Question

Which of the following gases will have least volume if \(10 \mathrm{~g}\) of each gas is taken at same temperature and presure? (a) \(\mathrm{CO}_{2}\) (b) \(\mathrm{N}_{2}\) (c) \(\mathrm{CH}_{4}\) (d) \(\mathrm{HCl}\)

Short answer

has the least volume for 10 grams at the same temperature and pressure.

Step-by-step solution

Understand the problem and Avogadro’s Law

The problem asks for the gas with the least volume when 10 grams of each gas is taken at the same temperature and pressure. According to Avogadro's law, equal volumes of all gases under the same conditions of temperature and pressure contain the same number of molecules. However, we are dealing with different masses, so the molar mass of each gas needs to be considered to find the moles of each gas.

Calculate the moles of each gas

Use the molar mass of each gas to calculate the moles for 10 grams. The number of moles () is calculated by dividing the mass () by the molar mass (). Formula: = / .

Identify the gas with the least moles

Since the volume is directly proportional to the moles at constant temperature and pressure (by Avogadro's law), the gas with the least moles will have the least volume.

Conclusion

Determine which gas has the least moles for 10 grams, and that will be the one with the least volume under the same conditions of temperature and pressure.

Key concepts

Molar Mass

Molar mass is a fundamental concept in chemistry that refers to the mass of one mole of a substance, typically measured in grams per mole (\text{g/mol}\text{).} It is an intrinsic property of each unique chemical substance, such as an element or compound. In essence, molar mass tells us how heavy a mole of molecules or atoms would be, which is critical for converting between grams of a substance and the number of moles.
To find the molar mass of a compound, one must sum the atomic masses of the individual elements that comprise the compound, taking into account their respective quantity in one molecule. For instance, the molar mass of carbon dioxide (CO\(_2\)) is calculated by adding the mass of one carbon atom (approximately 12.01 g/mol) to the mass of two oxygen atoms (approximately 16.00 g/mol each), totaling about 44.01 g/mol.

Why is Molar Mass Important?

Understanding molar mass is crucial when dealing with chemical reactions and stoichiometry. It allows us to determine how many moles of a substance we have in a given mass and vice versa, which in turn enables us to predict the volumes of gases produced or consumed in a reaction, provided the conditions of temperature and pressure are constant (according to Avogadro's law). Using the molar mass of each gas in the exercise, we can calculate the number of moles of the gas when given a specific mass like 10 grams.

Gas Volumes

Gas volumes in chemistry are directly related to the amount of gas present and the conditions under which the gas is measured. According to Avogadro's law, equal volumes of gases, at the same temperature and pressure, contain the same number of molecules. This implies that the volume of a gas is proportional to the number of moles present, provided the temperature and pressure remain constant.
When comparing different gases under identical conditions, a gas with a higher number of moles will occupy a greater volume. This is used in calculations involving gas-producing reactions or when collecting gases over water in laboratory experiments.

Applying Avogadro's Law

In the context of the given problem, understanding gas volumes helps to identify which gas will occupy the least volume when 10 grams of each is taken at the same temperature and pressure. By calculating the moles of each gas from the mass and molar mass, we can infer their volumes using Avogadro's law. This is fundamental to solving problems where you need to predict how the volume of a gas will change with variations in the amount (moles) of gas.

Stoichiometry

Stoichiometry is the quantitative aspect of chemical reactions and deals with the relationships between the amounts of reactants and products. This area of chemistry is based on the conservation of mass and the concept of the mole, making use of molar mass to convert between mass and number of moles.
The principles of stoichiometry are applied when calculating the expected yield of a chemical reaction, understanding reactant limiting and excess quantities, or determining the proportions of gases involved in reactions following the ideal gas law and Avogadro's principle.

Significance in Gas Volume Calculations

In problems like the exercise provided, stoichiometry comes into play to help determine the gas that will have the least volume, taking into account 10 grams of each gas. By applying stoichiometric calculations, we can convert the mass of each gas to moles using their respective molar masses, thus enabling us to compare their volumes through the lens of Avogadro's law. This is essential in predicting and understanding the outcomes of reactions involving gases.

Molecular Weight

Molecular weight is another term that is often used interchangeably with molar mass, though it traditionally refers to the mass of a single molecule of a compound measured in atomic mass units (amu). One atomic mass unit is defined as one-twelfth the mass of a carbon-12 atom. The molecular weight of a substance is obtained by summing the atomic weights of all the atoms present in a molecule of the substance.
Molecular weight becomes particularly useful when dealing with individual molecules in theoretical calculations. It offers a way to conceptualize the 'heaviness' of a molecule relative to others and plays a role in determining gas behavior under various conditions.

Molecular Weight in Relation to Gases

In regards to gases, the molecular weight can provide insight into the comparative densities of different gases. A gas with a higher molecular weight generally has a higher density, assuming ideal behavior. For the given exercise, even though we are more concerned with molar mass when finding moles and volumes, understanding the concept of molecular weight reinforces our comprehension of why a larger molar mass leads to fewer moles for the same mass of gas, affecting the volume it occupies under standard conditions of temperature and pressure.