Question

White phosphorus melts and then vaporizes at high temperatures. The gas effuses at a rate that is 0.404 times that of neon in the same apparatus under the same conditions. How many atoms are in a molecule of gaseous white phosphorus?

Short answer

A molecule of gaseous white phosphorus contains 4 atoms.

Step-by-step solution

Identify the known values

Given that the rate of effusion of white phosphorus is 0.404 times that of neon. This can be denoted as \( \frac{\text{Rate of effusion of P}}{\text{Rate of effusion of Ne}} = 0.404 \).

Recall Graham's law of effusion

Graham's law states that \[ \frac{\text{Rate of effusion of gas 1}}{\text{Rate of effusion of gas 2}} = \sqrt{\frac{M_{2}}{M_{1}}} \] where \( M_1 \) and \( M_2 \) are the molar masses of gas 1 and gas 2, respectively. Here, gas 1 is white phosphorus and gas 2 is neon.

Set up the relationship using the given data

According to Graham's Law and the given data: \[ 0.404 = \sqrt{\frac{M_{Ne}}{M_{P}}} \]

Square both sides to eliminate the square root

Squaring both sides yields: \[ (0.404)^2 = \frac{M_{Ne}}{M_{P}} \] \[ 0.163216 = \frac{M_{Ne}}{M_{P}} \]

Rearrange to find the molar mass of white phosphorus

Rearranging gives: \[ M_{P} = \frac{M_{Ne}}{0.163216} \]

Substitute the given molar mass of neon

The molar mass of neon \( M_{Ne} \) is approximately 20.18 g/mol. Substituting, we get: \[ M_{P} = \frac{20.18}{0.163216} \] \[ M_{P} \approx 123.6 \text{ g/mol} \]

Determine the molecular formula of gaseous white phosphorus

The molar mass of a single phosphorus atom is approximately 30.97 g/mol. To find the number of atoms in a molecule of white phosphorus, we divide the molar mass of white phosphorus by the molar mass of a single phosphorus atom: \[ \text{Number of atoms} = \frac{123.6}{30.97} \approx 4 \]

Key concepts

Effusion

Effusion is when gas particles escape through a small hole into a vacuum. Picture a balloon with tiny holes. Gas inside escapes slowly through these holes. The rate at which different gases effuse depends on their molar masses. Lighter gases effuse faster than heavier gases.

Graham's Law helps us understand this by comparing the effusion rates of two gases. It reveals that the effusion rate is inversely proportional to the square root of their molar masses. This means if you know the molar masses of two gases, you can predict how fast they will effuse:

\[\frac{{\text{Rate of effusion of gas 1}}}{{\text{Rate of effusion of gas 2}}} = \sqrt{\frac{M_{2}}{M_{1}}}\]

In our exercise, white phosphorus effuses at a rate 0.404 times that of neon, telling us that phosphorus is a much heavier molecule.

Molar Mass

The molar mass is essentially the weight of one mole of a substance and is expressed in grams per mole (g/mol). It's a crucial concept for understanding reactions and properties of different substances.

For gases, the molar mass helps determine properties like the speed at which gas molecules move. This, in turn, affects how quickly they effuse. The molar mass is numerically similar to the molecular formula weight but refers to the mass of a mole of molecules rather than individual atoms.

In our problem: Neon has a molar mass of 20.18 g/mol. By using Graham's Law, we calculate the molar mass of gaseous white phosphorus like this:
\[ M_{P} = \frac{M_{Ne}}{0.163216} \]

Substitute neon's molar mass into the equation:
\[ M_{P} = \frac{20.18}{0.163216} \approx 123.6 \text{ g/mol}\]

This tells us the molar mass of white phosphorus is about 123.6 g/mol.

Molecular Formula

The molecular formula gives the exact number of each type of atom in a molecule. It's like a recipe that tells you how many atoms of each element combine to form a molecule.

In our example, we determined that white phosphorus has a molar mass of 123.6 g/mol. To find out how many atoms are in a molecule of white phosphorus, we compare this to the molar mass of a single phosphorus atom, which is about 30.97 g/mol.
\[ \text{Number of atoms} = \frac{123.6}{30.97} \approx 4\]

This means a molecule of gaseous white phosphorus contains 4 phosphorus atoms. So, the molecular formula for white phosphorus would be written as \( P_4 \). This formula helps to understand and predict the behavior of the substance in different reactions and conditions.