How many molecules of \(\mathrm{CO}_{2}\) gas at \(\mathrm{STP}\) are present in \(10.5 \mathrm{~L}\) ?

Avogadro's number is a fundamental constant in chemistry. It represents the number of particles, such as atoms or molecules, in one mole of a substance. The value of Avogadro's number is \(\text{6.022} \times \text{10}^{23}\). This number allows chemists to count particles by weighing them and converting these measurements to moles. It’s crucial for tasks like converting moles of gas to the actual number of molecules present. Avogadro's number helps us bridge the gap between the microscopic world of atoms and molecules and the macroscopic world we can measure and observe.
Imagine having \(\text{6.022} \times \text{10}^{23}\) marbles; that's how many molecules are in just one mole of a substance!

Moles are a basic unit in chemistry representing a specific quantity of a substance. One mole contains exactly \(\text{6.022} \times \text{10}^{23}\) particles, as defined by Avogadro's number. When we talk about gases, moles help us calculate how much gas is present in a given volume, especially under standard conditions of temperature and pressure (STP).
For example, in the given exercise, to find the number of moles of \(\text{CO}_{2}\) in 10.5 liters at STP, we use the molar volume at STP, which is 22.4 liters per mole. The number of moles is calculated by dividing the volume by the molar volume:
\[ \text{Number of moles} = \frac{10.5 \text{ L}}{22.4 \text{ L/mol}} \]
This calculation simplifies our understanding and measurement of gas quantities.

The Ideal Gas Law is a pivotal equation in chemistry, represented as \( PV = nRT \).
This equation connects several properties of a gas together:

• \(P\): pressure
• \(V\): volume
• \(n\): number of moles
• \(R\): the gas constant (0.0821 atm·L/(mol·K))
• \(T\): temperature in Kelvins

In our exercise, the Ideal Gas Law isn't directly used but forms the basis for understanding gas behaviors. It assumes that gases behave ideally, though real gases have slight deviations. Under standard temperature and pressure conditions, the equation simplifies volume calculations as moles can be directly related to volume using constant values, making it easier to predict gas behaviors in different conditions.

STP stands for Standard Temperature and Pressure. These conditions are defined as a temperature of 0°C (273.15 K) and a pressure of 1 atm. Under these conditions, 1 mole of any ideal gas occupies 22.4 liters. Understanding STP is crucial because it provides a standard reference for gas calculations, allowing easy comparison and conversion of different gas samples.
In the exercise, we use STP to determine the volume of \(\text{CO}_{2}\) gas in liters per mole. Knowing that 1 mole of gas occupies 22.4 liters at STP simplifies the calculation to find how many moles are in 10.5 liters of \(\text{CO}_{2}\). This known constant makes molecular calculations more manageable.

Molecular calculations involve converting between different units and quantities, such as volume, moles, and molecules. Understanding these conversions is essential in chemistry:

• First, calculate the number of moles from the volume using the molar volume at STP.
• Then, multiply the number of moles by Avogadro's number to get the number of molecules.

For example, in the exercise, we start with 10.5 liters of \(\text{CO}_{2}\) gas. We find the moles using:
\[ \text{Number of moles} = \frac{10.5 \text{ L}}{22.4 \text{ L/mol}} \]
After getting the number of moles (approximately 0.4696), we multiply it by Avogadro's number:
\[ \text{Number of molecules} = 0.4696 \text{ moles} \times 6.022 \times 10^{23} \text{ molecules/mole} = 2.828 \times 10^{23} \text{ molecules}\]
This step-by-step conversion from volume to molecules highlights the importance of knowing and applying fundamental concepts in molecular calculations.