Problem 54 - Chemistry: The Central Science in SI Units, Global Edition

Question

The molar mass of a volatile substance was determined by the Dumas-bulb method described in Exercise 10.53 . The unknown vapor had a mass of \(2.55 \mathrm{~g} ;\) the volume of the bulb was \(500 \mathrm{~mL}\), pressure \(101.33 \mathrm{kPa}\), and temperature \(37^{\circ} \mathrm{C.Calculate}\) the molar mass of the unknown vapor.

Step-by-Step Solution

Answer

The molar mass of the unknown vapor can be calculated using the ideal gas law and the given information. After converting the temperature to Kelvin and pressure to atm, we find the number of moles using the ideal gas law and then calculate the molar mass by dividing the mass by the number of moles. The molar mass of the unknown vapor is approximately 130.1 g/mol.

Step 1: Convert temperature to Kelvin

The given temperature is in Celsius, so we need to convert it to Kelvin by adding 273.15 to the Celsius value. Hence, the temperature in Kelvin is \(T = 37 + 273.15 = 310.15 \mathrm{K}\).

Step 2: Convert pressure to atm

The pressure is given in kPa, but the ideal gas constant R has the unit L atm mol^{-1} K^{-1}. To be consistent, we need to convert the pressure to atm using the conversion factor 1 atm = 101.325 kPa. Therefore, the pressure in atm is \(P = \frac{101.33 \mathrm{kPa}}{101.325\, \mathrm{kPa/atm}} = 1 \mathrm{atm}\).

Step 3: Calculate the number of moles of the unknown vapor

Now we can use the ideal gas law to find the number of moles \(n\): \(PV = nRT \Rightarrow n = \frac{PV}{RT}\). Plugging in the given values, we have: \(n = \frac{(1\, \mathrm{atm})(0.5\, \mathrm{L})}{(0.0821\, \mathrm{L\, atm\, mol^{-1}K^{-1}})(310.15\, \mathrm{K})} = 0.0196\, \mathrm{mol}\)

Step 4: Calculate the molar mass of the unknown vapor

We have the mass of the unknown vapor and the number of moles calculated in the previous step. We can now find the molar mass (M) by dividing the mass by the number of moles: \(M = \frac{\mathrm{mass}}{\mathrm{moles}} = \frac{2.55 \mathrm{g}}{0.0196\, \mathrm{mol}} = 130.1\, \mathrm{g/mol}\) The molar mass of the unknown vapor is approximately 130.1 g/mol.

Key Concepts

Dumas-bulb method

The Dumas-bulb method is a nifty way to calculate the molar mass of a volatile substance. This method is known for its simplicity and precision. It's named after the French chemist Jean-Baptiste Dumas. Here's the essence of how it works:
The substance is vaporized inside a glass bulb, and its physical conditions such as temperature, pressure, and volume are carefully measured. These parameters are key to finding the molar mass. The bulb is first filled with the vapor of the unknown substance, and it is weighed. Then, the bulb is also weighed when it’s completely evacuated to find the mass of the vapor.

• Put simply, the mass of the vapor is found by subtracting the mass of the empty bulb from the mass of the bulb filled with vapor.
• The volume of the bulb gives the volume occupied by the vapor.

By using these values and substituting them into the ideal gas law, we can find the molar mass of the vapor. It's a practical and efficient technique, especially useful in identification and analysis of gaseous substances.

ideal gas law

The ideal gas law is an equation of state that describes how gases behave under various conditions. It combines several fundamental gas laws. In its most common form, the equation is expressed as \(PV = nRT\). Let's break it down:

• \(P\) denotes the pressure of the gas.
• \(V\) stands for the volume it occupies.
• \(n\) is the number of moles of the gas.
• \(R\) is the ideal gas constant which makes sure our units are consistent.
• \(T\) represents the temperature in Kelvin.

What makes this equation powerful is its ability to relate these properties of gases, allowing us to calculate any one of them if the others are known. For the Dumas-bulb method, it's crucial because it allows us to determine the number of moles of gas from measurable quantities like the volume, pressure, and temperature. Once we have the moles, calculating the molar mass becomes straightforward by dividing the mass by the number of moles.

unit conversion

When performing calculations involving the ideal gas law and the Dumas-bulb method, unit conversion is essential. Conversions ensure consistency in calculations, which is critical to obtaining correct results. Here's how it typically comes into play:

• Temperature needs to be in Kelvin for usage in the ideal gas law.
• Pressure often needs conversion to atmospheres, aligning with the gas constant's units. This can involve converting from kPa or mmHg.
• Volume is usually kept in liters, as it directly corresponds with how the gas constant is defined.

Ensuring these conversions are done correctly helps avoid errors in calculating the number of moles and hence, determining the molar mass. In our original exercise, we converted temperature from Celsius to Kelvin and pressure from kPa to atm to use the ideal gas law effectively. Understanding and applying accurate unit conversions is a vital skill in chemistry!