Question

Consider the following reaction: $$ \mathrm{C}(s)+\mathrm{O}_{2}(g) \rightarrow \mathrm{CO}_{2}(g) $$ What volume of oxygen gas at 25 ' \(\mathrm{C}\) and 1.02 atm would be required to react completely with \(1.25 \mathrm{~g}\) of carbon?

Short answer

The volume of oxygen gas required to react completely with 1.25 g of carbon at 25 °C and 1.02 atm is approximately 2.54 L.

Step-by-step solution

Balanced Chemical Equation

The balanced chemical equation for the given reaction is: \[ \mathrm{C}(s) + \mathrm{O}_{2}(g) \rightarrow \mathrm{CO}_{2}(g) \] #Step 2: Convert mass of carbon to moles#

Moles of Carbon

First, we need to convert the mass of carbon to moles of carbon. The molar mass of carbon is 12.01 g/mol. Moles of carbon = (1.25 g) / (12.01 g/mol) = 0.1041 mol #Step 3: Find moles of oxygen gas using stoichiometry#

Moles of Oxygen Gas

According to the balanced chemical equation, one mole of carbon reacts with one mole of oxygen gas. Therefore, the moles of oxygen gas required to react completely with 0.1041 mol of carbon are also 0.1041 mol. #Step 4: Convert moles of oxygen gas to volume using the ideal gas law#

Ideal Gas Law

The ideal gas law is PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature. We will use this equation to find the volume of oxygen gas needed. First, convert the given temperature to Kelvin: T = 25 °C + 273.15 = 298.15 K Next, use the ideal gas law with the given pressure, moles of oxygen, and temperature: V = (nRT) / P We will use the ideal gas constant R = 0.0821 L atm/mol K: V = (0.1041 mol · 0.0821 L atm/mol K · 298.15 K) / 1.02 atm Solve for the volume V: V ≈ 2.54 L #Step 5: Conclusion#

Volume of Oxygen Gas

The volume of oxygen gas required to react completely with 1.25 g of carbon at 25 °C and 1.02 atm is approximately 2.54 L.

Key concepts

Ideal Gas Law

The ideal gas law is a fundamental equation in chemistry, represented by the formula \( PV = nRT \). It helps us calculate variables such as pressure, volume, number of moles, and temperature for a given gas. Here's what each component stands for:

• P is the pressure of the gas.
• V is the volume occupied by the gas.
• n is the number of moles of the gas.
• R is the ideal gas constant, typically valued at 0.0821 L atm/mol K.
• T is the temperature of the gas in Kelvin.

To use this law, always make sure that the units are compatible with each other, ensuring pressure is in atmospheres, volume in liters, temperature in Kelvin, and the gas constant appropriately chosen. This equation is invaluable when determining the volume of a gas produced or consumed in a chemical reaction.

Balanced Chemical Equation

Balancing a chemical equation is like maintaining a budget: the number of atoms for each element should be equal on both sides of the reaction. This ensures the conservation of mass principle, which states that matter cannot be created or destroyed.

Consider the reaction: \( \mathrm{C}(s) + \mathrm{O}_2(g) \rightarrow \mathrm{CO}_2(g) \). Here, one carbon atom reacts with one molecule of oxygen to produce one molecule of carbon dioxide. This simple, yet elegant, representation ensures that atoms are accounted for across the reaction process.

Balanced equations are crucial for stoichiometric calculations, allowing chemists to determine the proportionate amounts of reactants and products. Ensuring equations are balanced is foundational before you can delve into converting masses, volumes, or energy changes.

Moles Calculation

Calculating moles is a key aspect of stoichiometry, which often starts with converting a given mass of a substance to moles using its molar mass. Molar mass has the units of grams per mole (g/mol) and represents the mass of one mole of a substance.

For example, the molar mass of carbon is 12.01 g/mol. If you have a sample of 1.25 g of carbon, the moles of carbon can be calculated as:\[ \text{Moles of carbon} = \frac{1.25 \text{ g}}{12.01 \text{ g/mol}} \approx 0.1041 \text{ mol} \]This conversion is essential when applying the stoichiometric coefficients from balanced equations, enabling the determination of amounts for other substances involved in the reaction.

By understanding the relationships between moles, mass, and volume, you can effectively predict and analyze chemical behavior in reactions.

Pressure and Temperature Conversions

Conversions of pressure and temperature are often necessary for using the ideal gas law. To ensure calculations are accurate, each quantity must be in the proper units.

- **Temperature Conversion:** Typically, you need to convert Celsius to Kelvin because the Kelvin scale avoids negative temperatures and uses absolute zero as its null point. The conversion is straightforward: add 273.15 to the Celsius temperature. For instance, 25 °C becomes 298.15 K.
- **Pressure Conversion:** Pressure might be given in units other than atm, such as mmHg or Pascals. Commonly, 1 atmosphere is equivalent to 760 mmHg. Ensure you use the same unit for pressure as represented by the gas constant in the ideal gas law, which is typically atm. Converting these units correctly allows for precise calculations and helps avoid potential errors in stoichiometric and gas law applications.