Problem 54 - Chemistry The Central Science

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 0.846 g; the volume of the bulb was \(354 \mathrm{cm}^{3},\) pressure 752 torr, and temperature \(100^{\circ} \mathrm{C}\) . Calculate the molar mass of the unknown vapor.

Step-by-Step Solution

Answer

The molar mass of the unknown vapor is approximately \(27.8 \: g/mol\).

Step 1: Convert given units to appropriate SI units

First, we need to convert all given values to appropriate SI units to use the ideal gas law equation. The pressure needs to be in pascals (Pa), volume needs to be in meters cubed (m³), and the temperature needs to be in kelvin (K). The given values are: - Pressure (P) = 752 torr - Volume (V) = 354 cm³ - Temperature (T) = 100°C - Mass (m) = 0.846 g Convert them to SI units: - P = 752 torr × (101325 Pa / 760 torr) ≈ 100662 Pa - V = 354 cm³ × (1 m³ / 1000000 cm³) = 3.54 × 10⁻⁴ m³ - T = 100°C + 273.15 = 373.15 K

Step 2: Apply ideal gas law equation to find the number of moles

The ideal gas law equation is: PV = nRT We can solve for the number of moles (n) using the equation: n = PV / RT Substitute the converted values of P, V, and T, and use the R value for SI units (8.314 J/(mol·K)): n = (100662 Pa * 3.54 × 10⁻⁴ m³) / (8.314 J/(mol·K) * 373.15 K) ≈ 0.0304 moles

Step 3: Calculate the molar mass

The molar mass (M) of the substance can be calculated by dividing the mass by the number of moles: M = mass (g) / n (moles) Substitute the values for mass and the number of moles we just calculated: M = 0.846 g / 0.0304 moles ≈ 27.8 g/mol The molar mass of the unknown vapor is approximately 27.8 g/mol.

Key Concepts

Molar Mass Calculation

Calculating the molar mass of a substance is fundamental in chemistry for understanding the properties and behavior of molecules. The molar mass is simply the mass of one mole of a substance. It helps to determine how much of a substance is needed to achieve a desired chemical reaction. To find the molar mass, you divide the given mass of a substance by the number of moles.
For example, if you have a substance weighing 0.846 grams and have determined via the ideal gas law that it contains 0.0304 moles, the molar mass is calculated as follows:

• Molar Mass (\( M \)) = Mass (in grams) / Number of moles (\( n \))
• \( M = 0.846 \text{ g} / 0.0304 \text{ moles} \approx 27.8 \text{ g/mol} \)

This simple division helps in various areas like comparing gases, determining yield in reactions, and converting between amounts of substances.

Ideal Gas Law

The ideal gas law is a crucial equation in chemistry that couples pressure, volume, temperature, and moles of a gas using the formula \( PV = nRT \). This equation is key to understanding gas behavior under various conditions and simplifies analyzing and predicting the state of a gas. Here's a quick breakdown of each component:

• P: Pressure exerted by the gas, which in SI units is measured in Pascals (Pa).
• V: Volume occupied by the gas, measured in cubic meters (m³).
• n: Number of moles of the gas.
• R: The universal gas constant, approximately 8.314 J/(mol·K) for SI units.
• T: Temperature in Kelvin (K), which must always be used to ensure accuracy.

By rearranging the equation to \( n = \frac{PV}{RT} \), you can calculate the number of moles of gas if pressure, volume, and temperature are known, as seen in solving for the molar mass in the Dumas Bulb Method.

Unit Conversion in Chemistry

Unit conversion plays a fundamental role in chemistry, allowing for consistent measurements across various systems. Here’s how to handle common conversions:

• Pressure: Dumas Bulb Method often involves converting pressure from torr to pascals (Pa). Use \( \frac{101325 \text{ Pa}}{760 \text{ torr}} \) as the conversion factor. So, \( 752 \text{ torr} \times \frac{101325}{760} \approx 100662 \text{ Pa} \).
• Volume: Convert volume from cubic centimeters (cm³) to cubic meters (m³) using \( \frac{1 \text{ m³}}{1000000 \text{ cm³}} \). For instance, \( 354 \text{ cm}^3 = 3.54 \times 10^{-4} \text{ m}^3 \).
• Temperature: Convert Celsius to Kelvin by adding 273.15. So \( 100^{\circ} \text{C} + 273.15 = 373.15 \text{ K} \).

Such conversions ensure compatibility with the ideal gas law and other calculations, enhancing accuracy in measuring and manipulating substances in chemical reactions.

Volatility in Chemistry

Volatility refers to how quickly a substance can vaporize, which is an important factor in chemical experiments and processes. Volatile substances have higher vapor pressures at given temperatures and thus evaporate more readily than less volatile substances. This concept plays a key role in various fields:

• Distillation Processes: Volatile components are more easily separated through distillation due to differing vapor pressures.
• Environmental Impact: Understanding volatility helps predict the spread and concentration of substances in air or water.
• Product Formulation: Volatility knowledge informs the creation of perfumes or pharmaceuticals, where controlled evaporation rates are needed.

In the Dumas Bulb Method, a substance's volatility allows for the determination of molar mass because only volatile substances form vapors that can be measured under standard conditions.