One mole of \(\mathrm{N}_{2}\) and three moles of \(\mathrm{H}_{2}\) are placed in a flask at \(375^{\circ} \mathrm{C}\). Calculate the total pressure of the system at equilibrium if the mole fraction of \(\mathrm{NH}_{3}\) is \(0.21 .\) The \(K_{P}\) for the reaction is \(4.31 \times 10^{-4}\).

Let's dive into the fascinating world of gas laws by starting with Dalton's Law of Partial Pressures. This law is pivotal when dealing with mixtures of gases, as it helps us understand how individual gases within the mixture behave. Dalton's Law states that in a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of individual gases.

Imagine you have a container filled with different types of marbles, each type representing a different gas. The pressure each type of marble exerts on the container represents the partial pressure of each gas. Dalton's Law tells us that the total pressure is just like counting all the pushes from each type of marble combined.

The magic of this law lies in its simplicity and its principle which is essential for calculating the equilibrium total pressure in chemical reactions involving gas-phase reactants and products.

The mole fraction is a way of expressing the concentration of a component in a mixture. It's essentially a ratio that compares the number of moles of one substance to the total number of moles in the mixture. Think of it like slicing a pie—the mole fraction tells you the size of one slice compared to the whole pie.

To calculate this, you would divide the number of moles of the substance by the sum of the moles of all components in the mixture. This dimensionless number is crucial when applying Dalton's Law to find out the pressure each gas contributes in a mixture. More importantly, when dealing with chemical equilibria, the mole fraction indicates how much of a substance is present at equilibrium, which is a key factor in calculating the total pressure in a system.

The equilibrium constant, denoted as K or Kc for concentrations and Kp for partial pressures, is a value that represents the ratio of product concentrations or partial pressures to reactant concentrations or partial pressures, each raised to the power of their stoichiometric coefficients in the balanced equation. It's a bit like a score that tells us how much the game of a chemical reaction favors the products or the reactants.

In the context of gases, we use Kp to capture the relationship between partial pressures of the reactants and products when the reaction reaches a stable point, known as equilibrium. For the reaction in our exercise, the equilibrium constant expression helps to relate the given Kp value to the partial pressures of 2, 3, and 3, leading us towards the total pressure of the system at equilibrium.

Chemical equilibrium is a state in a chemical reaction where the rate of the forward reaction equals the rate of the reverse reaction, resulting in no net change in the concentrations of reactants and products over time. We can picture it as two teams of equal strength playing tug-of-war; neither side wins because they are perfectly matched.

At chemical equilibrium, the reaction doesn't stop; rather, it continues to proceed in both directions at the same rate. Understanding equilibrium is not just about the point of balance, but also how the conditions of the reaction, such as pressure and temperature, affect the position of the equilibrium, which in turn influences the calculation of the total pressure in the system. Different conditions can shift the position of the equilibrium, impacting the mole fractions and consequently the partial pressures of the involved gases.