Understanding Vapour Density
Vapour density (V.D.) is defined as the ratio of the mass of a certain volume of a gas or vapour to the mass of the same volume of hydrogen gas, measured at the same temperature and pressure.
Derivation using Avogadro's Hypothesis
Avogadro's hypothesis states that equal volumes of all gases under the same conditions of temperature and pressure contain the same number of molecules.
Let the volume of the gas and hydrogen be $V$. According to Avogadro's Law, let $V$ contain $n$ molecules.
$$V.D. = \frac{\text{Mass of } V \text{ volume of gas}}{\text{Mass of } V \text{ volume of hydrogen}}$$
Since $V$ contains $n$ molecules:
$$V.D. = \frac{\text{Mass of } n \text{ molecules of gas}}{\text{Mass of } n \text{ molecules of } H_2}$$
$$V.D. = \frac{\text{Mass of 1 molecule of gas}}{\text{Mass of 1 molecule of } H_2}$$
Since the molecule of hydrogen is $H_2$, its mass is $2 \times \text{atomic mass of H} = 2$. Therefore:
$$V.D. = \frac{\text{Molecular Mass}}{2}$$
Or simply: Molecular Mass = 2 $\times$ Vapour Density
Solving the Problem
Given:
- Vapour density of the oxide = 15.
- Molecular mass = 2 $\times$ 15 = 30 amu.
- The oxide contains half its own volume of nitrogen ($N_2$).
Step-by-step Solution:
Determine the number of Nitrogen atoms: If the volume of the oxide is $V$, the volume of $N_2$ present is $V/2$. Let the gas be $N_xO_y$. The amount of nitrogen in one molecule is $(V/2) / V = 0.5$ molecules of $N_2$. Each molecule of $N_2$ has 2 atoms of N, so $0.5 \times 2 = 1$ atom of Nitrogen. So, $x = 1$.
Calculate the Oxygen content: Total Molecular Mass = 30. Mass of 1 Nitrogen atom = 14. Mass of Oxygen = $30 - 14 = 16$. Since 1 atom of Oxygen has a mass of 16, $y = 1$.
Conclusion: The molecular formula is NO.