Problem 62 - Chemistry: An Atoms-Focused Approach

Question

Above what temperature does nitrogen monoxide form from nitrogen and oxygen? $$\mathrm{N}_{2}(g)+\mathrm{O}_{2}(g) \rightarrow 2 \mathrm{NO}(g)$$ Assume that the values of \(\Delta H_{\mathrm{rxn}}^{\circ}\) and \(\Delta S_{\mathrm{rxn}}^{\circ}\) do not change appreciably with temperature.

Step-by-Step Solution

Answer

Answer: The temperature at which nitrogen monoxide begins to form from nitrogen and oxygen gas is 7,200 K.

Step 1: Convert ΔH and ΔS to compatible units

To make the units consistent in the Gibbs free energy equation, we need to convert ΔH and ΔS into compatible units. Since ΔH is given in kJ/mol and ΔS is given in J/(mol·K), we will convert ΔH to J/mol. $$\Delta H_{\mathrm{rxn}}^{\circ} = 180\,\mathrm{kJ/mol} \times 1000\,\mathrm{J/kJ} = 180{,}000\,\mathrm{J/mol}$$

Step 2: Set ΔG to 0 and solve for T

Now, we set the Gibbs free energy (\(\Delta G_{\mathrm{rxn}}^{\circ}\)) to 0 because we want to find the temperature at which the reaction changes from non-spontaneous to spontaneous, and solve for T. $$0 = 180{,}000\,\mathrm{J/mol} - T\left(25\,\frac{\mathrm{J}}{\mathrm{mol}\cdot\mathrm{K}}\right)$$ Rearrange the equation to solve for T: $$T = \frac{180{,}000\,\mathrm{J/mol}}{25\,\mathrm{J/(mol\cdot K)}}$$

Step 3: Calculate the temperature

Now, we will calculate T using the provided values of \(\Delta H_{\mathrm{rxn}}^{\circ}\) and \(\Delta S_{\mathrm{rxn}}^{\circ}\). $$ T = \frac{180{,}000\,\mathrm{J/mol}}{25\,\mathrm{J/(mol\cdot K)}} = 7{,}200\,\mathrm{K} $$ The temperature at which nitrogen monoxide begins to form from nitrogen and oxygen is 7,200 K.

Key Concepts

Gibbs Free Energy

Gibbs free energy is an important concept in thermodynamics. It tells us whether a process will happen on its own or not. The symbol for Gibbs free energy is usually \( \Delta G \).

The key equation for Gibbs free energy is:

• \( \Delta G = \Delta H - T \Delta S \)

Here, \( \Delta H \) is the change in enthalpy, \( T \) is the temperature in Kelvin, and \( \Delta S \) is the change in entropy. If \( \Delta G \) is negative, the process is spontaneous. If it's positive, the process is non-spontaneous.

In the provided exercise, we find the temperature where \( \Delta G = 0 \). This point is where the reaction starts becoming spontaneous. By setting \( \Delta G \) to zero and rearranging the equation, we solve for the temperature \( T \). This helps us find the critical temperature above which nitrogen monoxide (NO) starts forming from nitrogen and oxygen.

Enthalpy

Enthalpy, symbolized as \( \Delta H \), is a measure of energy in a thermodynamic system.

It reflects the heat content under constant pressure. Enthalpy changes tell us how much heat is absorbed or released during a reaction.

• If \( \Delta H \) is positive, heat is absorbed, and the reaction is endothermic.
• If \( \Delta H \) is negative, heat is released, making the reaction exothermic.

In our exercise, we know the reaction enthalpy \( \Delta H_{\mathrm{rxn}}^{\circ} \) for forming nitrogen monoxide is given as \( 180\, \mathrm{kJ/mol} \). This positive value indicates that the reaction requires energy—thus, it absorbs heat from its surroundings. Specifically, to perform calculations, it was necessary to convert this energy value to \( \mathrm{J/mol} \) for compatibility. Thus, understanding this helps us see how energy exchanges during chemical reactions.

Entropy

Entropy, denoted \( \Delta S \), is a measure of randomness or disorder in a system. It's a fundamental concept in thermodynamics and is an indicator of molecular randomness.

When molecules spread out and create more disorder, entropy increases. Conversely, when molecules become more ordered, entropy decreases.

• A positive \( \Delta S \) implies an increase in disorder.
• A negative \( \Delta S \) suggests a decrease in disorder.

In the exercise, entropy change \( \Delta S_{\mathrm{rxn}}^{\circ} \) is given as \( 25\, \mathrm{J/(mol \cdot K)} \). This positive value means that forming nitrogen monoxide from nitrogen and oxygen results in more disorder or randomness in the system. Moreover, entropy plays a significant role in determining reaction spontaneity because it's directly linked with temperature in the Gibbs free energy equation. Understanding entropy helps us comprehend how and why spontaneous changes occur in nature.