Question

52\. Nitrogen gas can be obtained by decomposing ammonium nitrate at high temperatures. The nitrogen gas is collected over water in a \(500-\mathrm{mL}\) (three significant figures) flask at \(19^{\circ} \mathrm{C}\). The ambient pressure is \(745 \mathrm{~mm}\) Hg. (Vapor pressure of water at \(19^{\circ} \mathrm{C}\) is \(\left.16.48 \mathrm{~mm} \mathrm{Hg} .\right)\) (a) What is the partial pressure of nitrogen? (b) How many moles of water are there in the wet gas? (c) How many moles of dry gas are collected? (d) If \(0.128 \mathrm{~g}\) of \(\mathrm{Ne}\) are added to the flask at the same temperature, what is the partial pressure of neon in the flask? (e) What is the total pressure after Ne is added?

Short answer

Answer: The partial pressure of nitrogen in the flask is 728.52 mm Hg (or 0.9584 atm), and the total pressure after neon is added is 1.0116 atm.

Step-by-step solution

Find the partial pressure of nitrogen

To find the partial pressure of nitrogen, we can use Dalton's Law of partial pressures, which states that the total pressure is the sum of the partial pressures of the individual gases. In this case, we have nitrogen and water vapor. The total pressure is given as 745 mm Hg, and the vapor pressure of water at 19°C is 16.48 mm Hg. We can calculate the partial pressure of nitrogen (P_N2) with the following formula: P_N2 = P_total - P_water P_N2 = 745 - 16.48 = 728.52 mm Hg

Find the number of moles of water in the wet gas

To find the number of moles of water, we'll use the Ideal Gas Law: PV = nRT We'll need to convert the pressures to atm: P_water = 16.48 mm Hg * (1 atm / 760 mm Hg) = 0.0217 atm The volume of the flask is 500 mL, which is equivalent to 0.5 L. Also, the temperature is given in Celsius, so we must convert it to Kelvin: T = 19°C + 273.15 = 292.15 K Now, we can solve for the number of moles (n) of water vapor: n_water = PV / RT = (0.0217 atm * 0.5 L) / (0.0821 L * atm / mol * K * 292.15 K) = 0.00216 mol

Find the number of moles of dry nitrogen gas collected

We'll use the Ideal Gas Law again, this time for nitrogen gas. Convert the pressure of nitrogen gas to atm: P_N2 = 728.52 mm Hg * (1 atm / 760 mm Hg) = 0.9584 atm Now, solve for the number of moles (n) of nitrogen gas: n_N2 = PV / RT = (0.9584 atm * 0.5 L) / (0.0821 L * atm / mol * K * 292.15 K) = 0.0202 mol

Calculate the partial pressure of neon in the flask

We are given that 0.128 g of Ne is added to the flask. First, convert the mass of Ne to moles: n_Ne = 0.128 g / 20.18 g/mol = 0.00635 mol Now, use the Ideal Gas Law to calculate the partial pressure of Ne (P_Ne) in the flask: P_Ne = nRT / V = (0.00635 mol * 0.0821 L * atm / mol * K * 292.15 K) / 0.5 L = 0.0315 atm

Calculate the total pressure after Ne is added

Now that we have the partial pressure of Ne, we can find the total pressure in the flask by adding the partial pressures of N2, water vapor, and Ne: P_total_after_Ne = P_N2 + P_water + P_Ne = 0.9584 atm + 0.0217 atm + 0.0315 atm = 1.0116 atm The total pressure after Ne is added is 1.0116 atm.

Key concepts

Dalton's Law of Partial Pressures

Understanding Dalton's Law of Partial Pressures is crucial when working with mixtures of gases, as it helps predict the behavior of gases under various conditions. This law states that in a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of the individual gases. A partial pressure is the pressure that one gas in a mixture would exert if it occupied the entire volume on its own.

As seen in the exercise, Dalton's Law can be used to calculate the partial pressure of nitrogen gas in a mixture. When nitrogen gas is collected over water, the vapor pressure of water contributes to the total pressure in the container. By subtracting the vapor pressure of water from the total pressure measured, we determine the partial pressure of nitrogen.

This calculation is essential for properly understanding not just nitrogen collection, but any scenario where multiple gases are present in a system. Remembering this law can simplify complex problems involving gas mixtures and is particularly useful for students studying chemistry or physics.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in understanding the behavior of gases under various conditions. It is expressed as PV = nRT, where P represents the pressure of the gas, V stands for volume, n is the number of moles, R is the universal gas constant, and T is the temperature in Kelvin.

When applying the Ideal Gas Law to calculate the number of moles of water vapor, as was done in the step-by-step solution, it's necessary to maintain consistency in units. This involves converting pressure to atmospheres (atm) and volume to liters (L). Additionally, temperature must be in Kelvin, which requires adding 273.15 to the Celsius temperature.

For students working with this equation, it's critical to understand each variable's role and how they interrelate. This understanding allows for the manipulation of the equation to solve for any one variable, given the others.

Moreover, through the Ideal Gas Law, we learn that gases tend to behave ideally under low pressure and high temperature, a vital fact for practical applications.

In essence, the Ideal Gas Law provides a reliable approximation for the behavior of real gases in many situations, making it an invaluable tool for solving a variety of problems in chemistry and physics.

Mole Calculations

The concept of moles is a cornerstone in chemistry, and mole calculations are used to quantify the amount of a substance. One mole is defined as Avogadro's number (approximately 6.022 x 10^23) of particles, which could be atoms, molecules, electrons, or any other particle or specified group of particles. The mole allows chemists to count particles by weighing them.

In the context of the given exercise, mole calculations are used to determine the amount of gas present in a container, whether it's dry nitrogen gas or neon. By converting the mass of Ne to moles using the molar mass of neon (20.18 g/mol), we can then use the Ideal Gas Law to find the partial pressure of neon.

Mole calculations often go hand-in-hand with the Ideal Gas Law and Dalton's Law of Partial Pressures.

Such calculations can predict the outcome of reactions and help in determining reactant and product quantities.

Prospective students should aim to get comfortable with converting between mass, moles, number of particles, and volume (for gases) to excel in chemical reaction calculations and stoichiometry. Understanding mole calculations is not just a theoretical exercise; it's a fundamental skill that opens up the vast and intriguing world of quantitative chemistry.