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 NO, in molecules/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 \(P_{\mathrm{O}_{2}}=0.2 \mathrm{atm}\). b. Typical concentrations of NO in relatively pristine environments range from \(10^{8}\) to \(10^{10}\) molecules/cm \(^{3}\). Why is there a discrepancy between these values and your answer to part a?

Short answer

In air at 25°C, the calculated equilibrium concentration of NO is found using the Ideal Gas Law, the given pressures of N2 and O2, and the given Kp value. The result is a very low concentration of NO, which is significantly lower than the typical concentrations found in relatively pristine environments (10^8 to 10^10 molecules/cm^3). The discrepancy is likely due to additional processes - such as kinetically controlled reactions - that prevent the system from reaching equilibrium under actual environmental conditions.

Step-by-step solution

Convert Pressures into Concentrations

To calculate the concentrations of N2 and O2 in air at 25°C, use the Ideal Gas Law equation (PV=nRT), express the number of moles per volume (n/V), and rearrange. The Ideal Gas Law in terms of concentration (C) is: $$C = \dfrac{P}{RT}$$ where P is the pressure in atm, R is the gas constant (0.0821 L·atm/mol·K), and T is the temperature in Kelvin. Convert 25°C to Kelvin: 25°C + 273.15 = 298.15 K Now, use the Ideal Gas Law equation to calculate the concentrations of N2 and O2 in air: $$C_{\mathrm{N}_{2}} = \dfrac{P_{\mathrm{N}_{2}}}{RT} = \dfrac{0.8\,\mathrm{atm}}{(0.0821\,\mathrm{L\cdot atm/mol\cdot K})(298.15\,\mathrm{K})}$$ $$C_{\mathrm{O}_{2}} = \dfrac{P_{\mathrm{O}_{2}}}{RT} = \dfrac{0.2\,\mathrm{atm}}{(0.0821\,\mathrm{L\cdot atm/mol\cdot K})(298.15\,\mathrm{K})}$$

Write the Equilibrium Expression and Calculate the Concentration of NO

The equilibrium expression for the given reaction is: $$K_{\mathrm{p}}=\frac{(P_{\mathrm{NO}})^2}{P_{\mathrm{N}_{2}}P_{\mathrm{O}_{2}}}$$ Given that Kp = 1 × 10^{-31}: $$1 \times 10^{-31}= \dfrac{(P_{\mathrm{NO}})^2}{(0.8\,\mathrm{atm})(0.2\,\mathrm{atm})}$$ Now, solve for P_NO (the pressure of NO): $$P_{\mathrm{NO}} = \sqrt{(1 \times 10^{-31} \times (0.8\mathrm{atm})(0.2\mathrm{atm}))}$$ Calculate the concentration of NO in equilibrium: $$C_{\mathrm{NO}} = \dfrac{P_{\mathrm{NO}}}{RT}$$

Compare the Calculated Equilibrium Concentration with Typical Concentrations in Pristine Environments

Compare the calculated equilibrium concentration of NO in air with the typical concentrations of NO in relatively pristine environments, which range from 10^8 to 10^10 molecules/cm^3.

Explain the Discrepancy

If there is a discrepancy between the calculated equilibrium concentration and the typical concentrations found in pristine environments, it might be due to other factors such as kinetically controlled processes that could prevent the reaction from reaching equilibrium in those environments.

Key concepts

Chemical Equilibrium

Chemical equilibrium is a fundamental concept in the study of chemical reactions. It is the state where the rate of the forward reaction equals the rate of the reverse reaction, and as a result, the concentrations of the reactants and products remain constant over time. Despite this balance of reactions, it doesn't mean the reactants and products are in equal concentrations, but rather that their ratios remain stable. Understanding equilibrium is crucial for predicting how changes in conditions, like temperature and pressure, will affect the composition of a reaction mixture.

In chemical equilibrium, the system is dynamic meaning that the reactants and products are continuously converted back and forth, but because their rates are identical, no net change is observed. This concept is essential when considering atmospheric gases and their reactions, as with nitrogen dioxide (NO), which is in equilibrium with nitrogen (N2) and oxygen (O2) in the air.

Ideal Gas Law

The ideal gas law is an equation of state for a hypothetical ideal gas, which provides a connection between pressure (P), volume (V), temperature (T), and the number of moles of gas (n). The law is commonly expressed as PV = nRT, where R is the universal gas constant. In terms of concentration, the law can be rearranged to express the molar concentration (C) as C = P/RT.

To apply the ideal gas law in calculating equilibrium concentrations, we first need to ensure that temperature is expressed in Kelvin. For instance, to convert 25°C to Kelvin, one would add 273.15, resulting in 298.15 K. By using the ideal gas law, we can then determine the concentration of each gas involved in a reaction based on its partial pressure and the given temperature.

Equilibrium Constant (Kp)

The equilibrium constant (Kp) for gas-phase reactions expresses the relationship between the partial pressures of the gases involved at equilibrium. It is a dimensionless value that quantifies the position of equilibrium for a reaction. The expression for Kp is derived from the balanced chemical equation and takes the form: Kp = (P_products)^coefficients / (P_reactants)^coefficients.

For the reaction involving nitrogen dioxide (NO), nitrogen (N2), and oxygen (O2), we would write Kp as a function of the partial pressures of NO, N2, and O2. Once Kp and the partial pressures of the reactants are known, we can solve for the unknown partial pressure of the product. In our example, the minute value of Kp suggests that, at equilibrium, the amount of NO would be extremely low, in accordance with Le Chatelier's principle.

Nitrogen Dioxide (NO) Concentration

Nitrogen dioxide (NO) concentration is crucial for understanding air quality and pollutant levels. In terms of equilibrium, the NO concentration can be calculated using its partial pressure and the ideal gas law. As NO is involved in equilibrium reactions in the atmosphere, its concentration is particularly sensitive to the conditions present.

When calculating equilibrium concentrations for a gas like NO, it's important to employ the ideal gas law in conjunction with the equilibrium constant expression. NO concentration is not only a matter of academic curiosity but is also key for environmental monitoring and regulatory purposes because changes in its concentration can indicate shifts in environmental conditions or pollutant levels.

Partial Pressure

Partial pressure is the pressure exerted by a single gas in a mixture of gases. Each gas in a mixture behaves independently and contributes to the total pressure of the system. The partial pressure is directly proportional to the mole fraction of the gas and the total pressure of the gas mixture.

In equilibrium calculations, partial pressures are used in the expression for the equilibrium constant (Kp). For a gas-phase reaction, knowing the partial pressures of reactants can allow us to determine the partial pressure of products at equilibrium. It's important to note that the ideal gas law assumes that these gases do not interact with each other and that they behave ideally. However, in the real world, we often find that other factors interfere with this assumption, leading to discrepancies between calculated and observed values.