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Chapter 6: Problem 45

The density of phosphorus vapor is \(2.64 \mathrm{g} / \mathrm{L}\) at \(310^{\circ} \mathrm{C}\) and \(775 \mathrm{mm}\) Hg. What is the molecular formula of the phosphorus under these conditions?

Short Answer

Expert verified
The molecular formula of phosphorus under the given conditions is \(P_2\).

Step by step solution

01

Convert Units

Firstly, convert the given conditions to the standard units used in the ideal gas law. Convert the temperature from Celsius to Kelvin by adding 273 to the Celsius temperature. This gives you the temperature as \(T = 310^{\circ}C + 273 = 583K\). Then, convert the pressure from mm Hg to atm using the conversion factor 1 atm = 760 mm Hg. This gives the pressure as \(P = 775mmHg * (1 atm / 760 mmHg) = 1.02 atm\).
02

Calculate Molar Mass

Next, rearrange the ideal gas law equation to solve for 'n', the number of moles by dividing both sides by 'RT'. This leaves us with \(n = PV/RT\). Then, substitute the pressure, volume, temperature, and the gas constant \(R = 0.0821 L·atm/K·mol\) into the equation. This gives \(n = (2.64 g/L) * (1.02 atm) / (0.0821 L·atm/K·mol * 583 K) = 0.055 mol/L\). Now to get the molar mass of the phosphorus, divide this calculated density by the given mass of phosphorus which gives you \((2.64 g/mol) / (0.055 mol/L) = 48 g/mol\).
03

Determine Molecular Formula

The atomic mass of phosphorus is about 31 g/mol. To get the molecular formula of phosphorus, divide the molar mass of the gas by the atomic mass of phosphorus. \(48 g/mol / 31 g/mol = 1.55 \approx 1.5\). Since the molecular form should be a whole number, we round 1.5 to the nearest whole number which gives '2'. Therefore, the molecular formula of phosphorus under the given conditions would be P2.

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