1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
S_A_V [24]
3 years ago
9

A certain radioactive isotope decays at a rate of 0.2​% annually. Determine the ​half-life of this​ isotope, to the nearest year

.
Chemistry
1 answer:
pychu [463]3 years ago
6 0

Answer:

The half-life of the radioactive isotope is 346 years.

Explanation:

The decay rate of the isotope is modelled after the following first-order linear ordinary differential equation:

\frac{dm}{dt} = -\frac{m}{\tau}

Where:

m - Current isotope mass, measured in kilograms.

t - Time, measured in years.

\tau - Time constant, measured in years.

The solution of this differential equation is:

m(t) = m_{o}\cdot e^{-\frac{t}{\tau} }

Where m_{o} is the initial mass of the isotope. It is known that radioactive isotope decays at a yearly rate of 0.2 % annually, then, the following relationship is obtained:

\%e = \frac{m(t)-m(t+1)}{m(t)}\times 100\,\% = 0.2\,\%

1 - \frac{m(t+1)}{m(t)} = 0.002

1 - \frac{m_{o}\cdot e^{-\frac{t+1}{\tau} }}{m_{o}\cdot e^{-\frac{t}{\tau} }}=0.002

1 - e^{-\frac{1}{\tau} } = 0.002

e^{-\frac{1}{\tau} } = 0.998

-\frac{1}{\tau} = \ln 0.998

The time constant associated to the decay is:

\tau = -\frac{1}{\ln 0.998}

\tau \approx 499.500\,years

Finally, the half-life of the isotope as a function of time constant is given by the expression described below:

t_{1/2} = \tau \cdot \ln 2

If \tau \approx 499.500\,years, the half-life of the isotope is:

t_{1/2} = (499.500\,years)\cdot \ln 2

t_{1/2}\approx 346.227\,years

The half-life of the radioactive isotope is 346 years.

You might be interested in
Why do we need a “mole”?
Ivan
Wdym mole like the animal or something else ?
4 0
2 years ago
Read 2 more answers
4. Which of the following have mass? Select all that apply.
valkas [14]
All of these EXCEPT helium in a balloon
4 0
3 years ago
When converting from moles to molecules OR molecules to moles you use what quantitative value?
miv72 [106K]
24 because it amen r amend
7 0
3 years ago
List three examples of a carbon reservoir.
romanna [79]

Answer:

Oceans, Fossil fuels, atmosphere

8 0
3 years ago
Science 7 Grade 7. Can anyone please help me answer these science questions?
masha68 [24]
A cellphone battery is a electrical energy 
and a piston in an engine is an thermal energy
6 0
3 years ago
Read 2 more answers
Other questions:
  • KClKCl has a lattice energy of −701 kJ/mol.−701 kJ/mol. Consider a generic salt, ABAB , where A2+A2+ has the same radius as K+,K
    13·1 answer
  • For the following scenarios what is the metal? A piece of metal weighing 59.047 g was heated to 100.0 degree C and then put it i
    10·1 answer
  • Interpret the equations in terms of particles moles and mass
    5·1 answer
  • If sock type A is pulled first, sock type B second, and sock type C third, the fourth sock type must match one of A, B, or C.
    7·1 answer
  • Milk of magnesia is a base. What happens when you drink milk of magnesia for an upset stomach?
    5·2 answers
  • When is a scientific theory formed?
    15·1 answer
  • What are interactions between flip flops and sulfuric acid
    8·1 answer
  • What is the chemical name of the compound Cu(NO3)2? Use the list of polyatomic ions and the periodic table to help you answer.
    13·2 answers
  • Find the molality of a solution of 58.5 g NaCl dissolved in 0.556 kg of water
    11·1 answer
  • Express each quantity in regular notation along with its appropriate unit. a. 3.60 × 1 0 5 s 3.60×10 5 s b. 5.4 × 1 0 − 5 g / c
    13·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!