Question

A flask has \(1.35 \mathrm{~mol}\) 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 0.801 moles of nitrogen gas were added.

Step-by-step solution

List the given information and convert the temperature to Kelvin

We have the following known facts: - Moles of hydrogen gas (n_H2) = 1.35 mol - Initial pressure (P_initial) = 1.05 atm - Final pressure (P_final) = 1.64 atm - Temperature (T) = 25°C, which is equal to 25 + 273.15 = 298.15 K.

Write down the Ideal Gas Law for the initial state

To solve this problem, we'll use the Ideal Gas Law, which states: PV = nRT where P = pressure (atm) V = volume (L) n = moles of gas (mol) R = ideal gas constant (0.0821 L atm / (mol K)) T = temperature (K) For the initial state (with only hydrogen), the equation looks like: P_initial * V = n_H2 * R * T

Write down the Ideal Gas Law for the final state

For the final state (with both hydrogen and nitrogen), the equation looks like: P_final * V = (n_H2 + n_N2) * R * T, where n_N2 = moles of nitrogen gas added.

Eliminate the volume (V) by dividing the two equations

Dividing the final state's equation by the initial state's equation: (P_final * V) / (P_initial * V) = (n_H2 + n_N2) * R * T / (n_H2 * R * T) The volume (V) and ideal gas constant (R) and temperature (T) terms cancel out: (P_final / P_initial) = (n_H2 + n_N2) / n_H2

Solve for the moles of nitrogen gas added (n_N2)

Now, we can solve for n_N2: n_N2 = n_H2 * ( (P_final / P_initial) - 1) Use the given values for n_H2, P_initial, and P_final: n_N2 = 1.35 * ( (1.64 / 1.05) - 1) n_N2 ≈ 0.801 mol Therefore, approximately 0.801 moles of nitrogen gas were added.

Key concepts

Chemical Reactions

Understanding chemical reactions is fundamental in the study of chemistry. They involve the transformation of substances through the breaking and forming of chemical bonds, resulting in the production of new substances with different properties. During a reaction, atoms are rearranged, but the total number of atoms of each element remains constant due to the Law of Conservation of Mass.

When dealing with gases, reactions often involve changes in volume, pressure, and temperature. These changes relate directly to the behavior described by gas laws. In the context of chemical reactions involving gases, it’s essential to understand stoichiometry, which provides the quantitative relationship between reactants and products. These relationships allow chemists to predict the amounts of products that will form under given conditions, as seen in our exercise where the addition of nitrogen gas changes the pressure in a flask.

Gas Laws

Gas laws are crucial for understanding the behavior of gases under different conditions of temperature, pressure, and volume.

One of the fundamental gas laws is the Ideal Gas Law, expressed as \( PV = nRT \), where \( P \) represents pressure, \( V \) the volume, \( n \) the amount of gas in moles, \( R \) the ideal gas constant, and \( T \) the absolute temperature in Kelvin.

The Ideal Gas Law is based on the assumptions that the molecules of a gas occupy negligible space and do not have intermolecular forces. It allows us to calculate the number of moles of gas in a container, as long as the other three variables are known. In our textbook exercise, we see it in action as we figure out the number of moles of nitrogen gas added to a flask containing hydrogen gas, thereby changing the pressure.

Stoichiometry

Stoichiometry is the section of chemistry that pertains to the calculation of the quantities of reactants and products in chemical reactions. It is based on the balanced chemical equation and the mole concept.

Stoichiometry enables us to predict how much reactants are needed to form a certain amount of product or how much product will form from given amounts of reactants. For gas-law problems involving reactions, stoichiometry helps relate the volume, pressure, and temperature of gases before, during, and after a reaction.

Returning to our exercise, once we know how many moles of hydrogen were present initially and the pressure changes, stoichiometry, combined with the Ideal Gas Law, allows us to calculate the number of moles of nitrogen gas added, by assuming that the temperature remains constant and the gases behave ideally.