Combustion of glucose ($C_6H_{12}O_6$) is the main source of energy for animal cells:
$C_6H_{12}O_6(s) + 6O_2(g) \rightarrow 6CO_2(g) + 6H_2O(l) \quad \Delta G_{rxn}(37^{\circ}C) = -2872. kJ$
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.
Answer (2)
Calculate the number of peptide bonds: 81 − 1 = 80 .
Calculate the energy required per mole of protein: 80 \tims 21 = 1680 kJ/mol.
Calculate the total energy required for 650 μ mol of protein: 1680 \tims 650 \tims 1 0 − 6 = 1.092 kJ.
Calculate the mass of glucose required: 2872 1.092 \tims 180.16 = 0.0685 g, which rounds to 0.07 g .
Explanation
Understanding the Problem First, let's break down the problem. We need to find the mass of glucose required to assemble a certain amount of protein. We know the energy required to form a peptide bond, the number of amino acids in the protein, the amount of protein to be assembled, and the energy released during glucose combustion.
Calculating Peptide Bonds Next, we calculate the number of peptide bonds in the protein. Since the protein is made of 81 amino acids, there are 80 peptide bonds.
Energy per Mole of Protein Now, let's find the total Gibbs free energy required to assemble one mole of the protein. We multiply the number of peptide bonds by the Gibbs free energy of formation of one peptide bond: E p ro t e in = 80 × 21 m o l k J = 1680 m o l k J
Total Energy Required We need to calculate the total Gibbs free energy required to assemble 650 μ m o l of the protein. We multiply the energy required per mole of protein by the amount of protein assembled: E t o t a l = 1680 m o l k J × 650 × 1 0 − 6 m o l = 1.092 k J
Moles of Glucose Required Now, we calculate the number of moles of glucose required to produce this amount of energy. We divide the total energy required by the Gibbs free energy released per mole of glucose: n g l u cose = 2872 k J / m o l 1.092 k J = 3.802 × 1 0 − 4 m o l
Mass of Glucose Required Next, we calculate the mass of glucose required. We multiply the number of moles of glucose by the molar mass of glucose (180.16 g/mol): m g l u cose = 3.802 × 1 0 − 4 m o l × 180.16 m o l g = 0.0685 g
Rounding the Answer Finally, we round the answer to 2 significant digits: 0.069 g.
Examples
In the human body, the energy from glucose combustion is used to synthesize proteins, which are essential for building and repairing tissues. This process is similar to constructing a building where glucose is the fuel, amino acids are the bricks, and peptide bonds are the mortar. Understanding the stoichiometry and energy requirements helps us appreciate how our bodies efficiently convert food into functional components.
To assemble 650 μ m o l of a protein made of 81 amino acids, a minimum of 0.07 g of glucose must be burned. This is determined by calculating the number of peptide bonds formed, the energy requirements for bonding, and the energy released from glucose combustion. The mass of glucose is derived from the total energy needed to synthesize the protein divided by the energy output from glucose combustion.
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