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Combustion of glucose $\left( C _6 H _{12} O _6\right)$ is the main source of energy for animal cells:
$C_6 H_{12} O_6(s)+6 O_2(g) \rightarrow 6 CO_2(g)+6 H_2 O(l) \quad \Delta G_{r \times n}\left(37^{\circ} C\right)=-2872 . kJ$
One of the most important uses to which this energy is put is the assembly of proteins out of amino acid building blocks. The Gibbs free energy of formation of one peptide bond, joining one amino acid to another, is $21 kJ / mol$.
Suppose some cells are assembling a certain protein made of 81 amino acids. (Note that the number of peptide bonds in the protein will be one less than the number of amino acids.) Calculate the minimum mass of glucose that must be burned to assemble $650 . \mu mol$ of this protein.
Round your answer to 2 significant digits.
$7.0 \times 10^{-2} . g$
Answer (1)
Calculate the number of peptide bonds: 81 − 1 = 80 .
Calculate the energy required to form one mole of protein: 80 × 21 = 1680 kJ/mol .
Calculate the energy required to form 650 μ mol of protein: 1680 × 650 × 1 0 − 6 = 1.092 kJ .
Calculate the moles of glucose required: 2872 1.092 = 0.00038022 mol .
Calculate the mass of glucose required: 0.00038022 × 180.16 = 0.06850 g .
Round to 2 significant digits: 0.069 g .
Explanation
Problem Analysis We are given the Gibbs free energy of combustion of glucose, the energy required to form a peptide bond, the number of amino acids in a protein, and the amount of protein to be assembled. We need to calculate the mass of glucose that must be burned to assemble the given amount of protein.
Calculating Peptide Bonds First, calculate the number of peptide bonds in one protein molecule. Since the protein is made of 81 amino acids, the number of peptide bonds is one less than the number of amino acids, which is 81 − 1 = 80 peptide bonds.
Energy per Mole of Protein Next, calculate the energy required to form one mole of the protein. Each peptide bond requires 21 kJ/mol , and there are 80 peptide bonds per protein molecule. Therefore, the energy required to form one mole of the protein is: 80 protein peptide bonds × 21 mol peptide bond kJ = 1680 mol protein kJ
Total Energy Required Now, calculate the energy required to form 650 μ mol of the protein. We have 650 μ mol = 650 × 1 0 − 6 mol . The energy required is: 1680 mol kJ × 650 × 1 0 − 6 mol = 1.092 kJ
Moles of Glucose Required Then, calculate the number of moles of glucose required to produce this amount of energy. The combustion of one mole of glucose releases 2872 kJ . Therefore, the number of moles of glucose required is: 2872 mol kJ 1.092 kJ = 0.0003802228481894157 mol
Mass of Glucose Required Finally, calculate the mass of glucose required. The molar mass of glucose ( C 6 H 12 O 6 ) is 180.16 g/mol . Therefore, the mass of glucose required is: 0.0003802228481894157 mol × 180.16 mol g = 0.0685009470752089 g
Final Answer Round the mass of glucose to 2 significant digits: 0.069 g .
Examples
The combustion of glucose to produce energy for protein synthesis is analogous to using fuel to power a manufacturing plant. Just as a plant needs energy to assemble products, cells require energy from glucose to assemble proteins. Understanding the stoichiometry and energy requirements of these processes helps us optimize energy usage and resource allocation in biological systems, similar to how industrial engineers optimize processes in manufacturing.
