Question

Methanol, \(\mathrm{CH}_{3} \mathrm{OH},\) can be produced in industrial plants by reacting carbon dioxide with hydrogen in the presence of a catalyst. Water is the other product. How many volumes of hydrogen are required for each volume of carbon dioxide when each gas is at the same temperature and pressure?

Short answer

3 volumes of hydrogen are needed for each volume of carbon dioxide.

Step-by-step solution

Write the balanced chemical equation

First, we need to write the balanced chemical equation for the reaction between carbon dioxide and hydrogen to form methanol and water. The equation is as follows: \[ \text{CO}_2 + 3\text{H}_2 \rightarrow \text{CH}_3\text{OH} + \text{H}_2\text{O} \] This equation is balanced because we have 1 carbon, 4 hydrogens, and 2 oxygens on both sides of the equation.

Identify the volume ratio from the equation

According to Avogadro's law, equal volumes of gases, at the same temperature and pressure, contain the same number of molecules. From the balanced equation, we see that 3 volumes of hydrogen are required for every 1 volume of carbon dioxide, since the coefficients before \(\text{CO}_2\) and \(\text{H}_2\) are 1 and 3, respectively.

Present the solution

Thus, according to the balanced chemical equation, 3 volumes of hydrogen react with 1 volume of carbon dioxide under the same temperature and pressure conditions.

Key concepts

Stoichiometry

Stoichiometry is the quantitative study of reactants and products in a chemical reaction. It allows chemists to predict the amount of each substance consumed and produced. In this exercise, stoichiometry helps us understand the relationship between carbon dioxide and hydrogen gases in the production of methanol.

### How Stoichiometry WorksThe coefficients in a balanced chemical equation represent the mole ratios of the reactants and products. These ratios are crucial for accurately predicting the amounts involved in the reaction.

• For example, the balanced equation given is: \[ \text{CO}_2 + 3\text{H}_2 \rightarrow \text{CH}_3\text{OH} + \text{H}_2\text{O} \].
• This tells us that 1 mole of \( \text{CO}_2 \) reacts with 3 moles of \( \text{H}_2 \) to produce methanol and water.

By using stoichiometry, we can calculate how much hydrogen is needed for a given amount of carbon dioxide. It serves as a blueprint for understanding the scale and proportions of a chemical reaction.

Avogadro's Law

Avogadro's Law is a key principle in chemistry that relates the volume of a gas to the number of molecules it contains. It states that equal volumes of gases, at the same temperature and pressure, have the same number of molecules. ### Application in the Given Reaction In the context of the methanol production problem, this law becomes quite useful.

• As per Avogadro's Law, if we maintain the same conditions of temperature and pressure for the gases involved, we can directly use the coefficients in the balanced equation to determine volume ratios.
• Since 3 volumes of hydrogen are needed for 1 volume of carbon dioxide, we apply Avogadro's Law to deduce the necessary volumes of each gas without needing further calculations.

This principle simplifies the analysis of gaseous reactions, allowing quick and efficient calculation of reactant and product quantities without converting volumes to moles or vice versa.

Balanced Chemical Equation

A balanced chemical equation is essential for accurately describing a chemical reaction. It provides the correct proportions of reactants and products. In the exercise, the balanced equation is crucial for determining the exact amount of hydrogen needed.### Balancing the EquationBalancing involves ensuring the number of atoms for each element is the same on both sides of the chemical equation.

• This ensures the conservation of mass, as matter cannot be created or destroyed.
• In the exercise, the balanced equation \( \text{CO}_2 + 3\text{H}_2 \rightarrow \text{CH}_3\text{OH} + \text{H}_2\text{O} \) shows that each carbon, hydrogen, and oxygen atom is accounted for on both sides.

This balanced equation provides the basis for determining that 3 volumes of hydrogen are required for every 1 volume of carbon dioxide, as reflected by the reactant coefficients. This establishes a clear and precise relation between the reacting volumes under consistent conditions of temperature and pressure.