## Conservation of mass in chemical reactions

Posted February 27, 2013 at 10:54 AM | categories: linear algebra | tags: reaction engineering

Updated March 06, 2013 at 04:27 PM

Atoms cannot be destroyed in non-nuclear chemical reactions, hence it follows that the same number of atoms entering a reactor must also leave the reactor. The atoms may leave the reactor in a different molecular configuration due to the reaction, but the total mass leaving the reactor must be the same. Here we look at a few ways to show this.

We consider the water gas shift reaction : \(CO + H_2O \rightleftharpoons H_2 + CO_2\). We can illustrate the conservation of mass with the following equation: \(\bf{\nu}\bf{M}=\bf{0}\). Where \(\bf{\nu}\) is the stoichiometric coefficient vector and \(\bf{M}\) is a column vector of molecular weights. For simplicity, we use pure isotope molecular weights, and not the isotope-weighted molecular weights. This equation simply examines the mass on the right side of the equation and the mass on left side of the equation.

import numpy as np nu = [-1, -1, 1, 1]; M = [28, 18, 2, 44]; print np.dot(nu, M)

0

You can see that sum of the stoichiometric coefficients times molecular weights is zero. In other words a CO and H_2O have the same mass as H_2 and CO_2.

For any balanced chemical equation, there are the same number of each kind of atom on each side of the equation. Since the mass of each atom is unchanged with reaction, that means the mass of all the species that are reactants must equal the mass of all the species that are products! Here we look at the number of C, O, and H on each side of the reaction. Now if we add the mass of atoms in the reactants and products, it should sum to zero (since we used the negative sign for stoichiometric coefficients of reactants).

import numpy as np # C O H reactants = [-1, -2, -2] products = [ 1, 2, 2] atomic_masses = [12.011, 15.999, 1.0079] # atomic masses print np.dot(reactants, atomic_masses) + np.dot(products, atomic_masses)

>>> ... >>> >>> >>> >>> >>> 0.0

That is all there is to mass conservation with reactions. Nothing changes if there are lots of reactions, as long as each reaction is properly balanced, and none of them are nuclear reactions!

Copyright (C) 2013 by John Kitchin. See the License for information about copying.