg of radioactive sodium (Na^24) and found that there was only 20 mg left 45 h later?

2 Answers
Jul 7, 2018

Answer:

#color(blue)(" The half life is 15 hours.")#

Explanation:

We need to find an equation of the form:

#A(t)=A(0)e^(kt)#

Where:

#bb(A(t))=# the amount after time t.

#bb(A(0)=# the amount at the start. i.e. t = 0.

#bbk=# the growth/decay factor.

#bbe=# Euler's number.

#bbt=# time, in this case hours.

We are given:

#A(0)=160#

#A(45)=20#

We need to solve for #bbk#:

#20=160e^(45k)#

Divide by 160:

#1/8=e^(45k)#

Taking natural logarithms of both sides:

#ln(1/8)=45kln(e)#

#ln(e)=1#

Hence:

#ln(1/8)=45k#

Dividing by 45:

#ln(1/8)/45=k#

#:.#

#A(t)=160e^(t(ln(1/8)/45))#

#A(t)=160e^(t/45(ln(1/8))#

#A(t)=160(1/8)^(t/45)#

Since by definition the half life is the time period when we have half of the starting amount:

#A(t)=80#

So we need to solve for t in:

#80=160*(1/8)^(t/45)#

#80/160=(1/8)^(t/45)#

#1/2=(1/8)^(t/45)#

Taking natural logarithms:

#ln(1/2)=t/45ln(1/8)#

#45*(ln(1/2))/(ln(1/8))=t=15#

The half life is 15 hours.

Jul 7, 2018

Answer:

15 hours

Explanation:

  • Quick Way

As the amount of a decaying substance halves over each half-life (hence the name), halving the amount in steps requires 3 steps to get from 160 to 20:

  • #160 to 80 to 40 to 20#

And #45 = 3 * 15#

So the half-life is 15 years.

  • More formal way

For half-life #tau#, where # X(t) # is the amount (mass/ number of particles/ etc) remaining at time t:

#X(t) = X_o (1/2)^(t/tau) qquad square#

So:

#X(0) = X_o, X(tau) = X_o/2, X(2tau) = X_o/4,...#

Plugging the values that are given into #square# :

#20 = 160 * (1/2)^(45/tau) #

#implies (1/2)^(45/tau) = 1/8 qquad qquad [= (1/2)^3]#

#implies 45/tau = 3 implies tau = 15#