Question

For the reaction, \(2 \mathrm{NO}_{2}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{NO}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{~g})\) \(\left(\mathrm{K}_{\mathrm{c}}=1.8 \times 10^{-6}\right.\) at \(\left.184^{\circ} \mathrm{C}\right)\) \((\mathrm{R}=0.0831 \mathrm{~kJ} /(\mathrm{mol} \mathrm{K}))\) when \(\mathrm{K}_{\mathrm{p}}\) and \(\mathrm{K}_{\mathrm{c}}\) are compared at \(184^{\circ} \mathrm{C}\) it is found that: (a) \(\mathrm{K}_{\mathrm{p}}\) is greater than \(\mathrm{K}_{\mathrm{c}}\) (b) \(\mathrm{K}_{\mathrm{p}}\) is less than \(\mathrm{K}_{\mathrm{c}}\) (c) \(K_{p}=K_{c}\) (d) Whether \(\mathrm{K}_{\mathrm{p}}\) is greater than, less than or equal to \(\mathrm{K}_{\mathrm{c}}\) depends upon the total gas pressure

Short answer

(a) \( \mathrm{K}_{\mathrm{p}} \) is greater than \( \mathrm{K}_{\mathrm{c}} \).

Step-by-step solution

Recall Kp and Kc Relationship Formula

To compare \( K_p \) and \( K_c \), we use the equation \( K_p = K_c (RT)^{\Delta n} \) where \( \Delta n \) is the change in moles of gas between products and reactants. \( R \) is the gas constant and \( T \) is the temperature in Kelvin.

Calculate Δn

The balanced reaction is \( 2 \text{NO}_2(g) \rightleftharpoons 2 \text{NO}(g) + \text{O}_2(g) \). Count the moles of gases: Products have \( 2 + 1 = 3 \) moles, Reactants have \( 2 \) moles, so \( \Delta n = 3 - 2 = 1 \).

Convert Temperature to Kelvin

Convert the given temperature from Celsius to Kelvin: \( 184^{\circ} \mathrm{C} = 184 + 273.15 = 457.15 \text{ K} \).

Substitute Values into Kp and Kc Formula

Substitute the known values into the equation: \( K_p = K_c \cdot (RT)^{\Delta n} = 1.8 \times 10^{-6} \cdot (0.0831 \cdot 457.15)^{1} \).

Calculate (RT)

Calculate \( RT \): \( 0.0831 \cdot 457.15 = 37.961165 \).

Find Kp

Now calculate \( K_p = 1.8 \times 10^{-6} \times 37.961165 = 6.8328 \times 10^{-5} \).

Compare Kp and Kc

Compare \( K_p = 6.8328 \times 10^{-5} \) with \( K_c = 1.8 \times 10^{-6} \). Since \( 6.8328 \times 10^{-5} > 1.8 \times 10^{-6} \), \( K_p \) is greater than \( K_c \).

Key concepts

Reaction Quotient

The reaction quotient, often referred to by the symbol \( Q \), is a vital concept in understanding chemical equilibria. To grasp the difference between \( K \) and \( Q \), remember that \( K \) is the equilibrium constant, characterizing the ratio of product and reactant concentrations at equilibrium. On the other hand, \( Q \) can be thought of as a snapshot, reflecting this ratio at any given moment—not necessarily at equilibrium.

When you compare \( Q \) and \( K \):

• If \( Q < K \): The reaction must shift to the right, towards the products, to reach equilibrium.
• If \( Q > K \): The reaction needs to shift to the left, towards the reactants, to achieve equilibrium.
• If \( Q = K \): The system is at equilibrium.

Thus, by calculating \( Q \), you can determine the direction a reaction will proceed to achieve equilibrium. This becomes crucial in chemical analysis and predicting how changes in concentration can affect reaction dynamics.

Le Chatelier's Principle

Le Chatelier's Principle is a powerful tool for predicting how a chemical system at equilibrium will respond to changes. According to this principle, if a system in equilibrium experiences a disturbance, such as a change in concentration, temperature, or pressure, the system will adjust itself to counteract the effect of the disturbance and restore a new equilibrium.

Consider these scenarios:

• Concentration Changes: Adding more reactants typically shifts the equilibrium towards the products, while removing products encourages the formation of more reactants.
• Pressure Changes (for gases): An increase in pressure will favor the side of the reaction with fewer gas moles. In our example, with a positive \( \Delta n = 1 \), increasing pressure would favor the reactants.
• Temperature Changes: For exothermic reactions, increasing the temperature shifts equilibrium towards the reactants, while for endothermic reactions, it shifts towards the products.

Understanding Le Chatelier's Principle helps predict and control the outcomes of chemical reactions, a fundamental skill in chemistry.

Gas Laws

In the realm of chemistry, gas laws provide critical insights into the behavior of gases. These laws relate the pressure, volume, temperature, and quantity of gas to each other through simple mathematical relationships.

• Ideal Gas Law: Given by \( PV = nRT \), this equation describes the state of an ideal gas. It forms the foundation for calculations involving gases, with \( P \) representing pressure, \( V \) for volume, \( n \) for moles, \( R \) for the gas constant, and \( T \) for temperature in Kelvin.
• Boyle's Law: States that pressure and volume are inversely proportional at constant temperature, or \( P_1V_1 = P_2V_2 \).
• Charles's Law: Explains how volume is directly proportional to temperature at constant pressure, expressed as \( V_1/T_1 = V_2/T_2 \).

In the context of equilibrium constants like \( K_p \), these gas laws help convert between concentration-driven constants (\( K_c \)) and pressure-driven constants by taking into account the changes in moles of gases, as dictated by the formula \( K_p = K_c (RT)^{\Delta n} \). This conversion is crucial in understanding the nature and extent of reactions in gaseous systems.