Q.45.4 L of dinitrogen reacted with 22.7 L of dioxygen and 45.4 L of nitrous oxide was formed. The reaction is given below: $2N_2(g) + O_2(g) \rightarrow 2N_2O(g)$ Which law is being obeyed in this experiment? Write the statement of the law?

Key Idea: Gay-Lussac's Law of Gaseous Volumes.

Reasoning:

1. The balanced chemical equation $2N_2(g) + O_2(g) \rightarrow 2N_2O(g)$ shows that $2$ volumes of dinitrogen react with $1$ volume of dioxygen to produce $2$ volumes of nitrous oxide. The stoichiometric volume ratio is $2:1:2$.
2. The experimental volumes given are $45.4 \text{ L}$ of $N_2$, $22.7 \text{ L}$ of $O_2$, and $45.4 \text{ L}$ of $N_2O$.
3. Dividing these experimental volumes by the smallest volume ($22.7 \text{ L}$) gives the ratio: $N_2: \frac{45.4}{22.7} = 2$ $O_2: \frac{22.7}{22.7} = 1$ $N_2O: \frac{45.4}{22.7} = 2$ The experimental volume ratio is $2:1:2$.
4. Since the experimental volume ratio ($2:1:2$) is a simple whole number ratio and matches the stoichiometric ratio from the balanced equation, Gay-Lussac's Law of Gaseous Volumes is obeyed. …

When chemical reactions involve gases, their volumes often behave in a very specific and predictable way, provided the temperature and pressure remain constant. This predictability is a cornerstone of early chemical understanding and helps us relate macroscopic observations (volumes) to microscopic stoichiometry (moles). The key idea here is that for gases, under identical conditions, the volume occupied is directly proportional to the number of moles.

Understanding the Law of Gaseous Volumes

Imagine you have a balanced chemical equation. The coefficients in this equation tell you the molar ratios of reactants and products. For example, in the given reaction $2N_2(g) + O_2(g) \rightarrow 2N_2O(g)$, it means 2 moles of dinitrogen react with 1 mole of dioxygen to produce 2 moles of nitrous oxide.

Now, if all these substances are gases and are measured at the same temperature and pressure, a remarkable simplification occurs. According to Avogadro's Law, equal volumes of all gases, at the same temperature and pressure, contain an equal number of molecules (or moles). This means that the ratio of moles is directly equivalent to the ratio of volumes for gases under these conditions.

Therefore, the stoichiometric coefficients in a balanced chemical equation for gaseous reactions directly represent the ratios of their reacting and product volumes. This observation was first made by Joseph Louis Gay-Lussac.

Step-by-Step Solution

1. Identify the balanced chemical equation and its stoichiometric ratios:

   The given reaction is:

   $2N_2(g) + O_2(g) \rightarrow 2N_2O(g)$

   From this equation, we can see the molar ratios of the reacting gases and the gaseous product:

   $N_2 : O_2 : N_2O = 2 : 1 : 2$

2. Recall Gay-Lussac's Law of Gaseous Volumes:

   This law states that when gases react together, they do so in volumes which bear a simple whole-number ratio to one another and to the volumes of the gaseous products, provided that all volumes are measured at the same temperature and pressure.

   For gaseous reactions at constant temperature and pressure, the ratio of volumes of reactants and products is equal to the ratio of their stoichiometric coefficients in the balanced chemical equation.

3. Compare the experimental volumes with the stoichiometric ratios:

   The experimental volumes provided are:

   Volume of dinitrogen ($N_2$) reacted $= 45.4 \text{ L}$

   Volume of dioxygen ($O_2$) reacted $= 22.7 \text{ L}$

   Volume of nitrous oxide ($N_2O$) formed $= 45.4 \text{ L}$

   Let's find the ratio of these experimental volumes: …

Concept: Gay-Lussac’s Law of Gaseous Volumes

Method: Volume Ratio Comparison (Gay-Lussac’s Law)

Steps:

1. Write the balanced chemical equation

2. Identify the given volumes (all at same temperature and pressure)

   • $N_2$ = 45.4 L
   • $O_2$ = 22.7 L
   • $N_2O$ = 45.4 L

3. Find the simplest whole-number volume ratio

   Divide each volume by the smallest volume (22.7 L):

   • $N_2$: $\frac{45.4}{22.7} = 2$
   • $O_2$: $\frac{22.7}{22.7} = 1$
   • $N_2O$: $\frac{45.4}{22.7} = 2$

   So the volume ratio is $2 : 1 : 2$

4. Compare with the balanced equation coefficients

   The coefficients in the equation are also $2 : 1 : 2$.

   This matches exactly.

5. Conclusion

   The volumes combine in the same ratio as the gaseous coefficients.

   Hence, Gay-Lussac’s Law of Gaseous Volumes is obeyed. …

Here are the common mistakes students make with this question, along with how to avoid each one.

Mistake 1: Confusing Gay-Lussac’s Law with Avogadro’s Law

• The Mistake: Students often see volumes of gases and immediately write Avogadro’s Law (equal volumes at same T & P contain equal number of molecules). While related, this question specifically asks for the law being obeyed in the experiment based on the volume ratios given in the balanced equation.
• How to Avoid: Remember the key distinction:
  • Avogadro’s Law explains why volume ratios work (because moles are proportional to volume).
  • Gay-Lussac’s Law of Gaseous Volumes is the experimental observation that gases combine in simple whole-number ratios by volume.
  • Exam Tip: If the question gives you experimental volumes (like 45.4 L, 22.7 L, 45.4 L) and a balanced equation, the law being demonstrated is Gay-Lussac’s Law.

Mistake 2: Not Simplifying the Volume Ratio

• The Mistake: Students look at the numbers 45.4, 22.7, and 45.4 and fail to see the simple ratio. They might think the ratio is 45.4 : 22.7 : 45.4, which looks messy.
• How to Avoid: Always divide all volumes by the smallest volume to find the simplest whole-number ratio.
  • Smallest volume = 22.7 L
  • $N_2$: $45.4 \div 22.7 = 2$
  • $O_2$: $22.7 \div 22.7 = 1$
  • $N_2O$: $45.4 \div 22.7 = 2$
  • Result: The ratio is 2 : 1 : 2, which perfectly matches the coefficients in the balanced equation: $2N_2 : 1O_2 : 2N_2O$.

Mistake 3: Writing the Wrong Statement of the Law

• The Mistake: Students write a vague or incorrect statement, such as "Volumes of gases combine in a fixed ratio" or they mix it up with the statement of Avogadro's Law.
• How to Avoid: Memorize the exact statement of Gay-Lussac’s Law of Gaseous Volumes as required for exams:

  "When gases combine or are produced in a chemical reaction, they do so in a simple ratio by volume, provided all volumes are measured at the same temperature and pressure."

  • Key phrases to include: "simple ratio by volume", "same temperature and pressure". …