Problem 45 - Fundamentals of Physics

Question

How many megacoulombs of positive charge are in \(1.00 \mathrm{~mol}\) of neutral molecular-hydrogen gas \(\left(\mathrm{H}_{2}\right) ?\)

Step-by-Step Solution

Answer

There are 0.1928 megacoulombs of positive charge in 1 mole of neutral \(\mathrm{H}_2\) gas.

Step 1: Calculate the number of molecules in one mole

To find the number of molecules in one mole, use Avogadro's number, which is \(6.022 \times 10^{23}\) molecules per mole. Since there is 1 mole of \(\mathrm{H}_2\) gas, there are \(6.022 \times 10^{23}\) molecules of \(\mathrm{H}_2\).

Step 2: Determine the number of protons in each molecule

Each \(\mathrm{H}_2\) molecule consists of 2 hydrogen atoms. Each hydrogen atom contains 1 proton. Therefore, a single molecule of \(\mathrm{H}_2\) contains 2 protons.

Step 3: Calculate the total number of protons in the sample

Multiply the number of molecules by the number of protons per molecule: \[6.022 \times 10^{23} \text{ molecules} \times 2 \text{ protons/molecule} = 1.2044 \times 10^{24} \text{ protons}.\]

Step 4: Convert protons to charge in coulombs

The charge of a single proton is approximately \(1.602 \times 10^{-19}\) coulombs. Multiply the total number of protons by the charge of one proton to get the net charge in coulombs:\[1.2044 \times 10^{24} \text{ protons} \times 1.602 \times 10^{-19} \text{ C/proton} = 1.928 \times 10^{5} \text{ C}.\]

Step 5: Convert coulombs to megacoulombs

Divide the total charge in coulombs by \(10^6\) to convert to megacoulombs (since 1 megacoulomb = \(10^6\) coulombs):\[1.928 \times 10^{5} \text{ C} \div 10^6 = 0.1928 \text{ megacoulombs}.\]

Key Concepts

Mole concept

The mole is a fundamental concept in chemistry that answers the basic question: How many molecules or atoms are present in a given sample? This concept allows chemists to count particles directly by weighing.

• 1 mole is equivalent to 6.022 x 10²³ particles. These particles can be atoms, molecules, ions, etc., depending on the context.
• The mole provides a bridge between the atomic scale and human-level measurements, allowing us to work with manageable amounts.
• The concept simplifies chemical equations and reactions, turning them into mole calculations that are more intuitive and practical.

In the given exercise, one mole of \( \mathrm{H}_{2} \) gas is considered. Understanding the mole allows us to easily determine that this represents 6.022 x 10²³ H₂ molecules.

Avogadro's number

Avogadro's number is a constant that is fundamental to the mole concept. It is named after the scientist Amedeo Avogadro and is defined as the number of constituent particles (usually atoms or molecules) in one mole of a given substance.

• Numerically, Avogadro's number is approximately 6.022 x 10²³ particles per mole.
• This number is crucial for converting between the number of moles and the number of atoms or molecules.
• It helps in understanding the scale at which the chemical processes happen, explaining phenomena significantly at a macro and micro level.

When encountering problems in chemistry, Avogadro's number allows us to translate mole quantities into absolute counts of atoms or molecules, as demonstrated in the problem exercise to find the number of H₂ molecules.

Charge of a proton

Understanding the charge of a proton is fundamental when dealing with electricity and chemistry, as it explains the nature of atomic particles.

• A single proton carries a positive charge which is equal in magnitude to the negative charge of an electron, quantified as approximately 1.602 x 10^-19 coulombs.
• This consistency in charge allows for neutral matter under normal conditions, as seen in our example with H₂.
• In the case of the hydrogen molecule, knowing the charge of a proton is essential for calculating the total electric charge resulting from a sample quantity, such as one mole of H₂ gas.

In the exercise, once the number of protons was determined, this charge value was used to find the overall charge in the sample of H₂.