Question

The enthalpy change involved in the oxidation of glucose is \(-2880 \mathrm{~kJ} \mathrm{~mol}^{-1}\). Twenty five percent of this energy is available for muscular work. If \(100 \mathrm{~kJ}\) of muscular work is needed to walk one kilometer, what is the maximum distance that a person will be able to walk eating \(120 \mathrm{~g}\) of glucose? (a) \(7.9 \mathrm{~km}\) (b) \(9.7 \mathrm{~km}\) (c) \(4.8 \mathrm{~km}\) (d) \(8.4 \mathrm{~km}\)

Short answer

The maximum distance is 4.8 km.

Step-by-step solution

Calculate moles of glucose

First, determine how many moles of glucose are in 120 g. The molar mass of glucose (\( \text{C}_6\text{H}_{12}\text{O}_6 \)) is approximately \(180 \text{ g/mol}\). Therefore, \[\text{Moles of glucose} = \frac{120 \text{ g}}{180 \text{ g/mol}} = 0.67 \text{ mol}\]

Calculate total energy from glucose

Next, calculate the total energy that can be obtained from these moles of glucose using the enthalpy change. The enthalpy change is \[-2880 \text{ kJ/mol}.\] So, \[\text{Total energy} = 0.67 \text{ mol} \times 2880 \text{ kJ/mol} = 1929.6 \text{ kJ}\]

Calculate energy available for work

Now, determine the energy available for muscular work, which is 25% of the total energy: \[\text{Energy for work} = 1929.6 \text{ kJ} \times 0.25 = 482.4 \text{ kJ}\]

Calculate maximum walking distance

Finally, calculate the maximum distance that can be walked using the available energy. If 100 kJ is needed to walk one kilometer, then: \[\text{Number of kilometers} = \frac{482.4 \text{ kJ}}{100 \text{ kJ/km}} = 4.824 \text{ km}\] Round this value to one decimal place gives us 4.8 km.

Key concepts

Enthalpy change

In the world of thermodynamics, enthalpy change is an essential concept. It refers to the total heat content in a system and is symbolized by the Greek letter delta (ΔH). Enthalpy change accounts for the heat absorbed or released during a chemical reaction when pressure is constant. This intrinsic property helps us understand the energy transfers within a system, such as when glucose undergoes oxidation.

In our exercise involving glucose, the enthalpy change is \(-2880 \text{ kJ/mol}\). This negative value tells us that the oxidation of glucose is an exothermic reaction, meaning it releases energy. Understanding whether a reaction is endothermic (absorbing heat) or exothermic (releasing heat) is crucial because it affects how the energy can be harnessed for work, like in muscle movement.

Glucose oxidation

Glucose oxidation is a significant biochemical process where glucose molecules are broken down to release energy. This process occurs in living organisms, including humans, every moment to power activities. During this process, glucose reacts with oxygen to produce carbon dioxide, water, and energy. The chemical equation for glucose oxidation is:

\[\text{C}_6\text{H}_{12}\text{O}_6 + 6 \text{O}_2 \rightarrow 6 \text{CO}_2 + 6 \text{H}_2\text{O} + \text{Energy} \]

The energy released during this reaction is crucial. For instance, in our exercise, some of this energy is used for muscular work. However, not all energy released is available for this purpose due to energy conversion efficiencies and other losses. It highlights why only a portion of the total energy from glucose can be utilized for activities such as walking.

Molar mass calculation

Molar mass is a concept in chemistry that refers to the mass of one mole of a given substance. It is expressed in grams per mole (g/mol). The molar mass allows chemists to convert between the mass of a substance and the number of moles, which are pivotal in stoichiometric calculations. This concept was necessary to find out how many moles of glucose were in \(120 \text{ g}\) of glucose.

Glucose has a chemical formula of \(\text{C}_6\text{H}_{12}\text{O}_6\) and a molar mass of approximately \(180 \text{ g/mol}\). Calculating the moles involves dividing the mass of glucose by its molar mass:

\[\text{Moles of glucose} = \frac{120 \text{ g}}{180 \text{ g/mol}} = 0.67 \text{ mol}\]

This step is foundational as it ensures that we comprehend how much glucose we are working with in the context of reactions.

Energy conversion

Energy conversion is the process of transforming energy from one form to another. When dealing with biological processes, such as metabolic reactions, it's crucial to understand that not all energy is usable for work. In biological systems, typically only a fraction is converted into a form that organisms can use.

Taking our exercise example, when glucose undergoes oxidation, the total energy released is not entirely available for physical activities. It is stated that only \(25\%\) of the energy is available for muscular work. This conversion takes into account energy "losses" due to inefficiencies in biological systems, where some energy is always lost as heat.

• Initially, the full reaction releases \(1929.6 \text{ kJ}\) from \(0.67 \text{ mol}\) of glucose.
• Only \(25\%\) of this, equating to \(482.4 \text{ kJ}\), is available for walking.

Realistically, energy that organisms can harness for performing work is often far less than the total energy content, highlighting the importance of understanding energy conversion in biology.