Problem 16 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

For a zero-order reaction: \(2 \mathrm{NH}_{3}(\mathrm{~g})\) \(\rightarrow \mathrm{N}_{2}(\mathrm{~g})+3 \mathrm{H}_{2}(\mathrm{~g})\), the rate of reac- tion is \(0.1 \mathrm{~atm} / \mathrm{s}\). Initially only \(\mathrm{NH}_{3}(\mathrm{~g})\) was present at 3 atm and the reaction is performed at constant volume and temperature. The total pressure of gases after \(10 \mathrm{~s}\) from the start of reaction will be (a) \(4 \mathrm{~atm}\) (b) \(5 \mathrm{~atm}\) (c) \(3.5 \mathrm{~atm}\) (d) \(4.5\) atm

Step-by-Step Solution

Answer

The total pressure of gases after 10 s will be 4 atm (option a).

Step 1: Understand Zero-Order Reaction Kinetics

In a zero-order reaction, the rate of reaction is constant and independent of the concentration of the reactants. The rate equation can be written as rate = k[Reactant]^0, which simplifies to rate = k, where k is the rate constant.

Step 2: Calculate Change in Pressure of Reactants

The rate of reaction is given as 0.1 atm/s. This means that the pressure of NH3 decreases by 0.1 atm each second. After 10 seconds, pressure decrease in NH3 will be 0.1 atm/s * 10 s = 1 atm.

Step 3: Calculate Change in Total Pressure

For each 2 moles of NH3 consumed, 1 mole of N2 and 3 moles of H2 are produced. As the volume and temperature are constant, the total pressure change will be due to the mole change. Since 1 atm pressure of NH3 is consumed, the total pressure change will be (1/2)*(1 + 3) = 2 atm increase.

Step 4: Calculate Final Total Pressure

The initial total pressure is 3 atm, all of which is NH3. After 10 seconds the decrease in NH3 contributes a drop of 1 atm, but the formation of N2 and H2 contributes an increase of 2 atm. Thus, the final total pressure = initial pressure - drop in NH3 pressure + increase in product pressure = 3 atm - 1 atm + 2 atm = 4 atm.

Key Concepts

Chemical Kinetics and Zero-Order Reactions

Chemical kinetics is a branch of physical chemistry that studies the rates at which chemical reactions occur and the factors affecting those rates. A fundamental concept in chemical kinetics is the reaction order, which describes how the rate of a reaction depends on the concentration of its reactants.

Zero-order reactions are a specific type where the rate is constant and \textbf{does not depend} on the concentration of the reactant. Mathematically, it's described by the equation: \[ \text{rate} = k[Reactant]^0 \]which simplifies to rate = k, where \textbf{k} is the rate constant. This is counterintuitive to many students, as this means that even if you increase the concentration of the reacting substance, the reaction rate remains the same. In our example with the decomposition of \( \text{NH}_3(g) \), the reaction continues at a steady rate of 0.1 atm/s, regardless of how much ammonia is left. This steady decline in reactant concentration over time is a hallmark of zero-order kinetics.

Reaction Rate and Total Pressure Change

The reaction rate is defined as the speed at which reactants convert into products. In the context of our problem, the rate has been given as 0.1 atm/s, indicating that the pressure of ammonia, which stands in for its concentration under constant volume and temperature, decreases by 0.1 atm each second.

Understanding how the reaction rate affects pressure changes in reactions is key to solving this kind of problem. Since chemical reactions under constant volume and temperature conditions comply with the ideal gas law \( PV = nRT \), a change in the number of moles of gas (n) will correspond to a change in pressure (P), as both the gas constant (R) and temperature (T) remain constant.

The problem provided uses this concept to calculate the total pressure of gases after 10 seconds. With the loss of \( \text{NH}_3 \) pressure and the gain from the \( \text{N}_2 \) and \( \text{H}_2 \), students can witness how stoichiometry and gas laws intertwine in chemical kinetics to determine the extent of pressure changes in a system.

Calculating Pressure Change in Gas Reactions

In reactions involving gases, particularly at constant volume and temperature, one can deduce the final pressure in the system by considering stoichiometry—in other words, the quantitative relationship between reactants and products in a chemical reaction.

In our zero-order reaction, the stoichiometry tells us that for every 2 moles of \( \text{NH}_3 \) that react, 4 moles of gas are produced (1 mole of \( \text{N}_2 \) and 3 moles of \( \text{H}_2 \)). If we consider the initial and final pressures as representations of the number of moles of gases, we see that as \( \text{NH}_3 \) pressure drops by 1 atm (after 10 seconds), the total gas pressure actually increases by 2 atm since the total moles of gas are increasing.

This counterintuitive result—where the consumption of a reactant leads to an overall increase in system pressure—is a great example of the importance of understanding the relationship between stoichiometry and gas laws in the context of reaction kinetics.