To calculate the value of $K_p$ for the reaction:

$\text{CH}_4(g) + \text{O}_2(g) \rightleftharpoons 2\text{CO}(g) + 2\text{H}_2(g)$

given that $K_p = 4.5 \times 10^2$ at 825 K, we first recognize that the equilibrium constant $K_p$ is already provided. So if you're looking for the equilibrium expression involving partial pressures, it is written as:

$K_p = \frac{P_{CO}^2 \cdot P_{H_2}^2}{P_{CH_4} \cdot P_{O_2}}$

Where:

• $P_{CO}$ is the partial pressure of carbon monoxide (CO),
• $P_{H_2}$ is the partial pressure of hydrogen (H$_2$),
• $P_{CH_4}$ is the partial pressure of methane (CH$_4$),
• $P_{O_2}$ is the partial pressure of oxygen (O$_2$).

Since $K_p$ is already provided as $4.5 \times 10^2$, this represents the ratio of the products' partial pressures to the reactants' partial pressures at equilibrium under the given temperature of 825 K.

Would you like to calculate or solve anything further based on this, such as solving for individual partial pressures? Or clarifications on the equilibrium expression?

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Follow-up Questions:

1. Would you like to solve for partial pressures of individual gases in this reaction?
2. Do you need to calculate $K_c$ from $K_p$, considering temperature and the ideal gas constant?
3. Should we explore how changes in temperature would affect the $K_p$ value?
4. Do you need help with understanding Le Chatelier's principle applied to this reaction?
5. Are there any other equilibrium concepts you'd like to explore?

Tip:

Remember, for reactions involving gases, the relation between $K_p$ and $K_c$ is governed by the equation $K_p = K_c(RT)^{\Delta n}$, where $\Delta n$ is the change in the number of moles of gas.