Question

If \(K_{c}=0.013 \mathrm{~L} / \mathrm{mol}\) for \(2 \mathrm{NO}(g)+\mathrm{Br}_{2}(g) \rightleftharpoons 2 \mathrm{NOBr}(g)\) at \(1000 \mathrm{~K}\), what is the value of \(K_{p}\) for this reaction at this temperature?

Short answer

The value of \(K_p\) for the reaction \(2 NO(g) + Br_2(g) \rightleftharpoons 2 NOBr(g)\) at 1000 K is approximately 0.125, given that \(K_c = 0.013 \mathrm{~L} / \mathrm{mol}\).

Step-by-step solution

Determine the value of ∆n

Using the balanced chemical equation, we can determine the change in the number of moles of gaseous products and reactants. \[2 NO (g) + Br_2 (g) \rightleftharpoons 2 NOBr (g)\] The number of moles of gaseous products is 2 (from 2 moles of NOBr), and the number of moles of gaseous reactants is also 2 (from 2 moles of NO and 1 mole of Br2). Thus, \[\Delta n = 2 - (2 + 1) = -1\]

Calculate Kp using the relationship between Kc and Kp

Now that we have the values of \(K_c\), \(\Delta n\), \(R\), and \(T\), we can calculate the value of \(K_p\). \[K_p = K_c(RT)^{\Delta n}\] \[K_p = (0.013)(0.0821 \times 1000)^{-1}\] Now, find the value of \(K_p\): \[K_p = (0.013)(8.21)^{-1}\] \[K_p ≈ 0.125\] Thus, the value of \(K_p\) for this reaction at 1000 K is approximately 0.125.

Key concepts

Equilibrium Constant (Kc and Kp)

In the realm of chemical equilibrium, understanding equilibrium constants is crucial. These constants help us establish the ratio of the concentrations of products to reactants in a reversible chemical reaction, at equilibrium.
Equilibrium constants can be expressed in terms of:

• Concentration ( K_c )
• Partial pressure ( K_p )

Distinguishing Between K_c and K_p

K_c is determined using the molarity of the reactants and products, while K_p is calculated from their partial pressures. The question arises when dealing with reactions involving gases: how do you convert between these two forms? The answer lies in the simple relationship:\[K_p = K_c (RT)^{\Delta n}\]Here, \(R\) is the ideal gas constant (0.0821 L atm/mol K), \(T\) is the temperature in Kelvin, and \(\Delta n\) represents the change in the number of moles of gas (moles of gaseous products - moles of gaseous reactants).
Grasping this conversion helps evaluate the nature and extent of reaction commitment without resorting to complex experimental methods. Understanding the equilibrium constant is essential for exploring the dynamics of chemical reactions and comprehending the balance between reactants and products.

Reaction Quotient

The reaction quotient, denoted as \(Q\), provides a snapshot of the current state of a reaction, allowing us to predict which direction the reaction needs to shift to reach equilibrium. Though it's closely related to the equilibrium constant, \(Q\) is calculated the same way but using the current concentrations or pressures.

Predicting Reaction Shifts

When comparing \(Q\) to \(K\), keep the following in mind:

• If \(Q < K\), the reaction will move forward, converting reactants into products until equilibrium is reached.
• If \(Q > K\), the reaction will shift backward, transforming products back into reactants.
• If \(Q = K\), the system is already at equilibrium, and no shift will occur.

The elegance of the reaction quotient lies in its ability to dynamically predict the progress of a reaction. Always remember that while \(Q\) tells us where the reaction is and what direction it needs to move, \(K\) informs us of where the reaction stands when it's reached equilibrium.

Thermodynamics

Thermodynamics is the fascinating branch of physical science that delves into the relationships between heat, temperature, and energy transformations. It plays a vital role in understanding chemical equilibrium and the behavior of reactions.
Chemical reactions are subject to the laws of thermodynamics, with equilibrium being a state where the forward and backward reaction rates match, leading to constant reactant and product concentrations over time.

The Role of Thermodynamics in Equilibrium

Thermodynamics helps predict equilibrium positions and the spontaneity of reactions:

• The Gibbs free energy change (\(\Delta G\)) indicates the willingness of a process to occur spontaneously. At equilibrium, \(\Delta G\) equals zero.
• The relation \(\Delta G = -RT \ln K\) links thermodynamics with equilibrium. A negative \(\Delta G\) suggests a favorable reaction, while a positive \(\Delta G\) signifies a less favorable one.

Thermodynamics equips us with the theoretical framework to predict whether a given reaction will naturally progress and determine the required conditions for equilibrium. This invaluable insight allows chemists to manipulate conditions for desired product formation efficiently.