Question

A flask has \(1.35\) mol of hydrogen gas at \(25^{\circ} \mathrm{C}\) and a pressure of \(1.05\) atm. Nitrogen gas is added to the flask at the same temperature until the pressure rises to \(1.64\) atm. How many moles of nitrogen gas are added?

Short answer

Answer: Approximately 0.72 moles of nitrogen gas are added to the flask.

Step-by-step solution

List down the given values.

We are given: - Initial moles of hydrogen gas (\(n_{H_2}\)) = 1.35 mol - Initial temperature (\(T\)) = 25°C = 298.15 K (convert the temperature to Kelvin by adding 273.15) - Initial pressure of hydrogen gas (\(P_{H_2}\)) = 1.05 atm - Final pressure after adding nitrogen gas (\(P_{final}\)) = 1.64 atm

Calculate the volume of the flask using Ideal Gas Law.

We use Ideal Gas Law equation, PV = nRT, to find the volume (V) of the flask: \(V = \frac{n_{H_2}RT}{P_{H_2}}\) Plug in the given values: \(V = \frac{1.35 \times 0.0821 \times 298.15}{1.05}\) \(V \approx 30.15 \, L\) (rounded to 2 decimal places) The volume of the flask is about 30.15 L.

Find the initial and final pressure of nitrogen gas.

We know that before adding nitrogen gas, the pressure inside the flask was solely due to the hydrogen gas. After adding nitrogen gas, the flask's final pressure is the sum of pressures due to both hydrogen and nitrogen gases. So, the initial pressure of nitrogen (\(P_{N_2 \,initial}\)) = 0 atm (as there was none initially) The final pressure of nitrogen (\(P_{N_2 \,final}\)) = Final pressure - Initial pressure of hydrogen gas \(P_{N_2 \,final} = 1.64 -1.05\) \(P_{N_2 \,final} = 0.59 \, atm\)

Calculate the moles of nitrogen gas

Again, we can use the Ideal Gas Law equation: \(P_{N_2 \,final}V = n_{N_2}RT\) Rearrange the equation to find \(n_{N_2}\): \(n_{N_2} = \frac{P_{N_2 \,final}V}{RT}\) Plug in the values: \(n_{N_2} = \frac{0.59 \times 30.15}{0.0821 \times 298.15}\) \(n_{N_2} \approx 0.72 \, mol\) (rounded to 2 decimal places) So, about 0.72 moles of nitrogen gas are added to the flask.

Key concepts

Moles of Gas

Understanding the concept of 'moles of gas' is crucial in chemistry, particularly in gas law problems. A mole is a unit of measurement that represents a large number of particles, typically atoms or molecules. For gases, one mole is equivalent to Avogadro's number, which is approximately \(6.02 \times 10^{23}\) particles.

The amount of substance in moles plays a central role when working with the Ideal Gas Law, as it connects the physical quantity of a gas (volume, pressure, and temperature) with the amount of substance measured in moles. In the given problem, the initial amount of hydrogen gas in the flask is \(1.35\) moles, which indicates that there are \(1.35 \times 6.02 \times 10^{23}\) molecules of hydrogen in the flask.

Gas Law Problems

Gas law problems involve mathematical relations between the four properties of a gas: pressure (P), volume (V), temperature (T), and the amount of gas in moles (n). The Ideal Gas Law, represented by the equation \(PV = nRT\), where R is the universal gas constant, is widely used to solve these types of problems. To accurately solve for one of the variables in the equation, it is essential to understand and properly apply this relationship.

When dealing with gas law problems, converting all measurements to their proper units is fundamental. For example, temperatures must be in Kelvins, and pressures in atmospheres or another consistent unit. This conversion ensures that the gas constant R is applied correctly. As demonstrated in the original exercise and solution, the volume of the flask was calculated using the given properties of the hydrogen gas before nitrogen was added.

Stoichiometry

Stoichiometry is the quantitative relationship between reactants and products in a chemical reaction. It is also a method used to relate the amounts of reactants and products using a balanced chemical equation. In the context of gas stoichiometry, the moles of gas can be converted to or from other units like liters or grams using the Ideal Gas Law and molar mass.

While the problem provided doesn't involve a chemical reaction per se, it employs stoichiometric principles by quantifying the amount of a substance—nitrogen gas—added to the flask. Stoichiometry is fundamental in predicting the results of chemical reactions and for finding the amounts of reactants needed or products formed, whether in lab settings or industrial processes.

Chemical Reactions and Pressure

In the realm of chemical reactions, pressure can significantly influence reaction rates and equilibrium. Gases are compressible, and changes in pressure can alter the volume they occupy, affecting how particles interact. In the context of this exercise, pressure is an essential variable as it relates to the amount of gas present in a closed system.

In a flask where multiple gases are present, such as the mixture of hydrogen and nitrogen in our problem, the total pressure is the sum of the partial pressures of each individual gas. This is known as Dalton's Law of Partial Pressures. When nitrogen gas is added to the flask containing hydrogen gas, the total pressure increases, which allows us to calculate the moles of nitrogen gas added by using the Ideal Gas Law, assuming constant temperature and volume.