Question

How many molecules of ammonia gas are present in 67.2 $\text{dm}^3$, measured at S.T.P.?

• A. $2.0 \times 10^{24}$
• B. $1.0 \times 10^{23}$
• C. $1.8 \times 10^{24}$
• D. $5.0 \times 10^{24}$

Hint:

Recognize common multiples of $22.4$ to speed up calculations! Notice that $22.4 \times 3 = 67.2$. Finding out that you have exactly 3 moles instantly simplifies the calculation to $3 \times 6 \times 10^{23} = 18 \times 10^{23} = 1.8 \times 10^{24}$.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The problem asks us to find the total number of ammonia ($\text{NH}_3$) molecules contained within a given volume of $67.2\ \text{dm}^3$ at Standard Temperature and Pressure (S.T.P.).

Step 2: Key Formula or Approach:
According to Avogadro's law, 1 mole of any ideal gas occupies exactly $22.4\ \text{dm}^3$ (or liters) at S.T.P. and contains Avogadro's number of molecules ($N_A \approx 6.022 \times 10^{23}$).
The formula to determine the number of molecules is:
$$\text{Number of molecules} = \frac{\text{Given Volume}}{\text{Molar Volume}} \times N_A$$

Step 3: Detailed Explanation:
Given parameters:
Given Volume = $67.2\ \text{dm}^3$
Molar Volume at S.T.P. = $22.4\ \text{dm}^3\text{mol}^{-1}$
First, find the number of moles ($n$):
$$n = \frac{67.2\ \text{dm}^3}{22.4\ \text{dm}^3\text{mol}^{-1}} = 3\ \text{moles}$$ Now, compute the total number of molecules by multiplying the number of moles by Avogadro's number:
$$\text{Number of molecules} = 3 \times 6.022 \times 10^{23}$$ $$\text{Number of molecules} = 1.8066 \times 10^{24} \approx 1.8 \times 10^{24}$$ This corresponds to option (C).

Step 4: Final Answer:
The total number of molecules present is $1.8 \times 10^{24}$, which corresponds to option (C).