Question

At \(25^{\circ} \mathrm{C}, K_{\mathrm{p}} \approx 1 \times 10^{-31}\) for the reaction $$ \mathrm{N}_{2}(g)+\mathrm{O}_{2}(g) \rightleftharpoons 2 \mathrm{NO}(g) $$ a. Calculate the concentration of \(\mathrm{NO}\), in molecules \(/ \mathrm{cm}^{3}\), that can exist in equilibrium in air at \(25^{\circ} \mathrm{C}\). In air, \(P_{\mathrm{N}_{2}}=0.8 \mathrm{~atm}\) and \(P_{\mathrm{o},}=0.2\) atm. b. Typical concentrations of NO in relatively pristine environments range from \(10^{8}\) to \(10^{10}\) molecules \(/ \mathrm{cm}^{3}\). Why is there a discrepancy between these values and your answer to part a?

Short answer

The calculated concentration of NO at equilibrium in air at 25°C is approximately \(8.1 \times 10^7\) molecules/cm³. However, typical concentrations of NO in relatively pristine environments range from \(10^8\) to \(10^{10}\) molecules/cm³. The discrepancy between the calculated value and observed values could be due to factors such as incomplete reactions, local variations of reactants' concentrations, and simplifications when considering equilibrium, as well as assumptions made under the ideal gas law.

Step-by-step solution

Write the reaction and equilibrium expression

We are given the reaction: \[ \mathrm{N}_{2}(g) + \mathrm{O}_{2}(g) \rightleftharpoons 2 \mathrm{NO}(g) \] The equilibrium expression for this reaction is given by: \[ K_p = \frac{P_{NO}^2}{P_{N2} \cdot P_{O2}} \]

Calculate the partial pressures of N2 and O2

We are given the partial pressures as: \(P_{N2} = 0.8\) atm \(P_{O2} = 0.2\) atm

Calculate the equilibrium concentration of NO

We know the value of \(K_p\): \[ K_p = 1 \times 10^{-31} \] Now we can substitute the values of \(K_p\), \(P_{N2}\), and \(P_{O2}\) into the equilibrium expression: \[ 1 \times 10^{-31} = \frac{P_{NO}^2}{0.8 \cdot 0.2} \] Solve for \(P_{NO}\): \[ P_{NO}^2 = 10^{-31} \times 0.8 \times 0.2 \] \[ P_{NO} = \sqrt{1.6 \times 10^{-32}} \] \[ P_{NO} \approx 4 \times 10^{-17} \mathrm{~atm} \]

Convert the partial pressure of NO to concentration in molecules/cm³

Now we need to convert the partial pressure of NO into concentration using the ideal gas law equation in terms of concentration: \[ P = n \frac{RT}{V} \] where \(n\) = number of moles \(R\) = gas constant = \(8.314 \times 10^{-3} \frac{L \cdot atm}{K \cdot mol}\) (in the units we need) \(T\) = temperature in Kelvin (\(273.15 + 25 = 298.15 K\)) \(V\) = volume in L Since we need concentration in molecules/cm³, let's convert the units: \[ 1 \mathrm{mol} = 6.022 \times 10^{23} \mathrm{molecules} \] \[ 1 \mathrm{L} = 1000 \mathrm{cm}^3 \] Now we can multiply the pressure by the conversion factor: \[ \frac{1 \mathrm{mol}}{R \cdot T} \times \frac{6.022 \times 10^{23} \mathrm{molecules}}{\mathrm{mol}} \times \frac{1000\mathrm{ ~cm}^{3}}{\mathrm{L}} \] Plug in the values and solve for concentration: \[ C_{NO} = P_{NO} \times \frac{1}{8.314 \times 10^{-3} \frac{L \cdot atm}{K \cdot mol}} \times \frac{1}{298.15K} \times 6.022 \times 10^{23} \mathrm{molecules/mol} \times \frac{1000\mathrm{ ~cm}^{3}}{\mathrm{L}} \] \[ C_{NO} \approx 8.1 \times 10^{7} \mathrm{molecules/cm^3} \]

Discuss the discrepancy between our calculated value and typical concentrations in pristine environments

From our calculation, we found that the concentration of NO at equilibrium is about \(8.1 \times 10^7\) molecules/cm³. However, typical concentrations of NO in relatively pristine environments range from \(10^8\) to \(10^{10}\) molecules/cm³. The discrepancy between the calculated value and the observed value might be due to various factors: incomplete reactions, local variations of the concentrations of reactants, and the simplifications made when considering equilibrium. Equilibrium predictions assume that the system has reached a stable state and does not account for any transitory chemical reactions that might cause deviations in the observed values. Additionally, our calculation assumes ideal conditions as per the ideal gas law, which might not apply to real, non-ideal environmental systems.

Key concepts

Equilibrium Constant

In the realm of chemical reactions, the equilibrium constant, represented as \( K_p \), plays a critical role in determining how readily products form from reactants under equilibrium conditions. For a reaction in a gaseous equilibrium, like the formation of nitric oxide (NO) from nitrogen (\( \mathrm{N}_2 \)) and oxygen (\( \mathrm{O}_2 \)), \( K_p \) is expressed specifically in terms of partial pressures. It is calculated using the formula: \[ K_p = \frac{P_{\text{products}}}{P_{\text{reactants}}} \]Here, for the reaction \( \mathrm{N}_2 + \mathrm{O}_2 \rightleftharpoons 2 \mathrm{NO} \), it results in:\[ K_p = \frac{P_{NO}^2}{P_{N2} \cdot P_{O2}} \]
The equilibrium constant signifies the position of equilibrium and the extent to which a reaction proceeds. Its small value, such as \( 1 \times 10^{-31} \), indicates that the formation of products (NO) is not favored, and the reactants (\( \mathrm{N}_2 \) and \( \mathrm{O}_2 \)) predominantly remain unchanged at equilibrium. Understanding this concept aids students in predicting and calculating concentrations in gaseous reactions.

Partial Pressure

Within a mixture of gases, each type of gas exerts its own pressure, known as partial pressure. These partial pressures are a critical component in calculating the equilibrium state of a gaseous reaction like the production of nitric oxide (NO). Partial pressure is defined as the hypothetical pressure of a gas if it alone occupied the entire volume of the mixture, and is part of Dalton's Law. For example, in air with \( \mathrm{N}_2 \) and \( \mathrm{O}_2 \), their partial pressures are given as 0.8 atm and 0.2 atm, respectively.
Partial pressure is calculated using the mole fraction of the gas and the total pressure:\[ P_{\text{gas}} = \text{mole fraction} \times \text{total pressure} \]These values are essential when using the equilibrium constant formula to calculate the pressure and thus the concentration of products or reactants in the reaction.

• The total pressure is the sum of all partial pressures in the mixture.
• Each gas contributes independently of the others.

In summary, understanding and accurately determining the partial pressures enables scientists and students alike to explore and predict the behavior of chemical reactions at equilibrium.

Ideal Gas Law

The ideal gas law is a fundamental equation that links the physical properties of gases. It can be expressed as:\[ PV = nRT \]Where:

• \( P \) = pressure
• \( V \) = volume
• \( n \) = number of moles
• \( R \) = ideal gas constant
• \( T \) = temperature in Kelvin

In the case of calculating the concentration of nitric oxide in molecules/cm³, this equation is transformed to find concentration (C):\[ C = \frac{P \, \text{(in atm)}}{R \, T} \times 6.022 \times 10^{23} \, \frac{\text{molecules}}{\text{mol}} \times \frac{1000 \, \text{cm}^3}{\text{L}} \]By inserting the pressure \( P_{NO} \) derived from the equilibrium constant, and conditions of temperature and the gas constant, the concentration is computed. In this exercise, it served to find the final concentration of NO at a given temperature and pressure under ideal conditions.This foundational knowledge is indispensable for anyone diving into the study of gaseous chemical reactions.

Nitric Oxide Concentration

Nitric oxide (NO), being a diatomic molecule, can appear in various chemical contexts, often with implications in environmental science. In equilibrium chemistry, determining the concentration of NO is significant, as it dictates its availability and role in atmospheric or industrial settings. Calculate its concentration by first establishing its partial pressure using the equilibrium constant equation and known reactant pressures. Once this value is obtained, the ideal gas law allows conversion from pressure to molecular concentration, typically expressed in molecules per cubic centimeter (molecules/cm³).
This conversion accommodates contextual units crucial for real-world applications, such as assessing atmospheric NO levels and comparing them to typical or acceptable environmental conditions.

• Calculated NO concentration illustrates the balance between formation and breaking down of NO under given conditions.
• Discrepancies between calculated and real-world observed values may indicate non-ideal conditions, measurement variability, or dynamic environmental processes.

Understanding these variations helps in correlating theoretical chemistry with practical environmental chemistry.