Question

Acetylene The gas acetylene, often used for welding, burns according to the following equation. $$2 C_{2} H_{2}(\mathrm{g})+5 O_{2}(\mathrm{g}) \rightarrow 2 H_{2} O(\mathrm{g})+4 CO_{2}(\mathrm{g})$$ If you have a \(10.0-L\) tank of acetylene at \(25.0^{\circ} C\) and 1.00 atm pressure, how many moles of \(CO_{2}\) will be produced if you burn all the acetylene in the tank?

Short answer

If you burn all the acetylene in a $10.0-L$ tank at $25.0^{\circ} C$ and 1.00 atm pressure, you will produce 0.816 moles of \(CO_2\).

Step-by-step solution

Use the Ideal Gas Law to find the moles of acetylene in the tank

The Ideal Gas Law is given by the equation \(PV = nRT\), where: - \(P\) is the pressure in atm, - \(V\) is the volume in liters, - \(n\) is the number of moles, - \(R\) is the ideal gas constant (0.0821 L·atm/mol·K), and - \(T\) is the temperature in Kelvin. First, we need to convert the given temperature from Celsius to Kelvin: \(T = 25.0^{\circ}C + 273.15 = 298.15 K\) Now, we can rearrange the Ideal Gas Law to find the number of moles of acetylene (\(n_{C_2H_2}\)): \(n_{C2H2} = \frac{PV}{RT}\) Plugging in the given values, we have: \(n_{C2H2} = \frac{(1.00 \text{ atm})(10.0 \text{ L})}{(0.0821 \frac{\text{L·atm}}{\text{mol·K}})(298.15 \text{ K})} = 0.408 \text{ moles}\) So, there are 0.408 moles of acetylene in the tank.

Use stoichiometry to find the moles of \(CO_2\) produced

To find the moles of \(CO_2\) produced, we will use the stoichiometry from the balanced chemical equation: \(2 C_{2}H_{2}(g) + 5 O_{2}(g) \rightarrow 2 H_{2}O(g) + 4 CO_{2}(g)\) Looking at the coefficients, it's apparent that for every 2 moles of \(C_{2}H_{2}\) burned, 4 moles of \(CO_2\) are produced. We can set up a conversion factor using these coefficients: \(\frac{4 \text{ moles } CO_{2}}{2 \text{ moles } C_{2}H_{2}}\) Now, multiply the moles of acetylene by this conversion factor to find the moles of \(CO_2\) produced: \(n_{CO_2} = n_{C2H2} \times \frac{4 \text{ moles } CO_{2}}{2 \text{ moles } C_{2}H_{2}} = 0.408 \text{ moles } \times \frac{4 \text{ moles } CO_{2}}{2 \text{ moles } C_{2}H_{2}} = 0.816 \text{ moles}\) So, 0.816 moles of \(CO_2\) will be produced if all the acetylene in the tank is burned.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry that allows us to relate the pressure, volume, temperature, and number of moles of a gas. The law is commonly expressed as the formula:
\(PV = nRT\), where \(P\) represents pressure, \(V\) is volume, \(n\) signifies the number of moles of gas, \(R\) is the ideal gas constant, and \(T\) is temperature in Kelvin.

To use this law, if any three of the four variables (\(P\), \(V\), \(n\), \(T\)) are known, you can calculate the fourth. In the context of the textbook problem, this law is used to determine the number of moles of acetylene gas in a welding tank when given the volume, temperature, and pressure.

This calculation is crucial because it sets the stage for the next part of the problem, which is figuring out how much carbon dioxide, \(CO_2\), will be produced when the acetylene is burned. Understanding how to manipulate the Ideal Gas Law equation is essential for solving a wide range of problems in chemistry.

Chemical Reactions

Chemical reactions involve the transformation of one set of chemical substances (reactants) into another (products). These changes occur with strict adherence to the law of conservation of mass, which states that matter cannot be created or destroyed in an isolated system. The reaction of acetylene \(C_2H_2\) with oxygen \(O_2\) to form water \(H_2O\) and carbon dioxide \(CO_2\) follows this rule.To fully understand chemical reactions, one must be able to read and interpret a chemical equation, which uses symbols and formulas to represent the reactants and products. For the acetylene welding equation: \(2 C_2H_2(g) + 5 O_2(g) \rightarrow 2 H_2O(g) + 4 CO_2(g)\)Each reactant and product has a coefficient, which represents the number of moles of each substance that participate in the reaction. These coefficients are necessary for calculations involving stoichiometry, and they ensure the law of conservation of mass is obeyed. In our problem, these coefficients indicate that for every two moles of acetylene burned, four moles of carbon dioxide are produced, which is a key piece of information for finding the moles of \(CO_2\) produced from a given amount of acetylene.

Mole Concept

The mole concept is a bridge between the microscopic world of atoms and molecules and the macroscopic world we experience. One mole is defined as exactly 6.02214076 x \(10^{23}\) entities (Avogadro's number), whether they are atoms, molecules, ions, or other particles.

In chemistry, the mole serves as a unit to express amounts of a substance. It allows chemists to count particles by weighing them. For instance, in the textbook exercise, we calculate the moles of acetylene gas to eventually determine the moles of carbon dioxide produced. By using stoichiometry, which relies on the mole concept, we can relate quantities of reactants and products in a balanced chemical equation.

Understanding how many particles are involved in a chemical reaction is crucial because it helps us predict the amounts of substances needed or produced. This is why stoichiometry is often referred to as the 'mathematics of chemistry.' In this scenario, mastering the mole concept allows for precise predictions about the amount of \(CO_2\) generated from burning acetylene, which is of practical importance for applications such as welding, where acetylene is commonly utilized.