Question

Nitrogen monoxide reacts with oxygen to give nitrogen dioxide: $$ 2 \mathrm{NO}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g}) \rightarrow 2 \mathrm{NO}_{2}(\mathrm{g}) $$ (a) Place the three gases in order of increasing rms speed at \(298 \mathrm{K}\) (b) If you mix \(\mathrm{NO}\) and \(\mathrm{O}_{2}\) in the correct stoichiometric ratio and NO has a partial pressure of \(150 \mathrm{mm} \mathrm{Hg}\) what is the partial pressure of \(\mathrm{O}_{2} ?\) (c) After reaction between \(\mathrm{NO}\) and \(\mathrm{O}_{2}\) is complete, what is the pressure of \(\mathrm{NO}_{2}\) if the NO originally had a pressure of \(150 \mathrm{mm} \mathrm{Hg}\) and \(\mathrm{O}_{2}\) was added in the correct stoichiometric amount?

Short answer

(a) NO\(_2\), O\(_2\), NO; (b) 75 mmHg; (c) 150 mmHg.

Step-by-step solution

Calculations for RMS Speed Order

To identify the order of the root-mean-square (RMS) speed of gases, we use the formula \[ v_{\text{rms}} = \sqrt{\frac{3RT}{M}} \] where \(R\) is the gas constant, \(T\) is the temperature, and \(M\) is the molar mass. Gases with lower molar mass will have higher RMS speeds because \(v_{\text{rms}}\) is inversely proportional to the square root of \(M\). Calculate molar masses: \(\text{Molar mass of NO} = 30 \text{ g/mol}\), \(\text{Molar mass of } O_2 = 32 \text{ g/mol}\), \(\text{Molar mass of NO}_2 = 46 \text{ g/mol}\). Thus, the order from slowest to fastest is NO\(_2\) (slowest), O\(_2\), NO (fastest).

Stoichiometric Ratio Calculation for \(\mathrm{O}_{2}\) Partial Pressure

For the reaction \(2\, \mathrm{NO}(\mathrm{g}) + \mathrm{O}_2(\mathrm{g}) \rightarrow 2\, \mathrm{NO}_2(\mathrm{g})\), the stoichiometric ratio is 2 moles of NO to 1 mole of \(\mathrm{O}_{2}\). Given the partial pressure of NO is \(150 \mathrm{mmHg}\), it implies \(2y = x = 150 \mathrm{mmHg}\) where \(y\) is the partial pressure of \(\mathrm{O}_{2}\) and \(x\) is \(2\cdot\) pressure of NO. So, \(y = \frac{1}{2}x = \frac{1}{2} \cdot 150 = 75 \mathrm{mmHg}\).

Calculation of \(\mathrm{NO}_{2}\) Pressure

For every 2 moles of NO reacting with 1 mole of \(\mathrm{O}_2\), 2 moles of \(\mathrm{NO}_2\) are formed. Given the initial partial pressure of NO is \(150 \,\mathrm{mmHg}\) and \(\mathrm{O}_{2}\) is \(75 \mathrm{mmHg}\), they react completely in a 2:1 ratio to produce an equal mole of \(\mathrm{NO}_2\). Thus, \(\text{Final Pressure of } \mathrm{NO}_2 = \text{Initial Pressure of NO} = 150 \,\mathrm{mmHg}\).

Key concepts

RMS Speed

Root-mean-square (RMS) speed is the measure of the speed of particles in a gas, which reflects the average velocity of the gas particles. This concept becomes crucial when understanding gas reactions. The formula for RMS speed is given by: \[ v_{\text{rms}} = \sqrt{\frac{3RT}{M}} \] where:

• \( R \) is the gas constant,
• \( T \) is the temperature in Kelvin,
• \( M \) is the molar mass of the gas in kilograms per mole.

Because RMS speed is inversely proportional to the square root of the molar mass, gases with lighter molar masses will move faster. In case of the gases involved in this exercise: NO, \( O_2 \), and \( NO_2 \), the RMS speed order from fastest to slowest at a given temperature of \( 298 \) K would be: NO > \( O_2 \) > \( NO_2 \). Understanding the relationship between RMS speed and molar mass helps predict how gases will behave under various conditions, which is crucial for mastering gas reactions.

Stoichiometry

Stoichiometry is the calculation of reactants and products in chemical reactions, allowing chemists to predict the outcomes of reactions. It is based on balanced chemical equations where the coefficients represent the molar ratios of each substance involved. In the reaction between nitrogen monoxide (NO) and oxygen \((O_2)\), the balanced equation is:\[ 2 \mathrm{NO}(\mathrm{g}) + \mathrm{O}_2(\mathrm{g}) \rightarrow 2 \mathrm{NO}_2(\mathrm{g}) \]From this equation, we derive the stoichiometric ratio of 2 moles of NO for every 1 mole of \( O_2 \). To find the partial pressure of \( O_2 \), note that NO has an initial partial pressure of \( 150 \mathrm{mmHg} \). Using the stoichiometric ratio, the partial pressure of \( O_2 \) would be half of NO's, which computes as: \\( y = \frac{1}{2} \times 150 = 75 \mathrm{mmHg} \). This calculation is vital for determining how much \( O_2 \) is needed for complete reaction with NO, a key step in predicting the outcome of reactions involving gases.

Partial Pressure

Partial pressure refers to the pressure that a single component of a mixture of gases contributes to the total pressure of the mixture. Imagine a sealed container where different gases are mixed; each gas exerts pressure as if it were alone in the entire volume of the container. In the reaction of NO with \( O_2 \), assigning partial pressures helps ensure the correct proportions of reactants. Initially, NO is given with a partial pressure of \( 150 \mathrm{mmHg} \), and using stoichiometry, we determine that \( O_2 \) should have a partial pressure of \( 75 \mathrm{mmHg} \). After the reaction is complete, the total pressure due to \( NO_2 \) formed can be directly related to the initial conditions. The complete reaction of NO and \( O_2 \) produces an equivalent pressure of \( NO_2 \) which is \( 150 \mathrm{mmHg} \), given both gases were initially mixed in the required stoichiometric ratio. Understanding these individual contributions simplifies the prediction of gaseous reaction outcomes.

Chemical Kinetics

Chemical kinetics explores the rates of chemical processes and the factors that influence these rates. This aspect of chemistry examines how reactants are transformed into products over time, which is central to manipulating reaction conditions for desired outcomes.Involving gas reactions like the one between NO and \( O_2 \), kinetics would consider how quickly \( NO_2 \) forms under specified conditions. While the problem doesn't explicitly detail kinetic data, knowing the stoichiometry, it's assumed that once mixed in proper ratio and under suitable conditions, the reaction proceeds to completion quickly.The rate at which \( NO_2 \) is formed can be influenced by temperature, concentration of reactants, and the presence of a catalyst. Although this specific exercise doesn't require kinetic calculations, understanding these concepts helps predict reaction dynamics and could aid in optimizing reaction conditions in practical applications, like pollution control where NO and \( NO_2 \) are concerned.