Question

Welders commonly use an apparatus that contains a tank of acetylene \(\left(\mathrm{C}_{2} \mathrm{H}_{2}\right)\) gas and a tank of oxygen gas. When burned in pure oxygen, acetylene generates a large amount of heat. $$ 2 \mathrm{C}_{2} \mathrm{H}_{2}(g)+5 \mathrm{O}_{2}(g) \rightarrow 2 \mathrm{H}_{2} \mathrm{O}(g)+4 \mathrm{CO}_{2}(g) $$ What volume of carbon dioxide gas at STP is produced if \(1.00 \mathrm{~g}\) of acetylene is combusted completely?

Short answer

When 1.00 g of acetylene is combusted completely, 1.72 L of carbon dioxide gas at STP is produced.

Step-by-step solution

Calculate the number of moles of acetylene

To find the number of moles of acetylene, use the formula: number of moles = (mass of compound) / (molar mass of compound) The molar mass of acetylene, C₂H₂, is 2 × molar mass of C + 2 × molar mass of H Molar mass of C = 12.01 g/mol Molar mass of H = 1.01 g/mol So, molar mass of acetylene = 2 × 12.01 + 2 × 1.01 = 26.04 g/mol number of moles of acetylene = (1.00 g) / (26.04 g/mol) = 0.0384 mol

Use the stoichiometry of the balanced chemical equation to find the number of moles of CO₂ produced

From the balanced chemical equation, we have: 2 C₂H₂ + 5 O₂ → 2 H₂O + 4 CO₂ We can see that 2 moles of acetylene react to produce 4 moles of CO₂. So, we can set up a proportion: moles of CO₂ / moles of acetylene = 4 / 2 moles of CO₂ = \( (4/2) × 0.0384 \) = 0.0768 mol

Use the ideal gas law to find the volume of CO₂ gas at STP

At STP (Standard Temperature and Pressure), the temperature (T) is 273.15 K, and the pressure (P) is 1 atm. The ideal gas law states: PV = nRT Where P is the pressure, V is the volume, n is the number of moles, R is the gas constant (8.314 J⁄(mol·K) or 0.0821 L·atm⁄(mol·K)), and T is the temperature. We want to find the volume (V) of CO₂ gas produced. Since we are working at STP, we can use the molar volume of a gas at STP (22.4 L/mol): V = n × molar volume at STP V = 0.0768 mol × 22.4 L/mol = 1.72 L So, the volume of CO₂ gas produced at STP is 1.72 L.

Key concepts

Ideal Gas Law

The ideal gas law is a fundamental principle in chemistry and physics that describes the behavior of gases under ideal conditions.
It combines several important gas laws into one equation:

• Boyle's Law states that the pressure of a gas is inversely proportional to its volume when temperature is held constant.
• Charles's Law shows that the volume of a gas is directly proportional to its temperature when pressure is constant.
• Avogadro's Law indicates that the volume of a gas is directly proportional to the number of moles of gas, assuming pressure and temperature are constant.

The ideal gas law formula is: \[ PV = nRT \] In this equation:

• P represents pressure,
• V is the volume,
• n denotes the number of moles,
• R is the universal gas constant (0.0821 L·atm⁄(mol·K) for most practical purposes),
• and T is the temperature in Kelvin.

At Standard Temperature and Pressure (STP), which is defined as 273.15 Kelvin and 1 atmosphere of pressure, 1 mole of an ideal gas occupies 22.4 liters.
This is a handy approximation that simplifies calculations when you're dealing with gases.
In this exercise, the ideal gas law helps determine the volume of CO₂ produced by combustion.

Molar Mass

Molar mass is a critical concept when you're working with chemical reactions and stoichiometry.
It refers to the mass of one mole of a given substance.
Understanding molar mass is key to converting between moles—a measure of how many molecules or atoms there are—and grams, which is a measure of mass.
To calculate a compound's molar mass, you must sum the atomic masses of all atoms present in the molecule.

• For example, the molar mass of carbon ( C) is 12.01 g/mol,
• while hydrogen ( H) is 1.01 g/mol.

In acetylene ( C₂H₂), you calculate the total molar mass by adding together the weighted contributions of each element: 2 times the molar mass of C (carbon) plus 2 times the molar mass of H (hydrogen).
This gives you a molar mass of 26.04 g/mol for acetylene, essential for converting its mass in grams into moles for further calculations.
This conversion allows you to utilize stoichiometry to determine the amounts of reactants and products.

Chemical Reactions

Chemical reactions are processes where substances (reactants) change into new substances (products).
In this exercise, acetylene (C₂H₂) combusts with oxygen to produce water and carbon dioxide: \[2 \; \mathrm{C}_{2}\mathrm{H}_{2}(g) + 5 \; \mathrm{O}_{2}(g) \rightarrow 2 \; \mathrm{H}_{2}\mathrm{O}(g) + 4 \; \mathrm{CO}_{2}(g)\]This balanced equation tells us a lot about the reaction.

• The numbers in front of each molecule represent the stoichiometric coefficients.
• These coefficients indicate the ratios of moles of each substance involved in the reaction.

These are crucial in stoichiometry, a method used to calculate the relative quantities of reactants and products in chemical reactions.
By understanding this stoichiometric relationship, we can see that 2 moles of C₂H₂ yield 4 moles of CO₂.
For example, using this ratio, we calculate that when 1 mole of acetylene reacts, it will produce 2 moles of carbon dioxide.
This fundamental relationship allows chemists and students alike to predict how much product can be formed from a given amount of reactant, and is vital for both practical applications and theoretical calculations.