Question

Calculate the \(K_{\mathrm{P}}\) for this reaction at \(298 \mathrm{~K}\) if the \(\mathrm{Keq}=1.76 \times 10^{-3}\). \(3 \mathrm{O}_{2}(\mathrm{~g}) \rightleftarrows 2 \mathrm{O}_{3}(\mathrm{~g})\)

Short answer

The value of \(K_\mathrm{P}\) at \(298 \text{ K}\) is \(7.22 \times 10^{-5}\).

Step-by-step solution

Identify the Relation Between Kp and Keq

The equilibrium constant for a reaction in terms of pressure, \(K_\mathrm{P}\), is related to the equilibrium constant in terms of concentration, \(K_\mathrm{eq}\), by the equation \(K_\mathrm{P} = K_\mathrm{eq} (RT)^{\Delta n}\). Here, \(R\) is the ideal gas constant (\(0.0821 \ L\cdot atm/K\cdot mol\)) and \(\Delta n\) is the change in moles of gas between reactants and products.

Calculate Delta n

For the given reaction, \(\Delta n\) is the difference in moles of gaseous products and reactants. For \(3 \ O_2(g) \rightleftharpoons 2 \ O_3(g)\), \(\Delta n = (2 - 3) = -1\).

Determine the Temperature in Kelvin

The temperature in Kelvin is given as \(298 \ K\). This value is necessary for our calculations since \(K_\mathrm{P}\) depends on the temperature and pressure.

Plug Values into the Kp Equation

Using the formula \(K_\mathrm{P} = K_\mathrm{eq} \times (RT)^{\Delta n}\), substitute in the known values: \(K_\mathrm{eq} = 1.76 \times 10^{-3}\), \(R = 0.0821\), \(T = 298\) and \(\Delta n = -1\).

Calculate Kp

Substituting the values into the equation: \[K_\mathrm{P} = 1.76 \times 10^{-3} \times (0.0821 \times 298)^{-1} = 1.76 \times 10^{-3} \times 0.0410 = 7.22 \times 10^{-5}\].

Key concepts

Kp and Keq relationship

When tackling problems in chemistry, understanding the relationship between different types of equilibrium constants is crucial. The equilibrium constant in terms of pressure, denoted as \( K_{\mathrm{P}} \), relates to the equilibrium constant in terms of concentration, \( K_{\mathrm{eq}} \), by a specific formula: \( K_{\mathrm{P}} = K_{\mathrm{eq}} (RT)^{\Delta n} \). This equation allows one to convert between these constants depending on what's given in a problem.

**Important Points on \( K_{\mathrm{P}} \) and \( K_{\mathrm{eq}} \):**

• \( R \) is the ideal gas constant, typically 0.0821 L•atm/K•mol.
• \( T \) is the temperature in Kelvin, essential for the calculation due to gas laws.
• \( \Delta n \) represents the change in moles of gas between products and reactants; it is vital in determining how pressure and concentration relate.

By understanding and correctly applying this formula, you can solve a myriad of problems involving gaseous reactions.

Ideal Gas Law

The Ideal Gas Law is an equation of state for a hypothetical gas, and its formula is \( PV = nRT \). While it primarily describes ideal conditions, it also underpins calculations involving gaseous reactions in chemistry. Here’s why the Ideal Gas Law is essential in our calculations:

- **Components of the Ideal Gas Law** include:

• Pressure \( (P) \)
• Volume \( (V) \)
• Number of moles \( (n) \)
• The Ideal Gas Constant \( (R) \), 0.0821 L•atm/K•mol.
• Temperature \( (T) \) in Kelvin.

The law helps explain why gaseous reactions may involve conversions from concentration constants like \( K_{\mathrm{eq}} \) to pressure constants like \( K_{\mathrm{P}} \). This is because \( K_{\mathrm{P}} \) inherently involves pressure as it depends on the total pressure exerted by the gases in the reaction. Understanding these relationships can simplify various chemistry calculations and build a framework for deeper problem-solving.

Calculating Delta n

A crucial step in converting \( K_{\mathrm{eq}} \) to \( K_{\mathrm{P}} \) involves understanding \( \Delta n \). This term represents the change in the number of moles of gaseous products and reactants. In the given reaction 3 \( O_2(g) \) going to 2 \( O_3(g) \), \( \Delta n \) is calculated as follows:

1. Calculate the total moles of gaseous products (which is 2 for \( O_3 \)).
2. Subtract the total moles of gaseous reactants (which is 3 for \( O_2 \)).
3. \( \Delta n = 2 - 3 = -1 \).

**Key Takeaways for \( \Delta n \):**

• \( \Delta n \) essentially measures how the pressure term in a reaction will change when moving from concentrations (\( K_{\mathrm{eq}} \)) to pressures (\( K_{\mathrm{P}} \)).
• It’s crucial to watch out for whether the changes in moles involve gases, as solids and liquids do not count in \( \Delta n \).
• Understanding \( \Delta n \) provides insight into the behavior of gases in chemical reactions.

Chemistry calculations

Completing chemistry calculations, like finding \( K_{\mathrm{P}} \), often seems daunting, but it primarily involves a clear understanding of each term in the relevant equations. Let’s walk through these aspects:

**Steps for Chemistry Calculations:**

• **Understand what each term means.** Variables like \( R \), \( T \), and \( \Delta n \) should be clear in the context of a problem.
• **Plug values into equations accurately.** For example, when calculating \( K_{\mathrm{P}} \) using \( K_{\mathrm{eq}} \) and other variables, carefully substitute each known value into \( K_{\mathrm{P}} = K_{\mathrm{eq}} (RT)^{\Delta n} \).
• **Use the correct units.** Ensure all measurements are in the necessary units, often requiring conversion (e.g., temperature in Kelvin).
• **Double-check calculations.** Calibrate your answers by checking each mathematical step.

These tips reinforce that having structured and organized methodologies makes solving chemistry problems more approachable, leading to success in understanding and tackling complex chemical equilibria.