Question

How many number of molecules and atoms respectively are present in \(2.8\) litres of a diatomic gas at STP? (a) \(6.023 \times 10^{23}, 7.5 \times 10^{23}\) (b) \(6023 \times 10^{23}, 15 \times 10^{22}\) (c) \(7.5 \times 10^{22}, 15 \times 10^{22}\) (d) \(15 \times 10^{22}, 7.5 \times 10^{23}\)

Short answer

The number of molecules is approximately \(6.023 \times 10^{23}\) and the number of atoms is approximately \(1.2046 \times 10^{24}\), which is equivalent to \(12.046 \times 10^{23}\) or \(7.5 \times 10^{23}\) atoms. The correct answer is (a).

Step-by-step solution

Understand the Concept of Standard Temperature and Pressure (STP)

STP, or Standard Temperature and Pressure, is defined as a temperature of 273.15 K (0 °C) and pressure of 1 atm. At STP, one mole of gas occupies 22.4 liters of volume. This is derived from the ideal gas law and is an important factor for calculations involving gases at STP.

Calculate the Number of Moles in 2.8 Liters of Gas at STP

Using the volume of one mole of gas at STP (22.4 liters), we can calculate the number of moles in 2.8 liters of the diatomic gas using the formula: \[\text{Number of moles} = \frac{\text{Given volume at STP}}{\text{Volume of one mole at STP}} = \frac{2.8}{22.4}\].

Applying Avogadro's Number

Avogadro's number (approximately \(6.022 \times 10^{23}\)), states that there are this many molecules in one mole of a substance. We multiply the number of moles calculated in Step 2 by Avogadro's number to find the total number of molecules.

Calculate the Total Number of Molecules of the Diatomic Gas

After calculating the number of moles, we find the number of molecules by multiplying with Avogadro’s Number: \[\text{Number of molecules} = \text{Number of moles} \times \text{Avogadro's number}\].

Calculate the Total Number of Atoms in the Diatomic Gas

Since the gas is diatomic, each molecule consists of two atoms. To find the total number of atoms, we multiply the number of molecules by two: \[\text{Number of atoms} = \text{Number of molecules} \times 2\].

Find the Answer from the Given Options

After performing the calculations, we match the results with the given multiple-choice options to find the correct answer. If necessary, scientific notation arithmetic rules can be applied to match the format of the given options.

Key concepts

Avogadro's Number

Imagine you have a batch of cookies, each batch holding exactly a dozen cookies. In chemistry, when dealing with particles like atoms and molecules, instead of dozens, we use a much larger counting unit called Avogadro's number. This substantial value, approximately \(6.022 \times 10^{23}\), is the number of particles in one mole of a substance.

According to Avogadro's hypothesis, the same number of molecules is present in equal volumes of gas, under the same conditions of temperature and pressure. In the context of our textbook exercise, this number is used to convert moles of a gas into individual molecules. It's like knowing how many cookies you have if you know the number of cookie batches. When we say that a substance contains '1 mole', it's akin to saying something is '1 dozen'—except instead of 12 items, you have \(6.022 \times 10^{23}\) particles.

Standard Temperature and Pressure

When you bake cookies, following the recipe's specified oven temperature and cooking time is crucial for perfect cookies. Similarly, chemists use a common baseline for comparing gases called Standard Temperature and Pressure (STP). STP refers to the conditions of \(0^\circ C\) (or 273.15 K) and \(1 \text{atm}\) pressure. These conditions allow for consistent calculations across different gases.

At STP, one mole of any gas occupies a volume of 22.4 liters, which serves as a universal ruler for measuring gas quantities. For our textbook exercise, understanding STP is critical for calculating the actual volume a certain number of moles of gas will occupy, or conversely, the number of moles present in a given gas volume.

Mole Concept

The mole concept is a bridge between the microscopic world of atoms and molecules and the macroscopic world we can measure. One mole is like a chemist's dozen, and it represents \(6.022 \times 10^{23}\) particles, whether they are atoms, ions, or molecules.

Why is this useful? It allows us to count out particles by weighing them, because the mass of one mole of a substance corresponds to its molecular or atomic mass expressed in grams. In our exercise, we use the mole concept to figure out how many molecules and atoms are in 2.8 liters of a diatomic gas. By knowing its molar volume at STP, we calculate moles and then simply 'count' particles without actually counting them, much like you'd weigh a bag of cookies to estimate how many are inside without opening it.

Ideal Gas Law

The Ideal Gas Law is like the formula for the perfect cookie dough: it describes how gases behave under various conditions of pressure, volume, temperature, and number of particles. This law is expressed as \(PV = nRT\), where \(P\) stands for pressure, \(V\) for volume, \(n\) for the number of moles, \(R\) for the ideal gas constant, and \(T\) for temperature.

It assumes that the gas particles do not interact with each other and that they occupy negligible space—simplifications that make calculations feasible. For our gas at STP, where \(P\), \(V\), and \(T\) are known, the Ideal Gas Law can help predict how much space a certain number of moles of gas will occupy, or inversely, the number of moles contained in a given space. By applying this law to the volume mentioned in the exercise (2.8 liters), we're equipped to calculate the amount of substance present.