Problem 15 - Chemistry The Science in Context

Question

Under what conditions are the numerical values of \(K_{\mathrm{c}}\) and \(K_{\mathrm{p}}\) equal?

Step-by-Step Solution

Answer

The numerical values of \(K_{\mathrm{c}}\) and \(K_{\mathrm{p}}\) will be equal when the change in the number of moles of gas (\(\Delta n_g\)) in the balanced chemical equation is zero. This means that the number of moles of gaseous products must equal the number of moles of gaseous reactants.

Step 1: Converting \(K_{\mathrm{c}}\) to \(K_{\mathrm{p}}\)

First, let's recall the relationship between \(K_{\mathrm{c}}\) and \(K_{\mathrm{p}}\) for a general balanced chemical equation: \begin{align*} aA + bB \rightleftharpoons cC + dD \end{align*} The \(K_{\mathrm{c}}\) is the equilibrium constant in terms of concentration, and the \(K_{\mathrm{p}}\) is the equilibrium constant in terms of pressure. The conversion between them involves the change in the number of moles of gas (\(\Delta n_g\)) in the reaction: \begin{align*} K_{\mathrm{p}} = K_{\mathrm{c}} \cdot (RT)^{\Delta n_g} \end{align*} where \(R\) is the universal gas constant (0.08206 L·atm/mol·K) and \(T\) is the temperature in Kelvin. \(\Delta n_g\) is the difference in the number of moles of gaseous products and gaseous reactants: \begin{align*} \Delta n_g = (c + d) - (a + b) \end{align*}

Step 2: Identify when \(K_{\mathrm{c}}\) equals \(K_{\mathrm{p}}\)

Now, let's find the conditions that make \(K_{\mathrm{c}} = K_{\mathrm{p}}\). We have: \begin{align*} K_{\mathrm{p}} = K_{\mathrm{c}} \cdot (RT)^{\Delta n_g} \end{align*} For \(K_{\mathrm{c}}\) to be equal to \(K_{\mathrm{p}}\), the term \((RT)^{\Delta n_g}\) must equal 1. This can only happen when \(\Delta n_g = 0\), since any number to the power of 0 is equal to 1. So, our condition is: \begin{align*} \Delta n_g = 0 \end{align*} This means that the number of moles of gaseous products must equal the number of moles of gaseous reactants. To summarize, the numerical values of \(K_{\mathrm{c}}\) and \(K_{\mathrm{p}}\) will be equal under the condition that the change in the number of moles of gas (\(\Delta n_g\)) in the balanced chemical equation is zero.

Key Concepts

Kc and Kp relationship

In chemistry, we often discuss equilibrium constants to predict the behavior of a chemical reaction at equilibrium. There are two primary equilibrium constants: \( K_\mathrm{c} \) and \( K_\mathrm{p} \).

\( K_\mathrm{c} \) is the equilibrium constant when concentrations of reactants and products are used. Concentrations are typically expressed in molarity (moles per liter). \( K_\mathrm{p} \), on the other hand, is the equivalent constant but for pressures, used when dealing with gaseous reactions. Each of these constants gives us insight into where the equilibrium lies.

The relationship between \( K_\mathrm{c} \) and \( K_\mathrm{p} \) is given by the equation: \[ K_\mathrm{p} = K_\mathrm{c} \cdot (RT)^{\Delta n_g} \]Where:

• \( R \) is the universal gas constant,
  
• \( T \) is the temperature in Kelvin, and
  
• \( \Delta n_g \) is the difference in the moles of gaseous products and reactants.
  

This equation shows that \( K_\mathrm{c} \) and \( K_\mathrm{p} \) can be the same if \( \Delta n_g \) equals zero, meaning no change in moles of gas from reactants to products.

Moles of Gas in Reactions

The concept of moles of gas in reactions is central to understanding equilibrium constants in chemistry. During reactions involving gases, it's pivotal to account for the change in the number of moles of gas to predict how the reaction behaves under different conditions.

In many equilibrium reactions, especially those involving gases, the number of moles of reactants may not equal the number of moles of products. This discrepancy is described by \( \Delta n_g \). The value of \( \Delta n_g \) is found using the equation: \[ \Delta n_g = (c + d) - (a + b) \]Here:

• \( c + d \) represent moles of gaseous products, and
  
• \( a + b \) are moles of gaseous reactants.
  

If \( \Delta n_g \) equals zero, it signifies that there's no net change in the moles of gas, rendering \((RT)^{\Delta n_g} \) equal to 1, which leads to \( K_\mathrm{c} \) equaling \( K_\mathrm{p} \). This simplification proves useful in predicting reactions where gaseous moles remain constant.

Universal Gas Constant

The universal gas constant, \( R \), is a fundamental constant in chemistry, pivotal for calculations involving gases. It connects various gas laws and plays a critical role in the equation for relating \( K_\mathrm{c} \) and \( K_\mathrm{p} \).

\( R \) typically has a value of 0.08206 L·atm/mol·K. This value of \( R \) is especially useful when calculations involve both pressure in atmospheres and volume in liters.

In equilibrium reactions, \( R \) functions as the bridge between the pressure-based and concentration-based equilibrium constants, adjusting for temperature-dependent changes during reactions. The relation of \( K_\mathrm{p} \) to \( K_\mathrm{c} \) through \( R \) highlights the need for accurate temperature and pressure data to correctly predict reaction outcomes.