Problem 56 - Chemistry The Central Science

Question

The molar mass of a volatile substance was determined by the Dumas-bulb method described in Exercise \(10.55 .\) The unknown vapor had a mass of \(0.846 \mathrm{~g} ;\) 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 \(59.58 \frac{g}{mol}\).

Step 1: Write down the given information and convert it to proper units

We have the following data: Mass: \(0.846~g\) Volume: \(354~cm^3\) Pressure: \(752~torr\) Temperature: \(100 °C\) First, convert all the given values into the units that are most convenient for the ideal gas law equation. - Volume: Convert volume from \(cm^3\) to \(L\) by dividing by 1000. \(354~cm^3 = \frac{354}{1000}L = 0.354~L\) - Pressure: Convert pressure from \(torr\) to \(atm\) using the conversion factor 1 atm = 760 torr. \(752~torr = \frac{752}{760}~atm \approx 0.9895 ~atm\) - Temperature: Convert temperature from \(°C\) to \(K\) by adding 273.15. \(100 °C + 273.15 = 373.15 K\)

Step 2: Determine the number of moles using ideal gas law

Use the ideal gas law equation to find the moles of the unknown vapor: \[PV = nRT\] Where, P = Pressure = \(0.9895~atm\) V = Volume = \(0.354~L\) n = moles of the substance (to be determined) R = Ideal gas constant = \(0.0821 \frac{L \cdot atm}{mol \cdot K}\) T = Temperature = \(373.15~K\) Rearrange the ideal gas law equation to solve for the number of moles: \[n = \frac{PV}{RT}\] Substitute the known values of P, V, R, and T into the equation: \[n = \frac{(0.9895~atm)(0.354~L)}{(0.0821 \frac{L \cdot atm}{mol \cdot K})(373.15~K)}\] Solve for the number of moles: \[n \approx 0.0142~mol\]

Step 3: Calculate the molar mass of the unknown vapor

Molar mass is defined as the mass of the substance in grams divided by the number of moles of the substance. We can use the following equation to calculate the molar mass: \[Molar~mass = \frac{mass}{moles}\] From step 1, we know the mass of the unknown vapor is \(0.846~g\), and from step 2, we found that there are approximately \(0.0142~moles\) of the substance. Input these values into the equation: \[Molar~mass = \frac{0.846~g}{0.0142~mol}\] Solve for the molar mass: \[Molar~mass \approx 59.58 \frac{g}{mol}\] The molar mass of the unknown vapor is approximately \(59.58 \frac{g}{mol}\).

Key Concepts

Dumas-bulb method

The Dumas-bulb method is an integral technique used for determining the molar mass of a volatile liquid by using its vapor density. It involves vaporizing a known mass of a liquid in a flask, known as a Dumas bulb, and then measuring the volume, temperature, and pressure of the vapor. The flask is heated until all of the liquid has vaporized, and then the flask is sealed and cooled down to room temperature. The cooled flask's weight is then compared to the original weight, and the difference in mass gives you the mass of the vapor.

In order to ensure accuracy, it's crucial that the bulb is filled with vapor only and no air is present, as air would affect the pressure readings. Also, the volatilization must be done in such a way that the vapor does not escape, which would impact the mass measurement. Understanding the principles behind the Dumas-bulb method allows for accurate molar mass calculation of volatile substances, which is essential in many chemical processes and analyses.

Ideal gas law

The ideal gas law is a fundamental equation in chemistry that relates the pressure (P), volume (V), temperature (T), and number of moles (n) of an ideal gas. It is expressed as
\[PV = nRT\]
where R is the ideal gas constant (\(0.0821 \frac{L \cdot atm}{mol \cdot K}\)). This equation assumes that the particles of an ideal gas do not interact with each other and occupy no volume. Although no real gas perfectly aligns with these assumptions, the ideal gas law provides a good approximation for most gases under standard conditions.

When applying the ideal gas law, it is important to ensure that all measurements are in consistent units. Standard conditions for temperature and pressure (STP) are typically 0°C and 1 atm, but in specific conditions outlined by a problem, you may need to convert units appropriately. The law is widely used in chemical calculations and provides a clear relationship between physical properties of gases.

Molar mass calculation

Molar mass calculation is a fundamental concept in chemistry that allows one to determine the mass per mole of a substance. The molar mass can be thought of as the weight of 6.022 x \(10^{23}\) (Avogadro's number) of molecules of a substance and is typically expressed in grams per mole (g/mol). To calculate the molar mass, the mass of a sample in grams is divided by the number of moles of the sample. This is mathematically represented as:
\[Molar~mass = \frac{mass}{moles}\]
Here, the mass must be in grams and the moles must be dimensionless quantities. Calculating the molar mass is crucial for converting between grams and moles, which is a common practice in stoichiometric calculations and helps in understanding chemical reactions in terms of the mass of reactants and products.

Converting units

Converting units is an essential skill in the world of chemistry calculations, as it ensures accuracy when using different measurements. As seen in the ideal gas law, getting your units correctly aligned is key to finding the right results. Volume, for instance, should be converted from cubic centimeters (cm³) to liters (L), and pressure from torr to atmospheres (atm). Temperature must be in Kelvin for gas law calculations, requiring an addition of 273.15 to Celsius temperatures.

Without proper unit conversion, the calculated values could lead to incorrect interpretations, skewing experimental outcomes. Therefore, grasping how to seamlessly convert between units, using appropriate factors and understanding which units are appropriate for which equations, is a critical skill for any student in the field of chemistry.

Moles determination

Moles determination is a key aspect of chemical quantification. A mole is a unit that represents a specific number of particles (molecules, atoms, ions, etc.), which is Avogadro's number (approximately \(6.022 \times 10^{23}\)). To determine the number of moles in a sample, we apply formulas derived from the ideal gas law or other stoichiometric relationships.

In the context of the ideal gas law, moles are determined by rearranging the equation to solve for 'n': \[n = \frac{PV}{RT}\]
Every variable in the equation with the exception of 'n' must be known in order to calculate the moles. Moles are pivotal in understanding chemical reactions and relating the mass of a substance to its actual particle count, whether dealing with individual atoms or complex molecules.