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One molecule of water \(\left(\mathrm{H}_{2} \mathrm{O}\right)\) contains two atoms of hydrogen and one atom of oxygen. A hydrogen atom has a mass of \(1.0 \mathrm{u}\) and an atom of oxygen has a mass of \(16 \mathrm{u}\), approximately. (a) What is the mass in kilograms of one molecule of water? (b) How many molecules of water are in the world's oceans, which have an estimated total mass of \(1.4 \times 10^{21} \mathrm{~kg}\) ?

Short Answer

Expert verified
One water molecule has a mass of \(2.99 \times 10^{-26} \mathrm{kg}\), and there are about \(4.68 \times 10^{46}\) molecules in the oceans.

Step by step solution

01

Calculate the Molecular Mass of Water

Each molecule of water contains two hydrogen atoms and one oxygen atom. The molecular mass is computed by adding the mass of these atoms. Given the mass of a hydrogen atom is \(1.0 \ \mathrm{u}\) and the mass of an oxygen atom is \(16 \ \mathrm{u}\), the molecular mass of water is calculated as:\[2 \times 1.0 \ \mathrm{u} + 16 \ \mathrm{u} = 18 \ \mathrm{u}\]This means the molecular mass of one water molecule is \(18\ \mathrm{u}\).
02

Convert the Molecular Mass to Kilograms

To convert the molecular mass from unified atomic mass units (amu) to kilograms, use the conversion factor \(1 \ \mathrm{u} = 1.66053904 \times 10^{-27} \, \mathrm{kg}\). Thus, the mass of one water molecule is:\[18 \ \mathrm{u} \times 1.66053904 \times 10^{-27} \, \mathrm{kg/u} = 2.98897027 \times 10^{-26} \, \mathrm{kg}\]Therefore, one water molecule has a mass of approximately \(2.99 \times 10^{-26} \, \mathrm{kg}\).
03

Determine the Number of Molecules in the Oceans

Given the total mass of the world's oceans is approximately \(1.4 \times 10^{21} \, \mathrm{kg}\), we need to find how many water molecules this mass contains. Divide the total ocean mass by the mass of one water molecule:\[\frac{1.4 \times 10^{21} \, \mathrm{kg}}{2.99 \times 10^{-26} \, \mathrm{kg/molecule}} \approx 4.68 \times 10^{46} \, \text{molecules}\]This calculation shows there are approximately \(4.68 \times 10^{46}\) water molecules in the oceans.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Unified Atomic Mass Unit (u)
In chemistry and physics, the unified atomic mass unit (u) is a standardized unit of mass that reflects the mass of atoms and molecules. It provides a convenient measure to express atomic and molecular masses across elements and compounds.
This unit is defined as one twelfth of the mass of an unbound neutral atom of carbon-12, which is approximately equal to 1.66053904 脳 10鈦宦测伔 kilograms. By using this standardized measure, scientists can easily communicate and calculate the masses of atoms and molecules.
Some key points about this unit include:
  • It offers a straightforward way to represent small masses.
  • Enables comparison of atomic and molecular weights simply.
  • The unit is sometimes referred to as an "amu" (atomic mass unit) in older texts.
Water Molecule
Water, a vital substance for life, is composed of molecules represented by the chemical formula \(\mathrm{H}_{2}\mathrm{O}\). This means that each water molecule consists of two hydrogen atoms and one oxygen atom.
These components are bonded together in a bent or "V" shape, with the oxygen atom in the center. This configuration is vital for water's unique properties, including its polarity and the capability to dissolve many substances.
In quantitative terms, the mass of a single water molecule in unified atomic mass units is calculated by summing up the masses of the atoms within it:
  • Two hydrogen atoms each with a mass of approximately 1.0 u.
  • One oxygen atom with a mass of approximately 16 u.
So, the total molecular mass of water is \(2 \times 1.0 \, \mathrm{u} + 16 \, \mathrm{u} = 18 \, \mathrm{u}\). Water's molecular mass is crucial for determining various properties, such as boiling and freezing points, and its interaction with other molecules.
Conversion Factors
In chemistry, conversion factors play an essential role in translating measurements from one unit to another, facilitating calculations and applications across different systems of measurement.
When dealing with the molecular mass of water, the conversion factor used is transforming from unified atomic mass units (u) to kilograms. This is because kilogram is the standard unit of mass in the International System of Units (SI).
Here鈥檚 how the conversion works for water:
  • Molecular mass in u: 18 u
  • Conversion factor: 1 u equals approximately \(1.66053904 \times 10^{-27} \, \mathrm{kg}\)
Thus, the calculation is \(18 \, \mathrm{u} \times 1.66053904 \times 10^{-27} \, \mathrm{kg/u} = 2.99 \times 10^{-26} \, \mathrm{kg}\). This conversion is crucial to link the atomic scale with the macroscopic scale, enabling comparisons and understanding of biological and environmental processes.

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