Write a balanced equation for the given chemical reaction. CO2+H2 O →C6 H12 O6+O2 (photosynth

Write a balanced equation for the given chemical reaction....

Key Concepts

Balanced Chemical Equation

A balanced chemical equation accurately represents a chemical reaction by ensuring that the number of atoms of each element is the same in both the reactants and the products. This reflects the law of conservation of mass, where matter is neither created nor destroyed during the reaction.

Stoichiometry

Stoichiometry involves the quantitative relationships between the reactants and the products in a chemical reaction. It uses the coefficients in a balanced chemical equation to calculate the relative amounts of substances needed or produced, ensuring that the ratios abide by the conservation of mass.

Conservation of Mass

The principle of conservation of mass states that in any chemical reaction, the total mass of the reactants equals the total mass of the products. This concept underpins the need for balancing chemical equations, as the same atoms must be present on both sides of the equation.

Photosynthesis

Photosynthesis is the biological process used by plants, algae, and some bacteria to convert carbon dioxide and water into glucose and oxygen, usually with the assistance of sunlight. It is a key process in the energy cycle of ecological systems, linking inorganic and organic chemistry.

Transcript

00:01 Hi, so for this exercise we have a chemical reaction and we should work with the balancing problem.

00:07 So technically we can apply linear algebra to the balancing problem because we are trying to find the proportions that multiplies to these molecules x2, x3 and x4 in this case four unknowns that corresponds to these integral values that multiplies to the molecules to obtain the same number of atoms on the left and a number of atoms.

00:30 On the right okay so based on these property of the chemical chemical equation here that we should preserve the atoms we should work to construct our equations or set of linear equations so let's focus first on the carbon so what happened here is that the carbon we have only one atom of carbon multiplying x1 so we have x1 we have any other carbon here on the left part so this should be equal to the six atoms that are multiplying to x3.

01:07 That means six times x3.

01:12 Then we have the atom of hydrogen.

01:17 So the hydrogen on the left part, we have two atoms multiplying x2.

01:23 So two times x2.

01:26 And this should be equals to the 12 atoms that are multiplying to x3.

01:35 And finally we have the oxygen.

01:39 So on the left we have two atoms of oxygen that are multiplying x1, that means 2 times x1 plus the oxygen multiply in x2 so we have here x2 and this should be equals to 6 times x3 plus 2 times x4.

02:08 Okay so we can rewrite equations as x1 minus 6x3 equals to 0 to x2 minus 12 x3 equals to 0 and 2x1 was x2 minus 6x3 and minus 2x4 equals to 0 so you can see that we have an homogeneous system of linear equations so to solve that we have also we have four unknowns and three equations that means that this system will have an infinitely many solutions mathematically speaking of course so having this set of linear equations we can consider the extended matrix associated to this system to solve the problem but first you can see that from this equation from this yeah from this equation you can divide by two so we have this equation here okay so let's put this on the matrix one zero minus six zero zero one minus six zero and two one minus six minus two and the extended matrix so we have this matrix here.

03:49 So let's put a zero on this position...

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