Question

\(34.05 \mathrm{~mL}\) of phosphorus vapours weigh \(0.0625 \mathrm{~g}\) at \(546^{\circ} \mathrm{C}\) and \(0.1\) bar pressure. What is the molar mass of phosphorus? (a) \(124.77 \mathrm{~g} \mathrm{~mol}^{-1}\) (b) \(1247.74 \mathrm{~g} \mathrm{~mol}^{-1}\) (c) \(12.47 \mathrm{~g} \mathrm{~mol}^{-1}\) (d) \(30 \mathrm{~g} \mathrm{~mol}^{-1}\)

Short answer

The molar mass of phosphorus is 124.77 g/mol, so the correct answer is (a) 124.77 g mol^{-1}.

Step-by-step solution

Convert the temperature from Celsius to Kelvin

To use the Ideal Gas Law, temperatures must be in Kelvin. Convert the temperature from Celsius to Kelvin using the formula: T(K) = T(°C) + 273.15. T(K) = 546°C + 273.15 = 819.15 K

Convert the pressure from bar to atmospheres

The Ideal Gas Law uses pressure in atmospheres (atm). Convert the pressure from bar to atmospheres knowing that 1 bar = 0.986923 atm. Pressure in atm = 0.1 bar * 0.986923 atm/bar = 0.0986923 atm

Plug values into the Ideal Gas Law

Use the Ideal Gas Law in the form PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the gas constant (0.0821 L*atm/mol*K), and T is temperature in Kelvin. Since we have grams rather than moles, we'll solve for n first and then calculate the molar mass.

Calculate number of moles (n)

Rearrange the Ideal Gas Law to solve for n: n = PV / RT. Using the values P = 0.0986923 atm, V = 34.05 mL = 0.03405 L, R = 0.0821 L*atm/mol*K, and T = 819.15 K, plug them into the equation: n = (0.0986923 atm * 0.03405 L) / (0.0821 L*atm/mol*K * 819.15 K)

Calculate the molar mass (M)

Once you have n, the molar mass M can be found from the mass (m) given by m/n. The molar mass M = 0.0625 g / n. Finish calculating n and then use it to find M.

Determine the correct answer

Evaluate the previous steps' calculations to find the molar mass. Compare the result with the given options to determine the correct answer.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation that relates the pressure, volume, temperature, and number of moles of an ideal gas in a single, concise formula: \( PV = nRT \) where P represents pressure, V is volume, n stands for the number of moles of gas, R is the gas constant, and T is temperature in Kelvin. This law provides a basis for calculating various properties of gases, such as molar mass in this exercise. To apply the Ideal Gas Law, one must ensure that all units are consistent, typically with pressure in atmospheres (atm), volume in liters (L), temperature in Kelvin (K), and the gas constant \( R = 0.0821 \text{L atm mol}^{-1} \text{K}^{-1} \).

When given conditions such as weight and molar mass, the Ideal Gas Law allows for the determination of the number of moles (n). Once n is calculated, the molar mass can be derived by dividing the given mass of the gas by the number of moles. This is an essential step when working with various gaseous substances in chemical calculations.

Converting Temperature to Kelvin

Temperature conversions are crucial in chemistry, especially when working with gas laws. The Kelvin is the SI unit for temperature and is used in the Ideal Gas Law to avoid negative values and provide an absolute scale. To convert Celsius to Kelvin—a vital step in the exercise—the formula is \( T(\text{K}) = T(^{\circ}\text{C}) + 273.15 \).

In the given exercise, the temperature of phosphorus vapour is initially at \( 546^{\circ}\text{C} \) which must be converted to Kelvin before it can be used with the Ideal Gas Law formula. Adding \( 273.15 \) to the Celsius temperature yields \( 819.15 \text{K} \), providing the correct temperature value for the subsequent gas law calculations.

Gas Constant

The gas constant, symbolized by R, is a critical value in the Ideal Gas Law and other gas equations. It serves as a bridge between the physical properties of gases, incorporating the aspects of pressure, volume, and temperature into a single constant. For calculations involving the Ideal Gas Law, the constant \( R \) is often used as \( 0.0821 \text{L atm mol}^{-1} \text{K}^{-1} \).

This constant is derived from the equation R = PV/nT, where the values of P, V, n, and T for an ideal gas at standard conditions are known. The gas constant is universal, but its value can change depending on the units used for pressure, volume, and temperature. Therefore, it's essential to confirm that the gas constant matches the units of the other terms in the equation being used.

Pressure Unit Conversion

Pressure unit conversion is necessary when the pressure value given in a problem is not in the unit required by the formula or equation being applied. In the Ideal Gas Law, pressure is typically expressed in atmospheres (atm). However, pressure can be recorded in various units such as bar, Pascal (Pa), millimeters of mercury (mmHg), and Torr. For our exercise, the pressure provided in the bar must be converted to atmospheres.

The conversion factor between bar and atmospheres is that 1 bar is approximately equal to \( 0.986923 \text{atm} \). To convert bar to atmospheres, you multiply the pressure value in bar by this conversion factor. In this specific exercise, the pressure given is \( 0.1 \text{bar} \) which, when converted, gives \( 0.0986923 \text{atm} \)—the pressure value in atmospheres needed to use in the Ideal Gas Law equation.