Question

Which of the following pairs of gases contain equal number of molecules? (a) \(\mathrm{CO}_{2}\) and \(\mathrm{NO}_{2}\) (b) \(\mathrm{CO}\) and \((\mathrm{CN})_{2}\) (c) \(\mathrm{NO}\) and \(\mathrm{CO}\) (d) \(\mathrm{N}_{2} \mathrm{O}\) and \(\mathrm{CO}_{2}\)

Short answer

Answer is (c) \(\mathrm{NO}\) and \(\mathrm{CO}\).

Step-by-step solution

Understanding Mole Concept

The mole concept is used to convert between the number of molecules and moles for a given substance. According to Avogadro's law, 1 mole of any gas at the same temperature and pressure contains the same number of molecules, approximately \(6.022 \times 10^{23}\) molecules. Therefore, to find pairs of gases with equal number of molecules, they must be in the same molar quantity.

Compare Molar Masses

To compare gases, first consider molar masses. If the number of moles is the same and they exist under the same conditions, the number of molecules will also be equal. We don't have quantities specified for the gases, so we assume equal moles for comparison.

Analyze Each Pair

Examine each provided pair under the assumption they have the same molar amount under identical conditions.- For pair (a) \(\mathrm{CO}_{2}\) and \(\mathrm{NO}_{2}\): Different molar masses but do not affect molecular equality under equal moles.- For pair (b) \(\mathrm{CO}\) and \(\mathrm{CN}_{2}\): Different molar masses, but if equal moles are assumed, they have equal number of molecules.- For pair (c) \(\mathrm{NO}\) and \(\mathrm{CO}\): Close in mass and with equal moles, will have equal molecules.- For pair (d) \(\mathrm{N}_{2}\mathrm{O}\) and \(\mathrm{CO}_{2}\): Differing molar masses imply they are distinct in composition if equal moles.

Identify Answer

Since equal number of molecules come from equal moles, the pairs which most likely have equal molecules under same moles are those with similar speculated molar conditions: (c) \(\mathrm{NO}\) and \(\mathrm{CO}\) is the pair with equal molecular count under equal molar conditions.

Key concepts

Avogadro's Law

Avogadro’s Law states that equal volumes of gases, at the same temperature and pressure, contain an equal number of molecules. This concept is integral for understanding how gases behave under different conditions. According to this law, if you have 1 mole of any gas at standard temperature and pressure (STP), it will occupy the same volume as 1 mole of another gas under the same conditions. This constant number of molecules in a mole is Avogadro’s number, approximately \(6.022 \times 10^{23}\).

Avogadro's Law helps in comparing different gases. Even if their molar masses differ, as long as the volume, temperature, and pressure are consistent, the number of molecules will match. This makes it easier to predict and compare the behavior of gases in chemical reactions or other experimental settings.

Molar Mass

The concept of molar mass is crucial when working with chemical reactions and Avogadro’s Law. Molar mass is the mass of one mole of a given substance. It is expressed in grams per mole (g/mol) and is calculated by adding the atomic masses of the constituent elements found on the periodic table. For example, the molar mass of CO\(_2\) (carbon dioxide) is calculated as the sum of the atomic masses: carbon (12.01 g/mol) plus two times oxygen (16.00 g/mol), resulting in 44.01 g/mol.

Understanding molar masses allows for precise calculations in chemical equations. It helps determine how much of one reactant is necessary to completely react with another, or how much product is formed. In comparative aspects, even gases with different molar masses can have equal numbers of molecules if they are in equal moles, as indicated in the original exercise's discussion.

Chemical Molecules

Chemical molecules are groups of atoms bonded together, representing the smallest fundamental unit of a chemical compound that can take part in a chemical reaction. Molecules can consist of similar or different types of atoms. In the context of gases, molecules are what identify their chemical behavior and properties in reactions. For example, \(\mathrm{CO}\) comprises one carbon atom and one oxygen atom, making it distinct in behavior from \(\mathrm{CO}_2\), which includes one carbon and two oxygen atoms.

Pairing molecules by their amount, and thus ensuring equal numbers, ensures accurate comparisons in chemical behavior across different substances. As demonstrated in the exercise, by assuming equal moles, molecules like \(\mathrm{NO}\) and \(\mathrm{CO}\) have equal numbers, guiding predictability of reaction ratios and outputs.

Gas Laws

Gas Laws encompass several fundamental principles that describe how gases behave under various conditions. These include Boyle’s Law, Charles’s Law, and Avogadro’s Law, among others. Collectively known as the Ideal Gas Law, they are usually represented by the equation \(PV = nRT\), where \(P\) is pressure, \(V\) is volume, \(n\) is the number of moles, \(R\) is the gas constant, and \(T\) is temperature.

The Ideal Gas Law allows for the calculation of any one of the properties involved if the other three are known. It is essential for predicting how changes in pressure, temperature, or volume will affect a gas. Understanding these laws is key to analyzing gas behavior, such as identifying which pairs in the exercise contain equal molecules under the same conditions or predicting outcomes in chemical reactions.