Question

Apply Calculate the pressure of \(4.67 \times 10^{22}\) molecules of \(CO\) gas mixed with \(2.87 \times 10^{24}\) molecules of \(N_{2}\) gas in a 6.00 -L container at \(34.8^{\circ} C .\)

Short answer

The pressure of the gas mixture in the container, containing \(4.67 \times 10^{22}\) molecules of \(CO\) gas and \(2.87 \times 10^{24}\) molecules of \(N_{2}\) gas in a 6.00 L container at \(34.8^{\circ} C\), is approximately 20.75 atm.

Step-by-step solution

1. Convert the temperature to Kelvin

To calculate the pressure, we need the temperature in Kelvin, not degrees Celsius. To convert the given temperature from Celsius to Kelvin, we add 273.15. \( T_{K} = 34.8 + 273.15 = 307.95 K \)

2. Calculate the total number of moles

The Ideal Gas Law uses moles (n) instead of molecules. We are given the number of molecules of CO and N2 and can use Avogadro's number (\(6.022 \times 10^{23} \) molecules/mol) to convert to moles. Number of moles of CO: \( n_{CO} = \frac{4.67 \times 10^{22} \mathrm{molecules}}{6.022 \times 10^{23} \mathrm{molecules/mol}} = 0.07756 \mathrm{mol} \) Number of moles of N2: \( n_{N2} = \frac{2.87 \times 10^{24} \mathrm{molecules}}{6.022 \times 10^{23} \mathrm{molecules/mol}} = 4.767 \mathrm{mol} \) Total number of moles: \( n_{total} = n_{CO} + n_{N2} = 0.07756 \mathrm{mol} + 4.767 \mathrm{mol} = 4.8446 \mathrm{mol} \)

3. Apply the Ideal Gas Law to find the pressure

Now that we have the total number of moles, volume, and temperature in Kelvin, we can apply the Ideal Gas Law to calculate the pressure: \( P = \frac{nRT}{V} \) \( P = \frac{4.8446 \mathrm{mol} \times 0.0821 \frac{\mathrm{L \cdot atm}}{\mathrm{K \cdot mol}} \times 307.95 \mathrm{K}}{6.00 \mathrm{L}} \) \( P = 20.7451 \mathrm{atm} \) The pressure of the gas mixture in the container is approximately 20.75 atm.

Key concepts

Moles Conversion

When dealing with gases in chemistry, it's essential to convert molecules to moles. **Moles** are a standard unit in chemistry that help us understand and predict how substances behave. Moles convert large numbers of molecules into manageable figures using **Avogadro’s Number**.

• Avogadro’s Number: This is a constant equal to \(6.022 \times 10^{23}\) molecules/mol.
• Conversion: To convert molecules to moles, divide the number of molecules by Avogadro's Number.

**Example:** Given a certain number of molecules of a gas, like \(4.67 \times 10^{22}\) molecules of CO, we convert these by dividing by Avogadro’s Number:
\[ n_{CO} = \frac{4.67 \times 10^{22}}{6.022 \times 10^{23}} = 0.07756 \, \text{moles} \]
This process helps us use the **Ideal Gas Law** effectively by providing the moles needed for calculations.

Temperature Conversion

In order to perform gas law calculations, such as using the Ideal Gas Law, temperature needs to be in Kelvin. Kelvin is the standard unit for temperature in scientific calculations because it's an absolute scale based on absolute zero.

• Conversion to Kelvin: Add 273.15 to the Celsius temperature.

**Example:** In our exercise, the temperature is given as 34.8°C. To convert to Kelvin:
\[ T_K = 34.8 + 273.15 = 307.95 \, \text{K} \]
This temperature in Kelvin can now be used in the Ideal Gas Law formula to find pressure accurately.

Pressure Calculation

Pressure is a crucial aspect in understanding gases and is often calculated using the **Ideal Gas Law**. This law links the pressure, volume, number of moles, and temperature of a gas.

• Ideal Gas Law formula: \( P = \frac{nRT}{V} \)
• Constants: \( R \), the ideal gas constant, is \(0.0821 \, \text{L}\cdot\text{atm}/\text{K}\cdot\text{mol} \)

To calculate pressure, input the total moles, temperature in Kelvin, and the volume into the formula.
**Example:** Utilizing the values from our exercise:
\[ P = \frac{4.8446 \, \text{mol} \times 0.0821 \, \text{L} \cdot \text{atm}/\text{K} \cdot \text{mol} \times 307.95 \, \text{K}}{6.00 \, \text{L}} \]
\[ P \approx 20.75 \, \text{atm} \]
Thus, the pressure inside the container is approximately 20.75 atmospheres.

Gas Mixture Analysis

Understanding gas mixtures involves knowing how the different components interact. In a mixture, each gas behaves independently and contributes to the total pressure. This is explained by **Dalton’s Law of Partial Pressures**.

• Partial Pressure: The pressure exerted by one gas in a mixture is called its partial pressure.
• Total Pressure: It's the sum of the partial pressures of each component in the gas mixture.

In the given exercise, you calculate the total pressure of a mixture of CO and \(N_{2}\). For accuracy, we first convert the molecules of each to moles.
The **Ideal Gas Law** is then used to consider the total moles in the container, providing the total pressure contributed by both gases. Mixing emphasizes the independent behavior of gases, and the laws allow for straightforward analysis of pressure values.