Answered

Calculate the $\Delta H$ for the reaction: $CH _4(g)+ NH _3(g) \rightarrow HCN ( g )+3 H _2(g)$, from the following data:



$\begin{array}{ll}

N_2(g)+3 H_2(g) \rightarrow 2 NH_3(g) & \Delta H=-91.8\text{ kJ/mole} \\

C(s)+2 H_2(g) \rightarrow CH_4(g) & \Delta H=-74.9\text{ kJ/mole} \\

H_2(g)+2 C(s)+N_2(g) \rightarrow 2 HCN(g) & \Delta H=+270.3\text{ kJ/mole}

\end{array}$

Verified Answer :

  • Reverse Equation 2: \(CH_4(g) \rightarrow C(s) + 2H_2(g)\), \(\Delta H = +74.9 \text{ kJ/mol}\).
  • Reverse and divide Equation 1 by 2: \(NH_3(g) \rightarrow \frac{1}{2}N_2(g) + \frac{3}{2}H_2(g)\), \(\Delta H = +45.9 \text{ kJ/mol}\).
  • Divide Equation 3 by 2: \(\frac{1}{2}H_2(g) + C(s) + \frac{1}{2}N_2(g) \rightarrow HCN(g)\), \(\Delta H = +135.15 \text{ kJ/mol}\).
  • The total \(\Delta H\) for the reaction is the sum: \(74.9 + 45.9 + 135.15 = \boxed{255.95 \text{ kJ/mol}}\).

Explanation

  1. Problem Analysis and Data Overview

We aim to determine the standard enthalpy change (\(\Delta H\)) for the reaction:
\[CH_4(g) + NH_3(g) \rightarrow HCN(g) + 3H_2(g)\]
We are provided with three thermochemical equations:

  1. \(N_2(g) + 3H_2(g) \rightarrow 2NH_3(g)\), \(\Delta H_1 = -91.8 \text{ kJ/mol}\)
  2. \(C(s) + 2H_2(g) \rightarrow CH_4(g)\), \(\Delta H_2 = -74.9 \text{ kJ/mol}\)
  3. \(H_2(g) + 2C(s) + N_2(g) \rightarrow 2HCN(g)\), \(\Delta H_3 = +270.3 \text{ kJ/mol}\)

According to Hess's Law, the total enthalpy change for a reaction is the sum of the enthalpy changes for individual steps. We need to manipulate these equations so that they add up to the target reaction.

  1. Manipulating Thermochemical Equations

To get the reactants and products in the correct positions, we manipulate the given equations:

  1. Reverse Equation 2 to place \(CH_4\) as a reactant:

\[CH_4(g) \rightarrow C(s) + 2H_2(g) \quad \Delta H = -\Delta H_2 = +74.9 \text{ kJ/mol}\]

  1. Reverse and divide by 2 for Equation 1 to place \(NH_3\) as a reactant:

\[NH_3(g) \rightarrow \frac{1}{2}N_2(g) + \frac{3}{2}H_2(g) \quad \Delta H = -\frac{\Delta H_1}{2} = -\frac{-91.8}{2} = +45.9 \text{ kJ/mol}\]

  1. Divide by 2 for Equation 3 to maintain \(HCN\) as a product:

\[\frac{1}{2}H_2(g) + C(s) + \frac{1}{2}N_2(g) \rightarrow HCN(g) \quad \Delta H = \frac{\Delta H_3}{2} = \frac{270.3}{2} = +135.15 \text{ kJ/mol}\]

  1. Calculating the Total Enthalpy Change

Now, we sum the three manipulated equations:
\[(CH_4 + NH_3 + \frac{1}{2}H_2 + C + \frac{1}{2}N_2) \rightarrow (C + 2H_2 + \frac{1}{2}N_2 + \frac{3}{2}H_2 + HCN)\]
Canceling out common terms (\(C\), \(\frac{1}{2}N_2\), and part of the \(H_2\)) from both sides yields the target reaction:
\[CH_4 + NH_3 \rightarrow HCN + (2 + \frac{3}{2} - \frac{1}{2})H_2 = HCN + 3H_2\]
The total enthalpy change is the sum of the enthalpy changes of the manipulated equations:
\[\Delta H_{total} = 74.9 + 45.9 + 135.15 = 255.95 \text{ kJ/mol}\]
Thus, the enthalpy change for the reaction is \(\boxed{255.95 \text{ kJ/mol}}\).

Examples

The principles of Hess's Law are foundational in chemical engineering and industrial chemistry. For example, in the production of hydrogen cyanide (HCN) from methane and ammonia, engineers must determine whether the reaction is endothermic (absorbing heat) or exothermic (releasing heat) to maintain appropriate reactor temperatures and optimize energy consumption. By using thermochemical data, they can calculate the required energy input for the process without needing to measure the heat of reaction directly for every possible variation, ensuring efficient and safe industrial operation.

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