Question

Write the equation relating \(K_{\mathrm{c}}\) and \(K_{P}\) and define all the terms.

Short answer

The equation relating \(K_{P}\) and \(K_{\mathrm{c}}\) is \(K_{P} = K_{\mathrm{c}} (RT)^{\Delta n}\), where \(R\) is the ideal gas constant, \(T\) is the temperature (in Kelvin), and \(\Delta n\) is the difference in moles of gaseous products and gaseous reactants in the balanced chemical reaction.

Step-by-step solution

Understand and Define the Terms

The equilibrium constant, \(K\), for a chemical reaction is the ratio of the concentrations (if \(K_{\mathrm{c}}\)) or the partial pressures (if \(K_{P}\)) of the products to the reactants, each raised to a power equal to the stoichiometric coefficient in the balanced chemical reaction.

Write the general equation of \(K_{\mathrm{c}}\)

In the equilibrium constant \(K_{\mathrm{c}}\), the concentrations of products and reactants are used. \(K_{\mathrm{c}}\) is usually given in the form: \[ K_{\mathrm{c}} = \frac{[\text{{Products}}]}{[\text{{Reactants}}]} \]

Write the general equation of \(K_{P}\)

In the equilibrium constant \(K_{P}\), the partial pressures of products and reactants are used. \(K_{P}\) is usually given in the form: \[ K_{P} = \frac{(\text{{Pressure of Products}})}{(\text{{Pressure of Reactants}})} \]

Relate \(K_{P}\) and \(K_{\mathrm{c}}\)

The relationship between \(K_{\mathrm{c}}\) and \(K_{P}\) is given by the equation: \[ K_{P} = K_{\mathrm{c}} (RT)^{\Delta n} \] where \(R\) is the ideal gas constant, \(T\) is the temperature (in Kelvin), and \(\Delta n\) is the difference in moles of gaseous products and gaseous reactants in the balanced chemical reaction.

Key concepts

Kc and Kp relationship

In chemical equilibrium, understanding the different applications of equilibrium constants like \( K_c \) and \( K_p \) is essential. These constants help predict how concentrations or pressures of substances at equilibrium will interact. Specifically, \( K_c \) involves concentrations of substances, while \( K_p \) pertains to their partial pressures. They are intrinsically linked by the formula:\[ K_p = K_c (RT)^{\Delta n}\]Here, \( R \) represents the ideal gas constant, \( T \) is the absolute temperature, and \( \Delta n \) refers to the moles' difference between products and reactants. This formula is crucial because it demonstrates that the relationship between \( K_c \) and \( K_p \) depends on both temperature and the difference in the moles of gases involved. Understanding this relationship allows chemists to switch between the concentration-based and pressure-based perspectives of chemical equilibria according to the situation.

ideal gas constant

The ideal gas constant, symbolized as \( R \), plays an instrumental role in chemistry. This constant is used in various equations, most notably the ideal gas law, \( PV = nRT \). Here, \( P \) represents pressure, \( V \) is volume, \( n \) is the number of moles, and \( T \) is the temperature in Kelvin. The constant \( R \) provides the consistent conversion factor necessary for these variables to interrelate correctly.In the relationship between \( K_c \) and \( K_p \), \( R \) is vital as it allows for the adjustment of the equilibrium constant from concentration terms to pressure terms accounting for temperature and molecular change. It's value is usually 0.0821 L atm/mol K when pressure is measured in atmospheres. This helps ensure calculations remain standardized and comparable in the wide array of chemical reactions analyzed by chemists.

chemical equilibrium

Chemical equilibrium refers to the state where the concentrations or pressures of reactants and products remain constant over time, indicating that the forward and reverse reactions occur at the same rate. This equilibrium does not mean reactions have ceased but that the rate at which reactants convert to products is equal to the rate of the reverse conversion.In a state of equilibrium, understanding both \( K_c \) and \( K_p \) is crucial. The equilibrium constant provides a snapshot of the reaction balance under specific conditions. Without understanding equilibrium, predicting the behavior and distribution of chemical species in a reaction becomes challenging. Knowing the constants helps assess whether a reaction favors the formation of products or reactants, which is indispensable in industrial and laboratory settings.

partial pressure

Partial pressure is the pressure exerted by each individual gas in a mixture of non-reactive gases. It is a concept central to the understanding of gaseous equilibria. The total pressure of the mixture is the sum of the partial pressures of each individual gas. This is expressed mathematically by Dalton’s Law of Partial Pressures.In chemical equilibria involving gases, \( K_p \) is particularly useful because it utilizes the partial pressures of reactants and products at equilibrium. Calculations involving partial pressures require maintaining standard conditions and clear data on volume and moles of gases involved. These details enable accurate predictions and assessments of reaction direction and extent, especially for reactions occurring in closed systems, common in laboratory and industrial processes.