Question

A sealed tank at room temperature, \(25^{\circ} \mathrm{C}\), has \(22.0 \mathrm{~g}\) of \(\mathrm{CO}_{2} .\) The gas has a pressure of \(732 \mathrm{~mm} \mathrm{Hg} .\) The tank is moved to a room kept at \(12^{\circ} \mathrm{C}\) and an additional \(10.0 \mathrm{~g}\) of \(\mathrm{CO}_{2}\) are added to the tank. What is the pressure in the tank? Assume no loss of gas when more \(\mathrm{CO}_{2}\) is added.

Short answer

Answer: To find the final pressure inside the sealed tank, follow these steps: 1. Calculate the initial number of moles of CO₂ (n₁) using the mole formula, based on the given mass of CO₂ (22.0g) and the molar mass of CO₂ (44.01 g/mol). 2. Convert the given initial and final temperatures to Kelvin (K). 3. Calculate the initial pressure in Pascal (Pa) by converting the given pressure in mmHg. 4. Use the Ideal Gas Law to find the initial volume (V) of the tank. 5. Calculate the final number of moles of CO₂ (n₂) after adding 10.0g of CO₂. 6. Use the Ideal Gas Law again to find the final pressure (P₂) in Pascal. 7. Convert the final pressure from Pascal back to mmHg. By following these steps, you will find the final pressure inside the sealed tank in mmHg.

Step-by-step solution

Calculate the initial number of moles of CO₂

To find the initial number of moles of CO₂, we should use the mole formula \(n = \frac{m}{M}\), where n is the number of moles, m is the mass of the substance in grams, and M is the molar mass of the substance. For CO₂, the molar mass is 44.01 g/mol. Given that the initial mass of CO₂ is 22.0g, we can find the initial number of moles: n₁ = \(\frac{22.0 \,\text{g}}{44.01 \,\text{g/mol}}\).

Convert the given temperatures to Kelvin

We need to work with temperatures in Kelvin for the Ideal Gas Law. To convert Celsius to Kelvin, add 273.15: Initial temperature, T₁ = 25°C + 273.15 = 298.15 K Final temperature, T₂ = 12°C + 273.15 = 285.15 K

Calculate the initial pressure in Pascal

We need to convert the given pressure in mmHg to Pascal. The conversion factor is 1 mmHg = 133.322 Pa: Initial pressure, P₁ = 732 mmHg * 133.322 Pa/mmHg

Use the Ideal Gas Law to find initial volume

By rearranging the Ideal Gas Law, we get the initial volume V: V = \(\frac{n₁RT₁}{P₁}\)

Calculate the final number of moles of CO₂

After adding 10.0g of CO₂ to the tank, we need to find the final number of moles of CO₂: n₂ = n₁ + \(\frac{10.0 \,\text{g}}{44.01 \,\text{g/mol}}\)

Use the Ideal Gas Law to find the final pressure

Now that we know the final number of moles and the final temperature, we can use the Ideal Gas Law again to find the final pressure: P₂ = \(\frac{n₂RT₂}{V}\) Substitute the values of n₂, R, T₂, and V found earlier to calculate the final pressure P₂ in Pascal.

Convert the final pressure to mmHg

Finally, convert the final pressure from Pascal back to mmHg using the conversion factor: Final pressure, P₂ = \(\frac{P₂ \,\text{Pa}}{133.322 \,\text{Pa/mmHg}}\) This will give the final pressure inside the sealed tank in mmHg.

Key concepts

Understanding Molar Mass

The concept of molar mass is a fundamental cornerstone in chemistry, particularly when dealing with gases and the Ideal Gas Law. Simply put, molar mass is the weight of one mole of a substance. In technical terms, it is defined as the mass (in grams) of one mole of atoms, molecules, or other entities. One mole corresponds to Avogadro's number, which is approximately 6.02 x 10²³ particles of the substance.

For example, carbon dioxide (CO₂) has a molar mass of 44.01 g/mol. This means that one mole of CO₂ molecules weighs 44.01 grams. To calculate the number of moles from a given mass, use the following formula:
\[ n = \frac{m}{M} \]
Where \( n \) represents the number of moles, \( m \) is the mass in grams, and \( M \) is the molar mass of the substance.

Converting Gas Pressure

Working with gas pressures often requires converting units to match those needed for the Ideal Gas Law equation, which are typically in Pascals (Pa) for the metric system. Common units for pressure include atmospheres (atm), millimeters of mercury (mmHg), and torr, each of which can be converted to Pascals.

To convert from mmHg to Pa, for instance, you can use the conversion factor 133.322 Pa/mmHg, as shown in the solution steps. Therefore:
\[ P_{\text{Pa}} = P_{\text{mmHg}} \times 133.322 \frac{\text{Pa}}{\text{mmHg}} \]

This conversion is important because it aligns with the standard unit for pressure in the Internal System of Units (SI), ensuring that all inputs in the Ideal Gas Law are consistent and accurate.

Calculating Moles in Chemistry

The concept of moles is fundamental to understanding chemical reactions, stoichiometry, and gas laws. Calculating the amount of a substance in moles allows chemists to connect the macroscopic world, which we can measure, to the microscopic world of atoms and molecules. When you calculate moles, you're essentially counting the number of particles using their weight.

To calculate the moles of a substance when given the mass, you divide the mass by the molar mass:
\[ n = \frac{m}{M} \]
Continuing with our sample problem involving carbon dioxide, when the mass of CO₂ is increased by adding more gas, it is necessary to calculate the new total moles of CO₂. This involves adding the moles of gas added to the initial moles:
\[ n_2 = n_1 + \frac{\text{additional mass}}{\text{molar mass}} \]

This mole calculation is critical in predicting how the addition of gas will affect the pressure within a container, applying the Ideal Gas Law to find the final pressure after temperature or volume changes.