Problem 53 - Chemistry: The Central Science in SI Units, Global Edition

Question

In the Dumas-bulb technique for determining the molar mass of an unknown liquid, you vaporize the sample of a liquid that boils below \(100^{\circ} \mathrm{C}\) in a boiling-water bath and determine the mass of vapor required to fill the bulb. From the following data, calculate the molar mass of the unknown liquid: mass of unknown vapor, \(1.012 \mathrm{~g}\); volume of bulb, \(354 \mathrm{~cm}^{3}\); pressure, \(98.93 \mathrm{kPa} ;\) temperature, \(99{ }^{\circ} \mathrm{C}\).

Step-by-Step Solution

Answer

The molar mass of the unknown liquid can be calculated using the Ideal Gas Law and the given data: mass of unknown vapor \(1.012 \: g\), volume of bulb \(354 \: cm^3\), pressure \(98.93 \: kPa\), and temperature \(99^\circ C\). First, convert the temperature to Kelvin (\(372 K\)) and the volume to liters (\(0.354 L\)). Then, calculate the number of moles (n) using the Ideal Gas Law: \(n \approx 0.0451 \: moles\). Finally, calculate the molar mass (M) using the formula \(M = \frac{mass}{moles}\), which gives \(M \approx 22.43 \: g/mol\).

Step 1: Convert the given data to appropriate units

We need to convert the temperature given in Celsius to Kelvin and the volume given in cm³ to L: Temperature in Kelvin: \(T(K) = T(^\circ C) + 273\) Volume in Liters: \(V(L) = \frac{V(cm^3)}{1000}\)

Step 2: Calculate the Temperature in Kelvin

Given temperature, \(T = 99^\circ C\) \(T(K) = 99 + 273 = 372 K\)

Step 3: Calculate the Volume in Liters

Given volume, \(V= 354 cm^3\) \(V(L) = \frac{354}{1000} = 0.354 L\)

Step 4: Calculate the number of moles using the Ideal Gas Law

Now, we will use the Ideal Gas Law to find the number of moles (n): \(PV = nRT\) Rearrange the equation to solve for n: \(n = \frac{PV}{RT}\) Plug in the given values: \(n = \frac{(98.93 kPa)(0.354 L)}{(8.314 J/(mol·K))(372 K)}\)

Step 5: Convert pressure from kPa to Pa

1 kPa = \(10^3\) Pa Pressure, \(P = 98.93 kPa \times 10^3 = 98,930 Pa\)

Step 6: Calculate the number of moles (n)

Plug in the converted values: \(n = \frac{(98,930 Pa)(0.354 L)}{(8.314 J/(mol·K))(372 K)}\) \(n \approx 0.0451\: moles\)

Step 7: Calculate Molar Mass

Now, we will use the formula for molar mass (M): \(M = \frac{mass}{moles}\) Given mass of the unknown vapor, \(m = 1.012\: g\) Molar Mass, \(M = \frac{1.012\: g}{0.0451\: moles} \approx 22.43\: g/mol\) The molar mass of the unknown liquid is approximately 22.43 g/mol.

Key Concepts

Molar Mass Calculation

Molar mass is a key concept when trying to identify an unknown substance. It is defined as the mass of a given substance divided by the amount of substance, and is expressed in units of grams per mole (g/mol).
In chemistry, determining the molar mass is essential for various applications, such as identifying chemical compounds or measuring substances in reactions.
To calculate molar mass using the Dumas-bulb technique, you follow these steps: - Measure the mass of the vapor. In our example, the mass is 1.012 g. - Find the number of moles of vapor using the ideal gas law. This involves converting the data to appropriate units and plugging them into the formula. - Lastly, divide the mass of the vapor by the number of moles calculated. This gives the molar mass, which in this scenario is approximately 22.43 g/mol.

Ideal Gas Law

The Ideal Gas Law is a cornerstone of chemistry, providing a mathematical relationship between different properties of gases. It is represented as: \[ PV = nRT \] where: - \( P \) is pressure in pascals (Pa), - \( V \) is volume in liters (L), - \( n \) is the number of moles, - \( R \) is the ideal gas constant (8.314 J/(mol·K)), - \( T \) is temperature in Kelvin (K).
Using this equation, you can solve for various unknowns, such as mole number \( n \), by rearranging the formula. In our exercise, after converting all units appropriately, we use the rearranged equation \( n = \frac{PV}{RT} \) to find the number of moles of the unknown vapor.

Unit Conversion

Unit conversion is crucial in chemistry to ensure that calculations are accurate. Working in the correct units helps maintain consistency and avoid mistakes, especially when using equations like the Ideal Gas Law.
Here are the typical conversions utilized in the problem we're discussing: - Convert temperature from Celsius to Kelvin using \( T(K) = T(°C) + 273 \). For our exercise, this converts 99°C to 372 K. - Convert volume from cubic centimeters (cm³) to liters (L) using \( V(L) = \frac{V(cm^3)}{1000} \), resulting in 354 cm³ being 0.354 L. - Convert pressure from kilopascals (kPa) to pascals (Pa) since \( 1 kPa = 10^3 Pa \). Thus, 98.93 kPa becomes 98,930 Pa.
These conversions allow us to plug numbers correctly into equations like \( PV = nRT \), ensuring our calculations are accurate.

Chemistry Problem Solving

Approaching chemistry problems efficiently requires breaking them down into understandable steps, as demonstrated in this example. The following strategies can improve your problem-solving skills: - Understand the given data thoroughly and what is being asked. - Convert all measurements to the correct scientific units before using any formula.
- Recognize which principles apply to the problem. For example, the Dumas-bulb technique involves gas laws for capturing data about the vapor. - Work systematically to avoid mistakes, checking each step. - Perform calculations step-by-step, write down intermediate results, and re-check your work.
By breaking the problem into smaller, manageable parts and applying foundational chemistry concepts, you can tackle complex chemistry exercises with confidence.