At \(25^{\circ} \mathrm{C}\), the equilibrium partial pressures of \(\mathrm{NO}_{2}\) and \(\mathrm{N}_{2} \mathrm{O}_{4}\) are \(0.15 \mathrm{~atm}\) and \(0.20 \mathrm{~atm}\), respectively. If the volume is doubled at constant temperature, calculate the partial pressures of the gases when a new equilibrium is established.

Understanding Le Chatelier's Principle is a fundamental part of grasping chemical equilibrium. This principle states that if a dynamic equilibrium is disturbed by changing the conditions, the position of equilibrium will move to counteract the change. This is a natural attempt to restore equilibrium in a system.

In context to the given exercise, increasing volume decreases the pressure of the system. According to Le Chatelier's Principle, the system will respond by favoring the direction that produces more moles of gas, thus increasing the pressure. In our chemical equation, this means shifting to the side with more gas particles, which is from dinitrogen tetroxide (2O_4) to nitrogen dioxide (2).

The reaction quotient, Q, plays a pivotal role in predicting the direction of a reaction. It is calculated using the same formula as the equilibrium constant (_p), but with the initial concentrations or partial pressures. The value of Q can be compared with the equilibrium constant (_p), to predict which way the reaction will proceed to reach equilibrium.

If Q is less than K, the forward reaction is favored to reach equilibrium, producing more products. Conversely, if Q is greater than K, the reverse reaction is favored, producing more reactants. In the volume doubling scenario described in the exercise, Q will change as the initial partial pressures change. This will influence the direction the reaction shifts to achieve a new equilibrium state.

The ideal gas law is a crucial equation in understanding the behavior of gases under various conditions. It is represented as V = nRT( = Pressure, V = Volume, n = Amount of substance in moles, R = Ideal gas constant, T = Temperature in Kelvin). This equation shows that for a fixed amount of gas at constant temperature, the pressure is inversely proportional to the volume.

When applied to the exercise, the doubling of volume leads to an initial halving of the equilibrium partial pressures as per the ideal gas law. This change creates a system no longer at equilibrium, necessitating a shift in the position of equilibrium to re-establish the balance between reactants and products.

Equilibrium partial pressures refer to the pressures exerted by individual gases in a mixture at equilibrium. The partial pressure of each gas is proportional to its mole fraction at a constant volume and temperature.

When the system is at equilibrium, the partial pressures do not change unless the system is disturbed. In the example provided, the disturbance is the change in volume. To restore equilibrium, the reaction shifts, altering the amounts of reactants and products, and hence, their respective partial pressures. The resulting equilibrium partial pressures will reflect the new balance the system has reached after the shift induced by the volume change. Solving algebraically for these pressures requires the use of the reaction quotient and stoichiometry of the balanced equation.