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Chapter 1: Problem 42

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 \(2.99 \times 10^{-26} \text{ kg}\) and there are approximately \(4.68 \times 10^{46}\) water molecules in the oceans.

Step by step solution

01

Calculate Mass of One Water Molecule in Atomic Mass Units

A molecule of water (\(\text{H}_2\text{O}\)) consists of 2 hydrogen atoms and 1 oxygen atom. Using the atomic masses given, we find the mass of the water molecule as:\[\text{Mass of 2 hydrogen atoms} = 2 \times 1.0 \text{ u} = 2.0 \text{ u}\]\[\text{Mass of 1 oxygen atom} = 16 \text{ u}\]Therefore, the total mass of one water molecule is:\[2.0 \text{ u} + 16 \text{ u} = 18.0 \text{ u}\]
02

Convert Mass from Atomic Mass Units to Kilograms

The atomic mass unit (u) is related to kilograms through the conversion factor \(1 \text{ u} = 1.660539 \times 10^{-27} \text{ kg}\). Thus, the mass of one molecule of water in kilograms is:\[18.0 \text{ u} \times 1.660539 \times 10^{-27} \text{ kg/u} = 2.98897 \times 10^{-26} \text{ kg}\]
03

Calculate Number of Water Molecules in the Oceans

Given the total mass of the world's oceans is estimated to be \(1.4 \times 10^{21} \text{ kg}\), we can find the number of water molecules by dividing this total mass by the mass of a single water molecule:\[\text{Number of molecules} = \frac{1.4 \times 10^{21} \text{ kg}}{2.98897 \times 10^{-26} \text{ kg/molecule}}\]\[= 4.68 \times 10^{46} \text{ molecules}\]

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Key Concepts

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

Atomic Mass Unit
In order to understand the mass of atoms and molecules, we use the atomic mass unit (u), a standard measure in chemistry. The atomic mass unit defines a scale by which the masses of atoms and subatomic particles can be compared. Rather than dealing with incomprehensibly small numbers, the atomic mass unit simplifies calculations.
  • 1 atomic mass unit (u) is equivalent to \(1/12^{ ext{th}}\) of the mass of a carbon-12 atom.
  • The mass of a hydrogen atom is approximately 1 u, and an oxygen atom approximately 16 u.
This measurement allows scientists to easily sum the masses of atoms in a molecule, like water, to determine molecular mass in atomic mass units. Understanding this basis in atomic mass is key to grasping how these small-scale calculations relate to larger, accessible numerical scales.
Conversion Factor
To convert measurements from atomic mass units to kilograms, we use a specific conversion factor. This is necessary because atomic mass units are not commonly used in daily calculations outside scientific contexts; hence, converting to kilograms allows for broader applicability.
The conversion factor for atomic mass unit to kilograms is \(1 ext{ u} = 1.660539 \times 10^{-27} ext{ kg}\).
  • This conversion is crucial, taking the molecular mass from 18 u (for water) to a more communicable mass of \(2.98897 \times 10^{-26} ext{ kg}\) per molecule.
  • Applying the conversion factor lets us work with numbers in a recognizable unit of mass, which helps in comparing with macroscopic quantities in chemistry and physics.
Grasping the conversion factor aids in understanding how microscopic measurements align with macroscopic observations.
Number of Molecules
Once you have the mass of a single molecule in a familiar unit like kilograms, you can calculate the number of molecules in a large sample. The number of molecules in a substance is often essential for comprehending phenomena in nature or industrial processes.
Calculating the number of molecules involves dividing the total mass of the substance by the mass of a single molecule.
  • In the exercise, the world's oceans have an estimated mass of \(1.4 \times 10^{21} ext{ kg}\).
  • By dividing this by \(2.98897 \times 10^{-26} ext{ kg/molecule}\), we calculate there are about \(4.68 \times 10^{46} \) water molecules.
This calculation highlights how even a tiny mass at the molecular level integrates into substantial quantities at the macro level, a fundamental principle in chemistry and engineering.
Water Molecule Calculation
Understanding the calculation of a water molecule's mass benefits from breaking down its atomic composition. A water molecule comprises two hydrogen atoms and one oxygen atom, each with given atomic masses.
  • For two hydrogen atoms, the combined mass is \(2 \times 1 ext{ u} = 2 ext{ u}\).
  • An oxygen atom contributes an additional 16 u.
  • Thus, the total atomic mass of one water molecule is \(2 ext{ u} + 16 ext{ u} = 18 ext{ u}\).
By calculating the mass of hydrogen and oxygen atoms separately and then adding them, we make clearer how chemists calculate the molecular mass from atomic masses. Understanding this breakdown is vital, as it plays a significant role in stoichiometry, chemical reactions, and the practical applications of chemistry.

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Most popular questions from this chapter

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