Combustion of glucose ($C_6H_{12}O_6$) is the main source of energy for animal cells:
$C_6H_{12}O_6(s) + 6O_2(g) \rightarrow 6CO_2(g) + 6H_2O(l) \quad \Delta \dot{G}_{r \dot{x} n}(37^{\circ}C) = -2872 . kJ$
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 forma one peptide bond, joining one amino acid to another, is $21 kJ / mol$.
Suppose some cells are assembling a certain protein made of 23 amino acids. (Note that the number of peptide bonds in the protein will be one less thar number of amino acids.) Calculate the minimum mass of glucose that must be burned to assemble 450. $\mu$mol of this protein.
Round your answer to 2 significant digits.
Answer (1)
Calculate the total Gibbs free energy required to assemble 450 μ mol of the protein: 462 mol protein kJ × 450 × 1 0 − 6 mol protein = 0.2079 kJ .
Calculate the number of moles of glucose required: 2872 mol glucose kJ 0.2079 kJ = 7.238 × 1 0 − 5 mol glucose .
Calculate the molar mass of glucose: ( 6 × 12.01 ) + ( 12 × 1.008 ) + ( 6 × 16.00 ) = 180.156 g/mol .
Calculate the mass of glucose required and round to 2 significant digits: 7.238 × 1 0 − 5 mol glucose × 180.156 mol glucose g × 1000 g mg = 13 mg .
13 mg
Explanation
Problem Setup We are given the balanced chemical equation for the combustion of glucose and the change in Gibbs free energy for the reaction. We are also given the Gibbs free energy required to form one peptide bond. Our goal is to find the minimum mass of glucose that must be burned to assemble a certain amount of a protein.
Energy for Protein Assembly First, we need to determine the total Gibbs free energy required to assemble the protein. The protein is made of 23 amino acids, so there are 22 peptide bonds (since the number of peptide bonds is one less than the number of amino acids). The Gibbs free energy for forming one peptide bond is 21 kJ/mol. Therefore, the total Gibbs free energy required to assemble one mole of the protein is: 22 protein peptide bonds × 21 mol peptide bond kJ = 462 mol protein kJ
Energy for Given Amount of Protein Next, we need to calculate the total Gibbs free energy required to assemble 450 μ mol of the protein. We have: 462 mol protein kJ × 450 × 1 0 − 6 mol protein = 0.2079 kJ
Moles of Glucose Required Now, we need to calculate the number of moles of glucose required to produce this amount of energy. The combustion of one mole of glucose releases 2872 kJ of energy. Therefore, the number of moles of glucose required is: 2872 mol glucose kJ 0.2079 kJ = 7.238 × 1 0 − 5 mol glucose
Molar Mass of Glucose We need to calculate the molar mass of glucose ( C 6 H 12 O 6 ). The molar masses of carbon, hydrogen, and oxygen are approximately 12.01 g/mol, 1.008 g/mol, and 16.00 g/mol, respectively. Therefore, the molar mass of glucose is: ( 6 × 12.01 ) + ( 12 × 1.008 ) + ( 6 × 16.00 ) = 72.06 + 12.096 + 96.00 = 180.156 g/mol
Mass of Glucose Required Now, we can calculate the mass of glucose required: 7.238 × 1 0 − 5 mol glucose × 180.156 mol glucose g = 0.01304 g
Final Answer Finally, we convert the mass to milligrams and round to 2 significant digits: 0.01304 g × 1000 g mg = 13.04 mg Rounding to 2 significant digits, we get 13 mg.
Conclusion Therefore, the minimum mass of glucose that must be burned to assemble 450 μ mol of the protein is 13 mg.
Examples
In the human body, glucose is broken down to provide energy for various cellular processes, including protein synthesis. This problem illustrates how much glucose is needed to produce a specific amount of protein. Understanding these calculations is crucial in fields like nutrition, where energy requirements are carefully monitored to ensure optimal health and performance. For example, athletes need to consume enough glucose to support muscle protein synthesis after intense workouts. Similarly, individuals recovering from illness may require tailored glucose intake to facilitate tissue repair and protein production. The calculations we performed here help to quantify these relationships and inform dietary recommendations.
