Question

State the equation relating \(K_{\mathrm{P}}\) to \(K_{\mathrm{c}}\) and define all terms. Which is the only value of \(R\) that can be properly used in this equation?

Short answer

The equation is \(K_{\mathrm{P}} = K_{\mathrm{c}}(RT)^{\Delta n}\). Kp is the equilibrium constant for partial pressure, Kc is for molar concentration, R is the ideal gas constant (0.0821 L atm/mol K when dealing with atm for pressure), T is temperature in Kelvin, and Δn is the change in moles of gas. Only the specific value of R matching the pressure units in Kp should be used.

Step-by-step solution

State the equation relating Kp to Kc

Write down the equation that connects the equilibrium constants for gases, Kp, to the equilibrium constant for concentrations, Kc. This equation is given by: \[K_{\mathrm{P}} = K_{\mathrm{c}}(RT)^{\Delta n}\] where Kp is the equilibrium constant in terms of partial pressure, Kc is the equilibrium constant in terms of molar concentration, R is the ideal gas constant, T is the temperature in Kelvin, and Δn is the change in moles of gas going from reactants to products.

Define all terms

Define each term in the context of the chemical equilibrium: - \(K_{\mathrm{P}}\): Equilibrium constant in terms of partial pressure (dimensionless) - \(K_{\mathrm{c}}\): Equilibrium constant in terms of molar concentration (dimensionless) - \(R\): Ideal gas constant whose value depends on the units used for pressure and can be 0.0821 L atm / (mol K) or 8.314 J / (mol K) - \(T\): Temperature in Kelvin (K) - \(\Delta n\): Change in the number of moles of gas (products minus reactants).

Identify the proper value of R

Identify the value of the ideal gas constant R that is to be used in the equation relating Kp to Kc. Use the value that matches the units of pressure in Kc. For Kp being in atm: \[R = 0.0821 \frac{L \cdot atm}{mol \cdot K}\]. This is the only value of R in atm that can be properly used in the equation when dealing with partial pressures in atmospheres.

Key concepts

Kp to Kc Equation

Understanding the relationship between the equilibrium constants in terms of partial pressure (\(K_{\text{P}}\)) and molar concentration (\(K_{\text{c}}\)) is crucial in chemistry, especially when dealing with gaseous reactions. The equation that bridges these two constants is \[K_{\text{P}} = K_{\text{c}}(RT)^{\Delta n}\]where

• \(K_{\text{P}}\) represents the equilibrium constant calculated using the partial pressures of the gases involved.
• \(K_{\text{c}}\) is the equilibrium constant based on the molar concentrations of reactants and products.
• \(R\) is the ideal gas constant whose value is chosen based on the units of pressure we're working with.
• \(T\) is the absolute temperature in Kelvin.
• \(\Delta n\) indicates the difference in the number of moles of gaseous products and reactants.

The exponent \(\Delta n\) is particularly important, as it reflects the net change in moles of gas throughout the reaction, affecting the relationship between \(K_{\text{P}}\) and \(K_{\text{c}}\). This is why understanding stoichiometry and the mole concept is vital when applying this equation.

Ideal Gas Constant (R)

The ideal gas constant, represented by \(R\), is a fundamental constant in the equations of state that describe the behavior of ideal gases. Its value is determined by the units of measurement used for pressure, volume, and temperature. In the context of converting \(K_{\text{P}}\) to \(K_{\text{c}}\), the ideal gas constant provides a bridge between these terms, effectively taking into account the effects of temperature and the universal properties of gases.

When working with \(K_{\text{P}}\) in atmospheres, the correct value for \(R\) is \(0.0821 \frac{L \cdot atm}{mol \cdot K}\), ensuring that the units for pressure (atm) are consistent with the units used in the equilibrium constant for partial pressures. In this scenario, other values of \(R\) would result in incorrect units and hence wrong calculations.

Molar Concentration

Molar concentration, often known as molarity and symbolized by the term \(C\), refers to the amount of a substance (in moles) dissolved in a unit volume of solution (typically expressed in liters). The mathematical representation is given as \(C = \frac{n}{V}\), where \(n\) is the number of moles of the solute and \(V\) is the volume of the solution in liters.

Molarity is a central concept when discussing the equilibrium constant \(K_{\text{c}}\), which is derived from the concentrations of reactants and products at equilibrium according to their stoichiometry. Understanding how to manipulate molar concentrations is essential for predicting the outcome of reactions and for making the conversion between \(K_{\text{P}}\) and \(K_{\text{c}}\) using the previously mentioned equation.

Partial Pressure

Partial pressure is the pressure that a gas in a mixture would exert if it occupied the entire volume alone at the same temperature. It's a vital concept when gases are involved in chemical equilibria. According to Dalton's law of partial pressures, the total pressure of a gas mixture is the sum of the partial pressures of each individual gas.

Mathematically, the partial pressure \(P_i\) of gas \(i\) can be calculated using the ideal gas law, \(P_i = \frac{n_iRT}{V}\), where \(n_i\) is the number of moles of the gas, \(R\) is the ideal gas constant, \(T\) is the temperature, and \(V\) is the volume.
Partial pressures are used directly in the equilibrium expression for \(K_{\text{P}}\), which is why understanding how to measure and calculate partial pressures is integral for solving equilibrium problems in gaseous systems.