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Nesterboy [21]
2 years ago
11

Situation:

Mathematics
2 answers:
densk [106]2 years ago
8 0

\\ \rm\Rrightarrow N=N_oe^{-kt}

  • We need half-life
  • put N=N_o/2

\\ \rm\Rrightarrow \dfrac{N_o}{2}=N_oe^{-kt}

  • Cancel N_o

\\ \rm\Rrightarrow \dfrac{1}{2}=e^{-kt}

\\ \rm\Rrightarrow 2^{-1}=e^{-kt}

  • Cancel negative

\\ \rm\Rrightarrow 2=e^{kt}

  • Apply natural log

\\ \rm\Rrightarrow ln2=lne^{kt}

\\ \rm\Rrightarrow ktlne=ln2

\\ \rm\Rrightarrow kt=ln2

\\ \rm\Rrightarrow kt=0.693

\\ \rm\Rrightarrow T=\dfrac{0.693}{k}

  • Where T denotes half-life

Put k from the question

\\ \rm\Rrightarrow T=\dfrac{0.693}{0.125}

\\ \rm\Rrightarrow T=8(0.693)

\\ \rm\Rrightarrow T=5.54days

Note:-

  • This equation is popularly known as Arrhenius equation
  • The decay constant for first order reactions and half-life don't depend upon the initial concentration so 11g we haven't used
Gala2k [10]2 years ago
5 0

Answer:

5.5 days (nearest tenth)

Step-by-step explanation:

<u>Given formula:</u>

\sf N=N_0e^{-kt}

  • \sf N_0 = initial mass (at time t=0)
  • N = mass (at time t)
  • k = a positive constant
  • t = time (in days)

Given values:

  • \sf N_0 = 11 g
  • k = 0.125

Half-life:  The <u>time</u> required for a quantity to reduce to <u>half of its initial value</u>.

To find the time it takes (in days) for the substance to reduce to half of its initial value, substitute the given values into the formula and set N to half of the initial mass, then solve for t:

\begin{aligned}\sf N & = \sf N_0e^{-kt}\\\\\implies \sf \dfrac{11}{2} & = \sf 11e^{-0.125t}\\\\\sf \dfrac{1}{2} & = \sf e^{-0.125t}\\\\\sf \ln \left(\dfrac{1}{2}\right) & = \sf \ln e^{-0.125t\\\\\sf \ln \left(\dfrac{1}{2}\right) & = \sf -0.125t \ln e\\\\\sf \ln \left(\dfrac{1}{2}\right) & = \sf -0.125t(1)\\\sf t & = \sf \dfrac{\ln \left(\dfrac{1}{2}\right)}{-0.125}\\\\\sf t & = \sf 5.545177444...\\\\\sf t & = \sf 5.5\:\:days\:\:(nearest\:tenth)\end{aligned}

Therefore, the substance's half-life is 5.5 days (nearest tenth).

Learn more about solving exponential equations here:

brainly.com/question/28016999

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