Combustion of glucose [tex]$\left( C _6 H _{12} O _6\right)$[/tex] is the main source of energy for animal cells:
[tex]$C_6 H_{12} O_6(s)+6 O_2(g) \rightarrow 6 CO_2(g)+6 H_2 O(l) \quad \Delta G_{r x n}\left(37^{\circ} C\right)=-2872 . kJ$[/tex]
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 forming one peptide bond, joining one amino acid to another, is [tex]$21 kJ / mol$[/tex].
Suppose some cells are assembling a certain protein made of 71 amino acids. (Note that the number of peptide bonds in the protein will be one less than the number of amino acids.) Calculate the maximum moles of this protein that can be assembled if [tex]$600 . mg$[/tex] of glucose are available for combustion.
Round your answer to 2 significant digits.
Answer (2)
Calculate the moles of glucose available: n g l u cose = 180.156 g / m o l 0.6 g = 0.00333 m o l .
Determine the total energy released from glucose combustion: Δ G co mb u s t i o n = 0.00333 m o l × 2872 k J / m o l = 9.565 k J .
Calculate the energy required to assemble one mole of the protein: E p ro t e in = 70 × 21 k J / m o l = 1470 k J / m o l .
Calculate the maximum moles of protein assembled: n p ro t e in = 1470 k J / m o l 9.565 k J = 0.00651 m o l ≈ 0.01 m o l .
The maximum moles of the protein that can be assembled is 0.01
Explanation
Problem Analysis We are given the combustion reaction of glucose and the Gibbs free energy change for this reaction. We also know the Gibbs free energy required to form one peptide bond. Our goal is to find the maximum moles of a protein, made of 71 amino acids, that can be assembled from a given mass of glucose.
Glucose Moles Calculation First, we need to calculate the molar mass of glucose ( C 6 H 12 O 6 ). M g l u cose = 6 × 12.01 + 12 × 1.008 + 6 × 16.00 = 180.156 m o l g Next, we convert the mass of glucose from milligrams to grams: 600 × m g = 0.6 g Now, we calculate the number of moles of glucose: n g l u cose = 180.156 m o l g 0.6 g = 0.00333 m o l
Energy Calculations Now we calculate the total energy released from the combustion of glucose: Δ G co mb u s t i o n = n g l u cose × ∣Δ G r x n ∣ = 0.00333 m o l × 2872 m o l k J = 9.565 k J Since the protein is made of 71 amino acids, the number of peptide bonds is 70. n p e pt i d e b o n d s = 71 − 1 = 70 We calculate the energy required to assemble one mole of the protein: E p ro t e in = 70 × 21 m o l k J = 1470 m o l k J
Protein Moles Calculation Finally, we calculate the maximum moles of protein that can be assembled: n p ro t e in = E p ro t e in Δ G co mb u s t i o n = 1470 m o l k J 9.565 k J = 0.00651 m o l Rounding to 2 significant digits, we get 0.01 mol.
Final Answer Therefore, the maximum moles of the protein that can be assembled is 0.01 mol.
Examples
Proteins are essential for building and repairing tissues, making enzymes and hormones, and supporting the immune system. The energy required to assemble these proteins comes from the food we eat, primarily through the combustion of glucose. Understanding how much protein can be synthesized from a given amount of glucose helps in optimizing diets for athletes, patients recovering from illness, and individuals with specific dietary needs. For example, knowing the energy requirements for protein synthesis can help dieticians design meal plans that maximize muscle growth or tissue repair.
The maximum moles of a protein made of 71 amino acids that can be synthesized from 600 mg of glucose is approximately 0.01 moles. This is calculated by determining the moles of glucose available, the energy released from combustion, and the energy needed to form peptide bonds. After calculations, we find the total energy from glucose allows for the synthesis of a small amount of protein.
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