Chemistry, 16.10.2020 09:01 dev804

Radioactive isotope X has a half-life of 500,000 years. This isotope may be found in some types of volcanic rocks. A particular sample of volcanic rock taken from a layer that covered up some of the earliest known human-like footprints contains 0.125 mg of isotope X. The volcanic rock sample originally contained 8.00 mg of isotope X. How long ago were these footprints made. I need the equation. Thanks. please I am stuck on this question. PLEASE

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Answer from: kalebboatman

The age of the earliest known human-like footprints is approximately 3000000 years.

Explanation:

Decay of isotopes is represented by the following ordinary differential equation:

$\frac{dm}{dt} = -\frac{m}{\tau}$ (Eq. 1)

Where:

$\frac{dm}{dt}$ - First derivative of mass with respect to time, measured in miligrams per year.

- Current mass of the isotope, measured in miligrams.

$\tau$ - Time constant, measured in years.

Now we proceed to obtain the solution of the differential equation:

$\int\limits {\frac{dm}{m} } = -\frac{1}{\tau}\int dt$

$\ln m = -\frac{t}{\tau}+C$

$m(t) = e^{-\frac{t}{\tau}+C }$

$m(t) = m_{o}\cdot e^{-\frac{t}{\tau} }$ (Eq. 2)

Where:

$m_{o}$ - Initial mass of the isotope, measured in miligrams.

- Time, measured in years.

$\tau$ - Time constant, measured in years.

We proceed to clear time within the formula presented above:

$\ln \frac{m(t)}{m_{o}} = -\frac{t}{\tau}$

$t = -\tau \cdot \ln \frac{m(t)}{m_{o}}$

In addition, time constant can be found as a function of half-life:

$\tau = \frac{t_{1/2}}{\ln 2}$ (Eq. 3)

If we know that $t_{1/2} = 500000\,yr$ , $m_{o} = 8\,mg$ and $m(t) = 0.125\,mg$ , the age of the earliest known human-like footprints is:

$\tau = \frac{500000\,yr}{\ln 2}$

$\tau \approx 721347.520\,yr$

$t = -(721347.520\,yr)\cdot \ln \left(\frac{0.125\,mg}{8\,mg} \right)$

$t \approx 3000000\,yr$

The age of the earliest known human-like footprints is approximately 3000000 years.

Answer from: Quest

Note that 2 mol dm⁻³ = 2 mol/(dm³)outline: we're given the concentration of hcl, which is 2 mol dm⁻³ (cubic meters in the denominator can represent volume, so i know this is concentration). we're an amount of calcium carbonate caco₃, which is 5 g. we're presented with a chemical equation which includes caco₃ and hcl. we are told to find the volume of hcl. from what we see so far, this seems to be a stoichiometry problem. we'll have to use the given information about calcium carbonate to find the amount of moles of hcl needed to fully react with calcium carbonate. then we'll have information about the moles of hcl and the concentration of hcl; that will be enough to figure out volume according to the formula concentration = moles / volume.======all the answer choices are not in dm.convert the given concentration so that it uses cm⁻³.we can use conversion factors. we have to cube our conversion factors so that dimensional analysis works.we can convert to meters first, using the fact that 10 dm = 1 m ⇔ 10³ dm³ = 1 m³then we use the fact that 10² cm = 1 m ⇔ (10² cm)³ = (1 m)³ ⇔ 10⁶ cm³ = 1 m³so converting: $\dfrac{\text{2 mol}}{\text{dm}^{3}} \cdot \dfrac{\text{10}^3\text{ dm}^3}{\text{1 m}^3} \cdot \dfrac{\text{1 m}^3}{10^6\text{ cm}^3} \\ \\ = \dfrac{\text{2 mol}}{1} \cdot \dfrac{\text{10}^3}{1} \cdot \dfrac{1}{10^6\text{ cm}^3} \qquad \text{\footnotesize cancelling units} \\ \\ = \dfrac{2\text{ mol}}{10^3 \text{ cm}^3} \\ \\ = 0.002\text{ mol/cm}^3$ now we have to find the moles of hydrochloric acid neededfind the moles of calcium carbonate. we have to find the molar mass of calcium carbonate, caco₃, which can be found by molar mass = (mass ca) + (mass c) + 3(mass o) = 40 + 12 + 3(16) = 100 g/molconvert the given 5 g to moles: mol caco₃ = 5 g caco₃ * 1 mol/(100 g) = 0.05 mol caco₃use stoichiometry (ratios) to figure out the moles of hcl needed. ratio wise, there's 2 hcls for every caco₃, so use dimensional analysis mol hcl = mol caco₃ * (2 mol hcl) / (1 mol caco₃) = 0.05 mol hcl * 2 = 0.1 mol hcl we have 0.1 mol hcl and concentration of hcl (0.002 mol/cm³).we need to find volumesince concentration = moles / volume, rearrange to get volume = moles / concentration volume = 0.1 mol / (0.002 mol/cm³) volume = 50 cm³

Answer from: Quest

The answer to your question would be h+ + cn– + li+ + oh– mc001-4.jpg li+ + cn– + h2o. the total ionic equation for the following reaction is h+ + cn– + li+ + oh– mc001-4.jpg li+ + cn– + h2o.

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Radioactive isotope X has a half-life of 500,000 years. This isotope may be found in some types of v...

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