Question

A mixture of \(1 \mathrm{kmol}\) of \(\mathrm{CO}\) and \(\frac{1}{2} \mathrm{kmol}\) of \(\mathrm{O}_{2}\) is held at ambient temperature and pressure. After 100 hours only an insignificant amount of \(\mathrm{CO}_{2}\) has formed. Why?

Short answer

Insufficient activation energy at ambient temperature and pressure prevents the reaction from occurring significantly.

Step-by-step solution

Understanding the Reactants and Products

The chemical reaction in question is the formation of \text{CO}_2 from CO and O_2. The balanced equation for this reaction is: \(2\text{CO} + \text{O}_2 \rightarrow 2\text{CO}_2\). This indicates that two moles of CO react with one mole of O_2 to produce two moles of CO_2.

Initial Moles Present

The mixture contains \(1\) kmol of CO and \(0.5\) kmol of O_2. Based on the balanced equation, all the O_2 would need \(2 \times 0.5 = 1\) kmol of CO to react completely. Hence, the provided quantities are in the correct stoichiometric ratio for complete reaction.

Temperature and Pressure Conditions

The problem states that the mixture is held at ambient temperature and pressure. Ambient temperature generally means 25°C (298 K) and 1 atm.

Activation Energy and Reaction Rate

Even if the reactants are in stoichiometric amounts, for the reaction \(2\text{CO} + \text{O}_2 \rightarrow 2\text{CO}_2\) to occur, sufficient activation energy is required to overcome the energy barrier. If no external source of energy (like heat or a catalyst) is provided to the mixture, the reaction will proceed at an extremely slow rate or might not proceed at all.

Conclusion

The negligible formation of CO_2 after 100 hours suggests that the ambient temperature and pressure were insufficient to provide the necessary activation energy for the reaction to proceed at a significant rate.

Key concepts

Stoichiometry in Chemical Reactions

Stoichiometry refers to the calculation of reactants and products in chemical reactions. It's a fundamental aspect of chemistry that helps us understand the proportions in which chemicals react. In the given exercise, the balanced equation \( 2\text{CO} + \text{O}_2 \rightarrow 2\text{CO}_2 \) explains the stoichiometric relationship. Here, two moles of carbon monoxide (\text{CO}) react with one mole of oxygen (\text{O}_2) to form two moles of carbon dioxide (\text{CO}_2).
In simpler terms:

• 2 moles of CO + 1 mole of O_2 = 2 moles of CO_2.

This balance ensures that the reactants and products are used up and formed in definite, predictable amounts.
In the problem, we have 1 kmol of CO and 0.5 kmol of O_2. Applying stoichiometry:

• We need exactly 1 kmol of CO to react with 0.5 kmol of O_2.

Thus, the reactants are in the correct amounts, but knowing stoichiometry alone isn't enough. We also need to understand factors that affect the rate of the reaction.

Activation Energy

Activation energy is the minimum energy required for a chemical reaction to occur. It's like a hurdle that the reactants must overcome to turn into products.
Even with the ingredients in the right proportions (like in the given problem), if the activation energy is not met, the reaction won't proceed significantly. The chemical reaction \( 2\text{CO} + \text{O}_2 \rightarrow 2\text{CO}_2 \) needs energy to break the bonds in CO and O_2 and to form new bonds in CO_2.
At ambient conditions (25°C and 1 atm), the energy provided may be too low to overcome this barrier, which explains the insignificant amount of CO_2 formed after 100 hours.
Adding an external source of energy (like heating the mixture or using a catalyst) helps in providing the necessary activation energy. A catalyst can lower the activation energy needed, making it easier for the reaction to proceed at a faster rate.

Reaction Rate

Reaction rate refers to the speed at which reactants are converted into products in a chemical reaction. Several factors influence the rate:

• Concentration of Reactants: Higher concentrations can increase the chance of collision between reactant molecules, leading to a faster reaction.
• Temperature: Increased temperature usually raises reaction rates because reactant molecules move faster and collide more often.
• Catalysts: Substances that lower the activation energy required for the reaction, making it proceed faster.

In the provided exercise, the ambient temperature is not enough to drive the reaction at a noticeable rate. Even though the reactants are in correct stoichiometric amounts, the low temperature means fewer molecules have the energy needed to react.
So, summarizing: the reaction \( 2\text{CO} + \text{O}_2 \rightarrow 2\text{CO}_2 \) under ambient conditions proceeds very slowly because the ambient energy isn't sufficient to overcome the activation energy barrier.