Problem 141 - General Chemistry

Question

Consider the Haber process: $$ \mathrm{N}_{2}(g)+3 \mathrm{H}_{2}(g) \longrightarrow 2 \mathrm{NH}_{3}(g) ; \Delta H^{\circ}=-91.8 \mathrm{~kJ} $$ The density of ammonia at \(25^{\circ} \mathrm{C}\) and \(1.00 \mathrm{~atm}\) is \(0.696 \mathrm{~g} / \mathrm{L}\). The density of nitrogen, \(\mathrm{N}_{2}\), is \(1.145 \mathrm{~g} / \mathrm{L}\), and the molar heat capacity is \(29.12 \mathrm{~J} /\left(\mathrm{mol} \cdot{ }^{\circ} \mathrm{C}\right)\). (a) How much heat is evolved in the production of \(1.00 \mathrm{~L}\) of ammonia at \(25^{\circ} \mathrm{C}\) and \(1.00 \mathrm{~atm} ?\) (b) What percentage of this heat is required to heat the nitrogen required for this reaction \((0.500 \mathrm{~L})\) from \(25^{\circ} \mathrm{C}\) to \(400^{\circ} \mathrm{C}\), the temperature at which the Haber process is run?

Step-by-Step Solution

Answer

(a) -1.877 kJ of heat is evolved. (b) 11.88% of the evolved heat is required to heat the nitrogen.

Step 1: Determine moles of ammonia

First, calculate the number of moles of ammonia in 1.00 L. Use the formula for density: \[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \] which rearranges to \[ \text{Mass} = \text{Density} \times \text{Volume} \]\[\text{Mass of NH}_3 = 0.696 \text{ g/L} \times 1.00 \text{ L} = 0.696 \text{ g} \] Now, find the moles of ammonia using its molar mass (\(17.03\text{ g/mol}\)):\[ \text{Moles of NH}_3 = \frac{0.696 \text{ g}}{17.03 \text{ g/mol}} = 0.0409 \text{ mol} \]

Step 2: Calculate heat evolved from ammonia production

The reaction gives \(-91.8 \text{ kJ}\) for every 2 moles of \(\text{NH}_3\). So, calculate heat evolved for 0.0409 moles:\[ \text{Heat evolved} = 0.0409 \text{ mol} \times \left(\frac{-91.8 \text{ kJ}}{2 \text{ mol}}\right) = -1.877 \text{ kJ} \] This is the heat evolved when producing 1 L of \(\text{NH}_3\).

Step 3: Calculate heat needed to warm nitrogen

Use the specific heat formula \( q = mc\Delta T \), where \(c = 29.12 \text{ J/mol}^\circ\text{C}\) and \(\Delta T = 400^\circ\text{C} - 25^\circ\text{C} = 375^\circ\text{C}\). First, find moles of \(\text{N}_2\) in 0.500 L:\[ \text{Mass of } \text{N}_2 = 1.145 \text{ g/L} \times 0.500 \text{ L} = 0.5725 \text{ g} \]\[\text{Moles of } \text{N}_2 = \frac{0.5725 \text{ g}}{28.02 \text{ g/mol}} = 0.0204 \text{ mol} \]Now, calculate the heat required: \[ q = 0.0204 \text{ mol} \times 29.12 \text{ J/mol}^\circ\text{C} \times 375^\circ\text{C} \times \frac{1 \text{ kJ}}{1000 \text{ J}} = 0.223 \text{ kJ} \]

Step 4: Calculate percentage of produced heat used to heat nitrogen

Determine what percentage the heat required to warm the \(\text{N}_2\) is of the heat evolved from ammonia production:\[ \text{Percentage} = \left(\frac{0.223 \text{ kJ}}{1.877 \text{ kJ}}\right) \times 100\% = 11.88\% \]

Key Concepts

Enthalpy Change

The concept of enthalpy change is pivotal in understanding chemical reactions, especially in processes like the Haber Process. Enthalpy change, represented by \( \Delta H \), is the amount of heat absorbed or released during a chemical reaction at constant pressure. In the Haber Process, the enthalpy change is \(-91.8 \text{ kJ} \), which indicates an exothermic reaction—heat is released when producing ammonia.

This helps us compute how much energy is involved in transforming reactants to products. Knowing the enthalpy change enables us to calculate the heat evolved for a given amount of ammonia produced. It's calculated using the standard equation:

• Divide the enthalpy change by the number of moles involved in the reaction
• Multiply the resulting value by the specific moles of product

This gives the heat evolved for the stoichiometry of interest, allowing for further calculations on energy applications, such as in the heating of reactants.

Ammonia Production

The focus of the Haber Process is to efficiently produce ammonia, a crucial fertilizer component. The criticality lies in converting nitrogen and hydrogen into ammonia under high pressure and temperatures. For our calculations, we consider the production of 1 liter of ammonia gas.

First, establish the mass of ammonia using its density:

• Density = Mass/Volume, rearranged to Mass = Density x Volume
• Density of ammonia is given as 0.696 g/L
• This provides the mass of ammonia which then converts to moles using its molar mass \( (17.03 \text{ g/mol}) \)

Ensuring accurate mole calculations facilitates energy assessments via the enthalpy change, solidifying ammonia's production in industry benchmarks.

Heat Calculation

Calculating heat in chemical calculations involves determining energy changes related to physical and chemical processes. In our scenario, we evaluate the heat evolved during the transformation in the Haber Process.

To achieve this:

• Use the enthalpy change per mole
• Multiply with actual moles of product - ammonia in this case
• Evaluate heat evolution\(-1.877 \text{ kJ}\) for making 1 L of ammonium

Additionally, when considering heating nitrogen from \(25^{\circ} \text{C}\) to \(400^{\circ} \text{C}\), apply the specific heat capacity formula \( q = mc\Delta T \) to find the heat required. This involves:

• Calculating mass and moles of nitrogen
• Applying the specific heat capacity \( 29.12 \text{ J/mol}^\circ\text{C} \)
• Finding the temperature change \( \Delta T\) as 375°C

These principles empower understanding of thermal dynamics within reaction systems.

Gas Density

Gas density is crucial for determining mass and moles in chemical processes. It describes how much mass of a gas is found in a given volume, providing a bridge to further stoichiometric and thermal calculations.

For ammonia, with a density of 0.696 g/L, it directly influences how we model and understand gas behavior in the Haber Process. Calculating mass involves:

• Using the relation Density = Mass/Volume
• For given volume, relate back to moles using the molar mass
• For nitrogen, density is 1.145 g/L

These distinct densities enable assessing related thermal energy requirements under varying reaction conditions. Understanding densities offers predictive power over reaction outcomes in industrial production.

Molar Heat Capacity

Molar heat capacity, \( c \), signifies a substance's capability to absorb heat per mole for each temperature increment \( (\text{J/mol}^\circ\text{C}) \). It fundamentally influences energy considerations in temperature changes, embroiling concepts in process engineering and thermodynamics.

In the Haber Process:

• Molar heat capacity of nitrogen is specifically noted as \( 29.12 \text{ J/mol}^\circ\text{C} \)
• Essential in calculating nitrogen's heating from 25°C to 400°C
• Applies via the equation \( q = mc\Delta T \) where \( q \) is the heat required

These insights anchor the energy transition across chemical phases, marking its importance in the precise control of chemical engineering processes. Understanding molar heat capacity influences reactions under specific conditions, boosting energetic efficiencies.