Question

\(1 \mathrm{cc} \mathrm{N}_{2} \mathrm{O}\) at NTP contains (a) \(\frac{1.32}{224} \times 10^{23}\) electrons (b) \(\frac{6.02}{22400} \times 10^{23}\) molecules (c) \(\frac{1.8}{224} \times 10^{22}\) atoms (d) all of these

Short answer

All options (a, b, c) are correct; hence, the answer is (d) all of these.

Step-by-step solution

Understand the Problem

We need to find out how many electrons, molecules, and atoms are present in 1 cc (cubic centimeter) of the gas \( \mathrm{N_2O} \) at normal temperature and pressure (NTP). The correct answer will have calculations matching one of the given options.

Apply Molar Volume Concept at NTP

At NTP, 1 mole of a gas occupies a volume of 22400 cc. This allows us to determine how many moles are present in 1 cc of \( \mathrm{N_2O} \).

Calculate Moles of \( \mathrm{N_2O} \)

Since 1 mole of gas occupies 22400 cc, the moles of \( \mathrm{N_2O} \) in 1 cc is given by \( \frac{1}{22400} \) mole.

Calculate Number of Molecules of \( \mathrm{N_2O} \)

Since 1 mole contains \( 6.022 \times 10^{23} \) molecules, 1 cc has \( \frac{6.022 \times 10^{23}}{22400} \) molecules, which simplifies closely to \( \frac{6.02}{22400} \times 10^{23} \).

Calculate Number of Atoms in \( \mathrm{N_2O} \)

Each molecule of \( \mathrm{N_2O} \) contains 3 atoms (2 of \( N \) and 1 of \( O \)). Thus, the number of atoms is \( 3 \times \frac{6.022 \times 10^{23}}{22400} \), which simplifies to \( \frac{18.066}{22400} \times 10^{23} = \frac{1.8}{224} \times 10^{22} \).

Calculate Number of Electrons in \( \mathrm{N_2O} \)

Nitrous oxide (\( \mathrm{N_2O} \)) has the formula weight that includes 7 electrons per nitrogen and 8 per oxygen, totaling 7 + 7 + 8 = 22 electrons per molecule. Thus, the number of electrons is \( 22 \times \frac{6.022 \times 10^{23}}{22400} \), simplifying to \( \frac{1.32}{224} \times 10^{23} \).

Compare Results to Given Options

After performing the calculations, each calculation matches one of the given options exactly, confirming that all the options (a, b, and c) are correct.

Key concepts

Mole Concept

The mole concept is an essential part of chemistry that helps bridge the gap between the macroscopic and atomic worlds. A mole is a unit used to measure the amount of a substance. This concept allows chemists and students to work with the enormous quantities of atoms, ions, or molecules found in a substance in a manageable way.
To put it simply, 1 mole of any substance contains the same number of entities (like atoms or molecules) as there are atoms in exactly 12 grams of carbon-12. This number is known as Avogadro's Number, approximately equal to \(6.022 \times 10^{23}\).
When dealing with gases, we often use the molar volume concept, which states that 1 mole of an ideal gas occupies 22.400 liters (or 22,400 cubic centimeters) at standard temperature and pressure (STP). For the exercise given, knowing that 1 cc of gas corresponds to \(\frac{1}{22400}\) of a mole is crucial for determining the number of particles like molecules, atoms, or electrons present. Using the mole concept, we can confidently calculate how many molecules are in a specific volume of gas, helping us understand how composition on a microscopic level relates to measurable quantities.

Gas Laws

Gas laws provide critical relationships between pressure, volume, temperature, and quantity of a gas. These laws describe how changes in one property can affect another. The most commonly referenced are Boyle's Law, Charles's Law, and Avogadro's Law, among others.
For the problem at hand, we particularly focus on the use of the concept of molar volume which is related to these gas laws. At Normal Temperature and Pressure (NTP), gases behave ideally, and we assume that interactions between particles are negligible. Under these conditions, all ideal gases occupy a molar volume of 22,400 cc per mole.
This realization allows us to perform calculations using the relationship between volume and amount of gas. If you have 1 cc of gas, knowing its relation to a mole using the molar volume is critical because it helps calculate how many moles of \(\mathrm{N_2O}\) are there, which is \(\frac{1}{22400}\) moles.
By understanding gas laws and their implications, students can predict how a gas will react when subjected to different conditions, enhancing comprehension of gas behavior in both theoretical and practical contexts.

Avogadro's Number

Avogadro's Number, named after the Italian scientist Amedeo Avogadro, is a fundamental constant in chemistry that describes the quantity of particles (atoms, ions, or molecules) in one mole of any substance. It is defined as \(6.022 \times 10^{23}\) entities per mole.
This number is incredibly significant because it provides a bridge between the any quantity of gas measureable in the macroscopic world and the tiny particles composing it. When solving exercises involving quantities on a molecular or atomic scale, Avogadro's Number allows us to convert between moles and the absolute number of entities in a sample.
In the context of the given exercise, knowing Avogadro's Number enables the calculation of the number of molecules in 1 cc of \(\mathrm{N_2O}\). Each calculation, involving number of molecules, atoms, and electrons, leverages Avogadro's Number to transition from moles to the actual count of particles.
Mastery of Avogadro's Number empowers students not only to solve quantitative problems in chemistry but also to understand the fundamental basis of chemical equations and reactions.