The Ammonia Equilibrium

The industrial synthesis of ammonia from its elements is a classic and practically very important example of a chemical equilibrium described by the law of mass action. In this chapter we focus on:

• The specific equilibrium expression for the ammonia-forming reaction
• The role of the equilibrium constant under industrial conditions
• How temperature, pressure, and gas composition affect this particular equilibrium
• Practical aspects of the Haber–Bosch process that are directly related to the ammonia equilibrium

General concepts such as “chemical equilibrium”, “law of mass action”, and “Gibbs free energy” are assumed to be known from the parent chapters and are only used here, not re-developed.

The Ammonia-Forming Reaction

The industrial preparation of ammonia is based on the synthesis reaction of nitrogen and hydrogen:

$$
\ce{N2(g) + 3 H2(g) <=> 2 NH3(g)}
$$

Key features of this equilibrium:

• All components are gases.
• The number of moles of gas decreases from left to right: $1 + 3 = 4$ mol gas $\to 2$ mol gas.
• The reaction is exothermic (releases heat).

These features strongly determine how pressure and temperature influence the equilibrium position.

Equilibrium Expression for the Ammonia System

For the general gas-phase reaction above, the law of mass action gives an equilibrium constant in terms of activities $a_i$. In an ideal gas approximation, activities can be replaced by partial pressures $p_i$ divided by a standard pressure $p^\circ$ (often 1 bar). The equilibrium constant expressed in terms of partial pressures $K_p$ is then:

$$
K_p(T) = \frac{\left(\dfrac{p_{\ce{NH3}}}{p^\circ}\right)^2}
{\left(\dfrac{p_{\ce{N2}}}{p^\circ}\right)\left(\dfrac{p_{\ce{H2}}}{p^\circ}\right)^3}
$$

For compactness it is common in technical contexts (with $p^\circ = 1\ \text{bar}$) to write:

$$
K_p(T) = \frac{p_{\ce{NH3}}^2}{p_{\ce{N2}}\,p_{\ce{H2}}^3}
$$

with the understanding that the pressures are referenced to the standard pressure.

Expressions in Terms of Concentrations

If the reaction is considered in a fixed volume (e.g. for theoretical treatment), the equilibrium constant can also be written in terms of molar concentrations $c_i$:

$$
K_c(T) = \frac{c_{\ce{NH3}}^2}{c_{\ce{N2}}\,c_{\ce{H2}}^3}
$$

Because the total number of moles of gas changes during the reaction, $K_c$ and $K_p$ are not equal, but are related by the usual expression connecting $K_p$ and $K_c$ for gas-phase reactions (treated in the more general equilibrium chapter).

In practice, for the ammonia synthesis, $K_p$ is usually more directly relevant, because pressure and gas composition (mole fractions) are the quantities controlled in industrial reactors.

Temperature Dependence of the Ammonia Equilibrium

The synthesis of ammonia is exothermic:

$$
\Delta_\text{r} H^\circ < 0
$$

For an exothermic reaction, the equilibrium constant $K$ decreases with increasing temperature. This can be derived quantitatively from the van ’t Hoff equation (covered generally elsewhere) and is expressed qualitatively by Le Châtelier’s principle:

• Increasing temperature favors the endothermic direction (here: decomposition of NH₃).
• Decreasing temperature favors the exothermic direction (formation of NH₃).

For the ammonia reaction this means:

• At low temperature, $K_p$ is large and the equilibrium strongly favors $\ce{NH3}$.
• At high temperature, $K_p$ becomes smaller and the equilibrium shifts toward $\ce{N2}$ and $\ce{H2}$.

However, low temperatures also cause very slow reaction rates (kinetic limitation), so the technically optimal temperature must balance:

• A sufficiently large equilibrium constant (thermodynamics).
• An acceptable rate of approach to equilibrium (kinetics).

Typical industrial temperatures for the Haber–Bosch process are in the range of about $400{-}550^\circ\text{C}$, where this compromise is reasonably good.

Pressure Dependence and the Role of Total Pressure

In the gas-phase equilibrium

$$
\ce{N2 + 3 H2 <=> 2 NH3}
$$

the total number of moles of gas decreases during formation of ammonia. Increasing total pressure therefore favors the product side. In a simplified picture (Le Châtelier’s principle):

• Higher pressure $\Rightarrow$ equilibrium shift to the side with fewer gas molecules (here: $\ce{NH3}$).
• Lower pressure $\Rightarrow$ equilibrium shift to the side with more gas molecules (here: $\ce{N2}$ and $\ce{H2}$).

In more detail, for a fixed temperature and fixed overall composition, raising the total pressure increases the partial pressures of all components. Because the equilibrium constant $K_p$ itself does not depend on pressure, the composition at equilibrium must adjust so that

$$
\frac{p_{\ce{NH3}}^2}{p_{\ce{N2}}\,p_{\ce{H2}}^3} = K_p(T)
$$

remains valid. The solution of this relationship at higher total pressure always yields a higher equilibrium mole fraction of $\ce{NH3}$.

From a technical standpoint:

• Very high pressures (e.g. $>200$ bar) give higher equilibrium ammonia yields per pass.
• But the cost and complexity of high-pressure equipment strongly increase with pressure.

Industrial plants therefore choose pressures that give sufficient equilibrium yield while remaining economically and technically manageable. Common operating pressures are roughly $100{-}300$ bar.

Stoichiometry, Reaction Quotient, and Direction of Shift

To discuss how far the reaction proceeds under given conditions, it is useful to consider the reaction quotient $Q_p$ for the ammonia system, analogous to $K_p$ but using the current partial pressures (not necessarily equilibrium values):

$$
Q_p = \frac{p_{\ce{NH3}}^2}{p_{\ce{N2}}\,p_{\ce{H2}}^3}
$$

For the ammonia formation:

• If $Q_p < K_p$
  The mixture contains “too little” ammonia relative to equilibrium. The reaction will shift to the right (net formation of $\ce{NH3}$) until $Q_p = K_p$.
• If $Q_p > K_p$
  The mixture contains “too much” ammonia relative to equilibrium. The reaction will shift to the left (net decomposition of $\ce{NH3}$) until $Q_p = K_p$.

This is particularly important in understanding continuous industrial operation, where the composition at the reactor inlet and outlet differ.

Effects of Gas Composition and Feed Ratio

Besides total pressure, the composition of the gas phase (especially the ratio $\ce{N2}:\ce{H2}$ and the presence of inert gases) influences the ammonia equilibrium.

Stoichiometric Ratio of Nitrogen and Hydrogen

The stoichiometric ratio for the reaction is:

$$
\ce{N2 : H2} = 1 : 3
$$

Supplying the feed gas in this ratio is advantageous because, at equilibrium, neither $\ce{N2}$ nor $\ce{H2}$ is present in large excess solely due to feed composition. In practice:

• Nearly stoichiometric or slightly hydrogen-rich feeds are used.
• Deviations from stoichiometry in the feed gas alter the composition shift at equilibrium and can reduce the ammonia mole fraction for a given total pressure and temperature.

From the viewpoint of the law of mass action, an excess of one reactant initially lowers $Q_p$ relative to $K_p$ and thereby drives the reaction toward greater ammonia production, but the final equilibrium is constrained by the fixed total amounts and the requirement $Q_p = K_p$.

Influence of Inert Components

Inert gases (e.g. $\ce{Ar}$, $\ce{CH4}$ from natural gas feed impurities) do not participate in the equilibrium expression, but they:

• Contribute to the total pressure.
• Reduce the partial pressures of the reacting gases for a given total pressure (they dilute the reaction mixture).

At constant total pressure and temperature, introducing inert gases lowers $p_{\ce{N2}}$, $p_{\ce{H2}}$, and $p_{\ce{NH3}}$. The condition $Q_p = K_p$ must still be fulfilled, but the presence of inerts generally leads to a lower maximum ammonia mole fraction achievable in the mixture. Industrial plants must therefore periodically purge an inert-rich gas fraction to prevent buildup.

The Ammonia Equilibrium in the Haber–Bosch Process

The Haber–Bosch process is the industrial implementation of ammonia synthesis using an iron-based catalyst at high pressure and elevated temperature. While kinetics and catalysts are addressed in other chapters, some operational choices are directly dictated by equilibrium considerations.

Operating Window: Compromise Between Equilibrium and Kinetics

From a purely equilibrium standpoint:

• Lower temperature $T$ and higher pressure $p$ would give the greatest ammonia yield.

In practice:

• At too low temperatures, reaction rates are too slow even with catalysts; the reactor would have to be extremely large to achieve significant conversion.
• At too high temperatures, although the reaction runs faster, the equilibrium strongly favors the reactants, limiting achievable ammonia concentration.

Therefore, the chosen operating window (roughly $400{-}550^\circ\text{C}$ and $100{-}300$ bar) reflects a compromise:

• Temperature high enough for reasonable kinetics.
• Temperature low enough that $K_p$ is not too small.
• Pressure high enough that, at the chosen $T$, the equilibrium composition contains significant $\ce{NH3}$.

Use of Recycling to Approach Equilibrium

In a single pass through the reactor, the reacting gas does not reach complete conversion to ammonia; this is limited by the equilibrium at the chosen $T$ and $p$. To increase the overall yield:

1. The gas leaving the reactor is cooled, causing much of the $\ce{NH3}$ to condense and be separated.
2. The remaining gas (rich in unreacted $\ce{N2}$ and $\ce{H2}$) is largely recycled back to the reactor inlet.
3. Fresh feed gas is mixed with the recycle stream; the combined stream then re-enters the reactor.

From an equilibrium perspective:

• Each pass approaches the equilibrium composition for the reactor conditions.
• By removing $\ce{NH3}$ between passes (cooling and condensation), the effective overall process is driven further toward the product side than a single equilibrium stage would allow, because the product is continually removed from the reaction mixture.

This use of product removal and recycle exemplifies an application of the law of mass action: removing a product from the reacting phase decreases $Q_p$ below $K_p$, forcing the reaction to move toward more product formation in the next pass.

The Role of $\Delta_\text{r} G^\circ$ and $K_p$ at Process Temperatures

For the ammonia reaction, the standard reaction Gibbs free energy change $\Delta_\text{r} G^\circ(T)$ is:

• Negative at sufficiently low temperatures (favoring product formation, $K_p \gg 1$).
• Becomes less negative or may even become positive at high temperatures (reducing $K_p$).

The relationship between $\Delta_\text{r} G^\circ$ and $K_p$:

$$
\Delta_\text{r} G^\circ(T) = -RT \ln K_p(T)
$$

allows one to compute $K_p$ at process temperatures from thermodynamic data. In plant design:

• $K_p(T)$ values at the chosen operating temperature inform expected equilibrium compositions under various pressures.
• This data is used to size reactors, specify recycle ratios, and estimate energy requirements.

The general form of this relationship was developed in the parent chapter; the ammonia equilibrium provides a concrete and economically crucial application.

Comparison of Ideal and Real Behavior

So far, we have implicitly treated the gas mixture as ideal, using partial pressures directly in the equilibrium expression. In real industrial conditions:

• Pressures are high (up to several hundred bar).
• Non-ideal gas behavior becomes significant.

To describe the equilibrium more accurately, activities are expressed through fugacities $f_i$ instead of simple partial pressures:

$$
K = \frac{a_{\ce{NH3}}^2}{a_{\ce{N2}}\,a_{\ce{H2}}^3}
= \frac{\left(\dfrac{f_{\ce{NH3}}}{f^\circ}\right)^2}
{\left(\dfrac{f_{\ce{N2}}}{f^\circ}\right)\left(\dfrac{f_{\ce{H2}}}{f^\circ}\right)^3}
$$

Here the fugacities incorporate non-ideal interactions via fugacity coefficients. For introductory treatment of the ammonia equilibrium, it is often sufficient to regard the gas as nearly ideal and use $K_p$ with partial pressures, but advanced process design uses real-gas thermodynamic models.

Summary of Key Points for the Ammonia Equilibrium

• The ammonia formation reaction $\ce{N2 + 3 H2 <=> 2 NH3}$ is exothermic and involves a decrease in the number of gas molecules.
• Its equilibrium constant in terms of partial pressures is:
  $$
  K_p(T) = \frac{p_{\ce{NH3}}^2}{p_{\ce{N2}}\,p_{\ce{H2}}^3}
  $$
• Lower temperatures and higher pressures favor ammonia at equilibrium, but low temperature slows the reaction; industrial conditions are chosen as a compromise.
• The reaction quotient $Q_p$ determines the direction of the shift relative to equilibrium; continual removal of ammonia and recycling unreacted gases effectively drive the overall process toward higher yields.
• Feed composition (especially the $\ce{N2}:\ce{H2}$ ratio) and the presence of inert gases influence the equilibrium concentrations without changing $K_p$.
• At high industrial pressures, non-ideal gas behavior becomes important and is treated with fugacity-based activities in more advanced analyses.

The ammonia equilibrium thus provides a central, practically important example of how the law of mass action is applied in real chemical processes.