Conversion (chemistry)

Conversion

Conversion can be separated into instantaneous conversion and overall conversion. For continuous processes the two are the same, for batch and semi-batch there are important differences. Furthermore, for multiple reactants, conversion can be defined overall or per reactant.

Instantaneous conversion

Semi-batch

In this setting there are different definitions. One definition regards the instantaneous conversion as the ratio of the instantaneously converted amount to the amount fed at any point in time:

${\displaystyle X_{i,\text{inst}}=\frac{{\dot{n}}_{i,\text{react}}}{{\dot{n}}_{i,\text{in}}}}$.

with ${\displaystyle {\dot{n}}_i}$ as the change of moles with time of species i.

This ratio can become larger than 1. It can be used to indicate whether reservoirs are built up and it is ideally close to 1. When the feed stops, its value is not defined.

In semi-batch polymerisation, the instantaneous conversion is defined as the total mass of polymer divided by the total mass of monomer fed:

${\displaystyle X_{\text{poly}}=\frac{m_{\text{Pol}}}{\sum _i{\int }_0^t{\dot{m}}_{i,\text{in}}(\tau )d\tau }}$.

Overall conversion

Batch (This is the generally stated form)

${\displaystyle X_i=\frac{n_i(t=0)-n_i(t)}{n_i(t=0)}=1-\frac{n_i(t)}{n_i(t=0)}}$

Semi-batch

Total conversion of the formulation:

${\displaystyle X_i=\frac{n_i(t=0)+{\int }_0^t{\dot{n}}_{i,\text{in}}(\tau )d\tau -n_i(t)}{n_i(t=0)+{\int }_0^{t_{\text{end}}}{\dot{n}}_{i,\text{in}}(\tau )d\tau }}$

Total conversion of the fed reactants:

${\displaystyle X_i=\frac{n_i(t=0)+{\int }_0^t{\dot{n}}_{i,\text{in}}(\tau )d\tau -n_i(t)}{n_i(t=0)+{\int }_0^t{\dot{n}}_{i,\text{in}}(\tau )d\tau }}$

Continuous (This is the generally stated form)

${\displaystyle X_i=\frac{{\dot{n}}_{i,in}-{\dot{n}}_{i,out}}{{\dot{n}}_{i,in}}=1-\frac{{\dot{n}}_{i,out}}{{\dot{n}}_{i,in}}}$

Yield

Yield in general refers to the amount of a specific product (p in 1..m) formed per mole of reactant consumed (Definition 1). However, it is also defined as the amount of product produced per amount of product that could be produced (Definition 2). If not all of the limiting reactant has reacted, the two definitions contradict each other. Combining those two also means that stoichiometry needs to be taken into account and that yield has to be based on the limiting reactant (k in 1..n):

Continuous

${\displaystyle Y_p=\frac{{\dot{n}}_{p,\text{out}}-{\dot{n}}_{p,\text{in}}}{{\dot{n}}_{k,\text{in}}\underset{\text{only for Definition 1}}{\underbrace{-n_{k,\text{out}}}}}\left|\frac{{\mu }_k}{{\nu }_p}\right|}$

The version normally found in the literature:

${\displaystyle Y_p=\frac{{\dot{n}}_{p,\text{out}}-{\dot{n}}_{p,\text{in}}}{{\dot{n}}_{k,\text{in}}}\left|\frac{{\mu }_k}{{\nu }_p}\right|}$

Selectivity

Instantaneous selectivity is the production rate of one component per production rate of another component.

For overall selectivity the same problem of the conflicting definitions exists. Generally, it is defined as the number of moles of desired product per the number of moles of undesired product (Definition 1). However, the definitions of the total amount of reactant to form a product per total amount of reactant consumed is used (Definition 2) as well as the total amount of desired product formed per total amount of limiting reactant consumed (Definition 3). This last definition is the same as definition 1 for yield.

Batch or semi-batch

The version normally found in the literature:

${\displaystyle S_p=\frac{n_p(t=0)-n_p(t)}{n_k(t=0)+{\int }_0^t{\dot{n}}_{k,\text{in}}(\tau )d\tau -n_k(t)}\left|\frac{{\mu }_k}{{\nu }_p}\right|}$

Continuous

The version normally found in the literature:

${\displaystyle S_p=\frac{{\dot{n}}_{p,\text{out}}-{\dot{n}}_{p,\text{in}}}{{\dot{n}}_{k,\text{in}}-{\dot{n}}_{k,\text{out}}}\left|\frac{{\mu }_k}{{\nu }_p}\right|}$

Combination

For batch and continuous reactors (semi-batch needs to be checked more carefully) and the definitions marked as the ones generally found in the literature, the three concepts can be combined:

${\displaystyle Y_p=X_i\cdot S_p}$

For a process with the only reaction

${\displaystyle \text{A}⟶\text{B}}$

this mean that S=1 and Y=X.

Abstract example

For the following abstract example and the amounts depicted on the right, the following calculation can be performed with the above definitions, either in batch or a continuous reactor.

${\displaystyle \text{A}⟶\text{B}}$

${\displaystyle \text{A}⟶\text{C}}$

B is the desired product.

There are 100 mol of A at the beginning or at the entry to the continuous reactor and 10 mol A, 72 mol B and 18 mol C at the end of the reaction or the exit of the continuous reactor. The three properties are found to be:

${\displaystyle X_{\text{A}}=\frac{n_{\text{A}}(t=0)-n_A(t)}{n_{\text{A}}(t=0)}=1-\frac{n_{\text{A}}(t)}{n_{\text{A}}(t=0)}=\frac{100-10}{100}=0.9=90\mathrm{\%}}$

${\displaystyle Y_{\text{B}}=\frac{n_{\text{B}}(t)-n_{\text{B}}(t=0)}{n_{\text{A}}(t=0)+{\int }_0^t{\dot{n}}_{\text{A},\text{in}}(\tau )d\tau }\left|\frac{{\mu }_k}{{\nu }_p}\right|=\frac{72-0}{100+0}\cdot \frac{1}{1}=0.72=72\mathrm{\%}}$

${\displaystyle S_{\text{B}}=\frac{n_{\text{B}}(t=0)-{\dot{n}}_{\text{B}}(t)}{{\dot{n}}_{\text{A}}(t=0)-n_{\text{A}}(t)}\left|\frac{{\mu }_k}{{\nu }_p}\right|=\frac{0-72}{100-10}\cdot \frac{1}{1}=0.8=80\mathrm{\%}}$

The property ${\displaystyle Y_p=X_i\cdot S_p}$ holds. In this reaction, 90% of substance A is converted (consumed), but only 80% of the 90% is converted to the desired substance B and 20% to undesired by-products C. So, conversion of A is 90%, selectivity for B 80% and yield of substance B 72%.

Literature

• Werner Kullbach: Mengenberechnungen in der Chemie. Verlag Chemie, Weinheim 1980, ISBN 3-527-25869-8.
• Eberhard Aust, Burkhard Bittner: Stöchiometrie - Chemisches Rechnen, CICERO-Verlag, Pegnitz, 4. Auflage, 2011, ISBN 978-3-926292-47-6.
• Uwe Hillebrand: Stöchiometrie, Eine Einführung in die Grundlagen mit Beispielen und Übungsaufgaben, 2. Aufl., Springer-Verlag, Berlin Heidelberg 2009, ISBN 978-3-642-00459-9.