Question

The preparation of \(\mathrm{NO}_{2}(g)\) from \(\mathrm{N}_{2}(g)\) and \(\mathrm{O}_{2}(g)\) is an endothermic reaction: $$ \mathrm{N}_{2}(g)+\mathrm{O}_{2}(g) \longrightarrow \mathrm{NO}_{2}(g) \text { (unbalanced) } $$ The enthalpy change of reaction for the balanced equation (with lowest whole- number coefficients) is \(\Delta H=67.7 \mathrm{~kJ}\). If \(2.50 \times\) \(10^{2} \mathrm{~mL} \mathrm{~N}_{2}(g)\) at \(100 .{ }^{\circ} \mathrm{C}\) and \(3.50\) atm and \(4.50 \times 10^{2} \mathrm{~mL} \mathrm{O}_{2}(g)\) at \(100 .{ }^{\circ} \mathrm{C}\) and \(3.50 \mathrm{~atm}\) are mixed, what amount of heat is necessary to synthesize the maximum yield of \(\mathrm{NO}_{2}(g)\) ?

Short answer

The amount of heat necessary to synthesize the maximum yield of NO2 is approximately 4.99 kJ.

Step-by-step solution

Balance the chemical equation

To balance the chemical equation, we need to ensure the number of atoms of each element is equal on both sides of the equation. The balanced equation is: N2(g) + 2O2(g) → 2NO2(g)

Determine the quantities of N2 and O2 in moles

Given the volumes, temperature and pressure of both N2 and O2, we will use the Ideal Gas Law (PV = nRT) to determine the number of moles. We will use the gas constant R = 0.0821 L atm / (K mol). For N2: Volume, V = 2.50 × 10² mL = 0.250 L Temperature, T = 100°C = 373 K Pressure, P = 3.50 atm R = 0.0821 L atm / (K mol) n(N2) = PV / RT = (3.50 atm × 0.250 L) / (0.0821 L atm / (K mol) × 373 K) = 0.0369 mol For O2: Volume, V = 4.50 × 10² mL = 0.450 L Temperature, T = 100°C = 373 K Pressure, P = 3.50 atm R = 0.0821 L atm / (K mol) n(O2) = PV / RT = (3.50 atm × 0.450 L) / (0.0821 L atm / (K mol) × 373 K) = 0.0664 mol

Calculate the limiting reactant

Based on the stoichiometry of the balanced equation, 1 mole of N2 reacts with 2 moles of O2. To determine the limiting reactant, we will compare the mole ratios of N2 and O2: Mole ratio of N2 to O2 = 0.0369 mol / 0.0664 mol = 0.556 Since the ratio is less than the required 1:2 ratio, N2 is the limiting reactant.

Compute the maximum yield of NO2

From the balanced equation, 1 mole of N2 produces 2 moles of NO2. Using the moles of the limiting reactant N2: n(NO2) = 2 × n(N2) = 2 × 0.0369 mol = 0.0738 mol

Calculate the amount of heat needed

Given the ΔH = 67.7 kJ for the balanced equation, we can calculate the amount of heat needed to synthesize the maximum yield of NO₂: q = n(NO2) × ΔH = 0.0738 mol × 67.7 kJ/mol ≈ 4.99 kJ Therefore, 4.99 kJ of heat is necessary to synthesize the maximum yield of NO2.

Key concepts

Chemical Equation Balancing

Understanding how to balance chemical equations is fundamental in the study of chemistry. It's like a seesaw that must be in equilibrium. In a chemical reaction, the number of atoms for each element must be the same on both the reactant and product sides of the equation. This conservation of mass is crucial because atoms cannot be created or destroyed in a chemical reaction.

To balance an equation, you'll adjust the coefficients, which are numbers placed in front of the chemical formulas. These coefficients tell you how many molecules or moles of each reactant and product are involved in the reaction. Getting the hang of balancing equations is much like solving a puzzle; with practice, it becomes second nature. Always start with elements that appear in a single reactant and product first, and then proceed to balance more complex elements.

Ideal Gas Law

The Ideal Gas Law is a cornerstone of chemical and physical sciences. It's an equation of state for a hypothetical 'ideal' gas, where the molecules do not interact with each other except for perfectly elastic collisions. The law combines several gas laws into one formula: \( PV = nRT \), where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin.

This law allows you to calculate one state variable of a gas if you know the other three. In chemistry problems, it's often used to find the number of moles of a gas under certain conditions of temperature, pressure, and volume. Remember, the temperature must always be in Kelvin for the Ideal Gas Law to be accurate.

Limiting Reactant

In chemical reactions, the limiting reactant is the substance that runs out first and thus limits the amount of products formed. It's quite similar to the concept of having only so many seats in a car; if you have more friends than seats, some friends will be left without a ride. In the same way, if one reactant is in a lesser quantity than needed according to the stoichiometry of the reaction, it will determine the maximum amount of product that can be formed.

To identify the limiting reactant, you must first determine the mole ratio of the reactants and compare that to the ratio given by the balanced equation. The reactant that provides the smallest amount of product as per this stoichiometric ratio is the limiting reactant. Knowing the limiting reactant is essential for calculating yields and understanding how much of each reactant is needed for a reaction to proceed to completion.

Enthalpy Change Calculation

Enthalpy change, symbolized as \( \Delta H \), is a measure of heat absorbed or released during a chemical reaction at constant pressure. It is a central concept in thermochemistry and indicates whether a reaction is endothermic (absorbs heat) or exothermic (releases heat).

To calculate the enthalpy change for a reaction, you need the stoichiometry of the balanced equation and the enthalpy change per mole of reaction. Multiply the number of moles of the product (based on the limiting reactant) by the enthalpy change per mole from the balanced equation. This computation gives you the total heat involved in the reaction. Understanding enthalpy changes is crucial for predicting the energy requirements of chemical processes and designing reactions that are energetically efficient.