React But Maintain Your Equilibrium
Part 1

DOFPro Team

Types of Reactor Models

Extent of Reaction and Fractional Conversion

  • \(\dot{\xi}\), \(\xi\), and \(f_\mathrm{A}\) are treated as parameters that are either given or calculated from other given information.
  • Does not deal directly with either chemical kinetics or chemical thermodynamics

Chemical Equilibrium

  • Assumes reaction reaches equilibrium
  • \(\dot{\xi}\), \(\xi\), and \(f_\mathrm{A}\) are determined by temperature, pressure, and feed concentrations

Homogeneous Chemical Kinetics

  • Starts with the three idealized chemical reactors
    • Batch, CSTR (mixed or stirred tank), or PFR (Plug Flow Reactor)
  • Uses chemical-kinetics expressions such as \(-r_\mathrm{A} = kC_\mathrm{A}^n\)
  • \(\dot{\xi}\), \(\xi\), and \(f_\mathrm{A}\) are determined by the kinetic expression, feed conditions, reactor volume, and temperature.

React But Maintain Your Equilibrium Part 1 & Part 2

  • Introduce chemical equilibrium methods
  • Part 1
    • Starts from advanced chemical engineering thermodynamics
    • Arrives at the equilibrium expressions for ideal gases

You are not expected to fully understand the derivation at this point. It’s just worth being exposed to what advanced-level work in chemical engineering is like.

Chemical Equilibrium

From Thermodynamics, equilibrium is reached when the total Gibbs Energy is minimized. Mathematically the criterion is:

\[ \sum \nu_i G^\circ_i + RT \sum \ln \hat{a}_i^{\nu_i} = 0 \]

or rearranged

\[ \prod \hat{a}_i^{\nu_i} = \exp \left( - \frac{\sum \nu_i G^\circ_i}{RT}\right) \equiv K \]

The \(\hat{a}_i\) are the activities of the various species where

Chemical Equilibrium (cont.)

\[ \hat{a}_i \equiv \frac{\hat{f}_i}{f^\circ_i} \]

\(\hat{f}_i\) is the fugacity of species \(i\) in solution.

\(f^\circ_i\) is the reference fugacity of species \(i\).

\(\sum \nu_i G^\circ_i\) is often written as \(\Delta G^\circ\) and is similar to \(\Delta H^\circ\).

\(K\) is the equilibrium constant. It varies only with temperature.

Chemical Equilibrium (cont.)

What are the units of \(K\)?

For ideal gases (see upcoming videos) the fugacity in solution is numerically equal to the partial pressure, and the reference fugacity is the ideal gas state at the system \(T\) and a pressure of 1 bar.

With these substitutions:

\[ K = \prod \hat{a}_i^{\nu_i} = \prod \left( \frac{p_i}{\mathrm{1\ bar}}\right)^{\nu_i} \]

or

\[ K\ (\mathrm{1\ bar})^\nu = \prod p_i^{\nu_i} \]

\(K\) is still dimensionless but may not appear to be.

Mole Fractions

Often problems are posed in mole fractions instead of partial pressures. Since for an ideal gas (see Ideal Gases EXPOSED: Are They Even Real?!)

\[ y_i = \frac{p_i}{P} \]

then

\[ p_i = y_i P \]

Substituting

\[ K\ (\mathrm{1\ bar})^\nu = \prod (y_i P)^{\nu_i} =P^\nu \prod (y_i)^{\nu_i} \]

or

Mole Fractions (cont.)

\[ K\ (\mathrm{1\ bar})^\nu P^{-\nu} =\prod (y_i)^{\nu_i} \]

which is where we will begin. There are later videos that go into more detail about the earlier steps and explain when you’d need them. For the present we’ll only deal with equilibrium reactions involving ideal gases and pure solids.

As a reminder, \(K\) depends only on temperature but the equilibrium composition depends on total pressure and the presence of inerts.

Temperature Dependence

With a fair amount of calculus one can show that

\[ \frac{d \ln K}{dT} = -\frac{\Delta \hat{H}^\circ_r}{RT^2} \]

which, if the enthalpy of reaction is constant with temperature integrates to

\[ \ln\frac{K}{K_0} = -\frac{\Delta \hat{H}^\circ_r}{R} \left(\frac{1}{T} - \frac{1}{T_0} \right) \]

The Takeaways

  1. The criterion for chemical equilibrium is that the total Gibbs energy of the system is minimized.
  2. The equilibrium constant, \(K\), is equal to the product of the species activities raised to their stoichiometric coefficients.
  3. The equilibrium constant is calculated as the exponential of the negative of the Gibbs energy of reaction over \(RT\).
  4. The activity of an ideal gas is the partial pressure over the reference pressure of 1 bar.






Thanks for watching!
The Full Story companion video is in the link in the upper left. The companion video, React But Maintain Your Equilibrium Part 2, in the series, is in the upper right. To learn more about Chemical and Thermal Processes, visit the website linked in the description.