The natural abundances of the two stable isotopes of...

The natural abundances of the two stable isotopes of hydrogen (hydrogen and deuterium) are $99.99$ percent ${ }_{1}^{1} \mathrm{H}$ and $0.01$ percent ${ }_{1}^{2} \mathrm{H}$. Assume that water exists as either $\mathrm{H}_{2} \mathrm{O}$ or $\mathrm{D}_{2} \mathrm{O} .$ Calculate the number of $\mathrm{D}_{2} \mathrm{O}$ molecules in exactly $400 \mathrm{~mL}$ of water (density $1.00 \mathrm{~g} / \mathrm{mL}$ ).

Key Concepts

Isotopic Abundance

This refers to the relative proportions of different isotopes of an element that occur naturally. In problems like these, understanding isotopic abundance is crucial because it allows one to determine the likelihood of a water molecule containing a specific isotope of hydrogen, based on the given percentages.

Mole Concept and Molar Mass

The mole is a fundamental unit in chemistry used to relate masses of substances to the number of particles (atoms, molecules). Molar mass conversion helps in calculating the number of moles from a given mass of a substance, which is essential when transitioning from macroscopic measurements (like grams) to microscopic counts (such as molecules).

Water Density and Volume Conversion

Density is the mass per unit volume of a substance. When the density and volume of a liquid (like water) are known, this concept allows one to calculate its mass, which is a necessary step before converting to moles using the molar mass.

Avogadro's Number

Avogadro's number (approximately 6.022 x 10^23) is used to convert moles to the actual number of molecules. This constant is pivotal in bridging the gap between the macroscopic amount of a substance and the microscopic number of molecules it contains.

Probability in Isotopic Distribution

Since isotopes are present in certain proportions, probability principles are used to calculate the likelihood of a molecule having a specific isotopic composition. In this context, the chance that a water molecule consists of two particular isotopes (e.g., both deuterium atoms) is determined by multiplying the individual probabilities for each hydrogen site.

Transcript

00:01 So for this question, we are told that we have 400 milliliters of water, and we are asked to calculate the number of d2o molecules present.

00:13 So we are also given the information that the water is going to be comprised of both h2o and d2o.

00:23 So the h2o is just the hydrogen atom, which has the atomic mass of one, and deuterium, which has an atomic mass of two.

00:43 So in order to find out how many d2o molecules are in 400 millioners of water, let's first convert this into grams.

00:52 So we're told that the density of water is 1 gram per 1 millimeter of water.

01:00 So if we have 400 milliliters of water, we can assume that we also have 400 grams of water, or 400 grams in 400 mils.

01:17 So from there, we know that the percentages of h2o and d2o that would be present.

01:27 So we are told that 99 .99 % of the molecules in that water are going to be h2o, and 0 .01 % of those molecules in the water are going to be d2o.

01:44 So we'll take that percentage times the total weight of 400 grams.

01:53 So we have to convert from the percentage.

01:57 So divided by 100, we'll get us 0 .0001 times the 400 grams.

02:15 And that will get us .04 grams of d2o that are present.

02:25 Now you can do the same thing if you wanted to figure out how many grams of h2 are present.

02:31 You would just take .9999 times 400, and you would get 399 .96 grams of h2o...

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