Problem 146 - Chemistry An Atoms First Approach

Question

A large flask with a volume of \(936 \mathrm{mL}\) is evacuated and found to have a mass of \(134.66 \mathrm{g}\). It is then filled to a pressure of 0.967 atm at \(31^{\circ} \mathrm{C}\) with a gas of unknown molar mass and then reweighed to give a new mass of 135.87 g. What is the molar mass of this gas?

Step-by-Step Solution

Answer

The molar mass of the gas is approximately \(32.97 \mathrm{g/mol}\).

Step 1: Convert the given values to appropriate units

We need to convert the given temperature from Celsius to Kelvin and the given volume from milliliters to liters: Temperature in Kelvin (K): T(K) = T(°C) + 273.15 T(K) = 31 + 273.15 = 304.15 K Volume in liters (L): V(L) = V(mL) / 1000 V(L) = 936 / 1000 = 0.936 L

Step 2: Find the mass of the gas inside the flask

Subtract the mass of the empty flask from the mass of the filled flask: Mass of gas = Mass of filled flask - Mass of empty flask Mass of gas = 135.87 g - 134.66 g = 1.21 g

Step 3: Use the Ideal Gas Law to find the number of moles of gas

PV = nRT Solve for the number of moles (n): n = PV / RT We know the values of P, V, and T as well as the universal gas constant R (0.0821 L atm/mol K): n = (0.967 atm * 0.936 L) / (0.0821 L atm/mol K * 304.15 K) n = 0.0367 mol

Step 4: Calculate the molar mass of the gas

Now that we have the number of moles and the mass of the gas, we can calculate the molar mass (M) using the following formula: M = mass (g) / moles M = 1.21 g / 0.0367 mol M = 32.97 g/mol The molar mass of the gas is approximately 32.97 g/mol.

Key Concepts

Understanding the Ideal Gas Law

The Ideal Gas Law provides a clear relationship between pressure (P), volume (V), temperature (T), and the number of moles (n) of a gas. Represented as the equation PV = nRT, this principle allows us to calculate one variable as long as the others are known.

When we reference the universal gas constant (R), which has a value of 0.0821 L atm/mol K, we are able to use the Ideal Gas Law across various units such as liters for volume, atmospheres for pressure, and kelvins for temperature. It's essential to ensure that all measurements comply with these units prior to applying the law, involving conversions such as moving from Celsius to Kelvin for temperature and milliliters to liters for volume.

Moles Calculation

The process of calculating moles, often represented as 'n' in chemical equations, is crucial in understanding the amounts of substances involved. In the context of the Ideal Gas Law, n can be isolated and calculated using the rearranged formula n = PV / RT. This step is fundamental in determining other properties such as molar mass.

The number of moles is a gateway to understanding the stoichiometry of chemical reactions, which is the quantitative relationship between reactants and products. Grasping moles calculation allows students to advance into more complex exercises, such as predicting the outcomes of chemical reactions or calculating the yields of products formed.

The Essence of Chemical Thermodynamics

Chemical Thermodynamics focuses on the study of energy changes during a chemical reaction. Crucial concepts include enthalpy, entropy, and Gibbs free energy. While this exercise deals with the molar mass of a gas, understanding the broader context of thermodynamics is valuable.

Thermodynamics helps explain why certain reactions occur spontaneously while others do not. It also underlines the importance of temperature as a factor influencing the behavior of gases, as seen in the necessity to convert Celsius to Kelvin while solving the provided exercise. Mastery of thermodynamics will ultimately enable students to predict the feasibility of chemical processes and comprehend the energy dynamics at play.