David Bauer Physics & Astronomy · UCLA

Physics 5A Chapter 21

Molecular, chemical, and body energy

Application: molecular energy

Energy diagrams are not just a tool for blocks and springs. One of the most useful applications is to the interaction between two atoms in a molecule. Consider the following energy diagram, which shows a model for the interaction between two atoms in a diatomic molecule (for example, CO or N₂).

A graph of the interaction potential energy U of r between two atoms versus their separation r. The blue curve drops steeply from the upper left (a strong repulsion at small separation), crosses below zero, reaches a single minimum at an intermediate separation (the bond length), then rises gently back toward zero as r increases (a weak attraction at large separation, with U approaching zero as the atoms move far apart).

The coordinate $r$ is the distance between the atoms. We choose to set the potential energy $U = 0$ when the atoms are separated infinitely far apart ($r \to \infty$).

Recall that the force is minus the slope of the potential energy curve. Reading the two branches of the curve tells us how the atoms interact at different separations. When the atoms are close together (small $r$), the interaction is repulsive.

The same atom-atom potential energy curve U of r. The steep left branch, at small separation r, is highlighted: a callout labels this region as a repulsive interaction, where the potential energy rises sharply as the atoms are pushed together.

When the atoms are farther apart (large $r$), the interaction is attractive.

The same atom-atom potential energy curve U of r. The gently rising right branch, at large separation r, is highlighted: a callout labels this region as an attractive interaction, where the potential energy increases back toward zero as the atoms are pulled apart.

In between, the interaction has a point of stable equilibrium at the bond length of the molecule.

The same atom-atom potential energy curve U of r, with a black dot marking its minimum. A dotted guide rises from the minimum to the horizontal axis, where the separation is labeled the bond length. A callout identifies the minimum as a point of stable equilibrium: the slope of U is zero there, so the net interatomic force vanishes.

Bound and unbound molecules

The minimum energy the molecule needs in order for the atoms to completely separate is $E_\mathrm{mech} = 0$, since this is the potential energy when the atoms are very far apart.

If the total mechanical energy is less than zero, the atoms cannot separate. Since there is a maximum distance between the atoms, the molecule is bound.

The atom-atom potential energy curve U of r with a horizontal orange total-energy line drawn below the axis, at a negative value of the mechanical energy. The line spans the full width of the plot and crosses the curve at two points, marked with black dots, which are the turning points of the relative motion. Because the energy is negative, the atoms cannot separate to large r: the molecule is bound.

Since the molecule sits near a stable equilibrium, a bound molecule oscillates just like a mass on a spring or a pendulum.

The bound-molecule energy diagram: the atom-atom potential curve U of r with a negative horizontal total-energy line, spanning the full width, meeting the curve at two turning points marked by black dots. A callout notes that the molecule oscillates back and forth between these two turning points, like a mass on a spring.

This means we can model the atoms in a molecule as being connected by microscopic springs.

Two atoms, drawn as blue spheres each labeled m, connected by a horizontal coil spring. This represents the model of a diatomic molecule near its bond length as two masses joined by a microscopic spring.

If the energy is greater than zero, the molecule is no longer bound and can dissociate.

The atom-atom potential energy curve U of r with a horizontal orange total-energy line drawn above the axis, at a positive value of the mechanical energy. The line spans the full width and meets the steep repulsive branch at a single turning point on the left but never meets the curve again at large r. Because the energy is positive, the atoms can separate to infinity: the molecule is unbound and can dissociate.

Ground state energy and bond energy

The minimum energy the molecule can have is called the ground state energy. For quantum-mechanical reasons, the molecule cannot sit exactly at rest at the equilibrium point. In the ground state, it still vibrates slightly about the bond length.

The atom-atom potential energy curve U of r with a horizontal orange total-energy line drawn just above the bottom of the well, at the lowest energy the molecule can have. The full-width line meets the curve at two closely spaced turning points near the bond length, marked with black dots, indicating that even in its ground state the molecule vibrates slightly about the bond length.

The energy needed to separate the molecule from its ground state is called the bond energy.

The atom-atom potential energy curve U of r with the ground-state energy line drawn just above the bottom of the well, spanning the full width. A vertical double-headed arrow runs from the ground-state level up to zero (the energy of the free, separated atoms); this gap is labeled the bond energy, the energy needed to dissociate the molecule from its ground state.

Chemical energy

The greater the bond energy, the more stable the molecule.

An atom-atom potential energy curve U of r with a shallow well: the minimum is only a little below zero. A shallow well means a small bond energy and a less stable, more easily broken bond.

Lower bond energy $\Rightarrow$ less stable bond.

An atom-atom potential energy curve U of r with a deep well: the minimum lies well below zero. A deep well means a large bond energy and a more stable, harder-to-break bond.

Greater bond energy $\Rightarrow$ more stable bond.

Energy is not stored in chemical bonds — a bound system always has lower energy than an unbound system. However, a reaction that takes a system from a less stable state to a more stable state releases energy.

A reaction energy diagram comparing two molecules. A dashed horizontal line labeled Free atoms marks zero energy, with a vertical Energy axis and a zero mark at each side. On the left is a shallow potential well (the reactants) whose minimum, the reactant energy E sub reac, lies a short distance below zero; on the right is a deeper well (the products) whose minimum, the product energy E sub prod, lies farther below zero. A vertical double-headed arrow in the middle, labeled Delta E reaction, spans the gap between the two energy levels. Labels on the curves show kinetic energy converting to potential on the left and potential converting to kinetic on the right. Captions read: reactants are less tightly bound; Delta E reaction is the net energy change of the reaction; products are more tightly bound.

When a macroscopic number of chemical reactions take place, we call the total energy released or absorbed by the system a change in chemical energy $\Delta E_\mathrm{chem}$.

For example, combustion of one glucose molecule releases about $5 \times 10^{-18}\ \mathrm{J}$. The chemical energy released by combusting $1\ \mathrm{mol}$ of glucose is

\[|\Delta E_\mathrm{chem}| \approx 2800\ \mathrm{kJ}.\]

Energy in the body

One thing that makes energy conservation so powerful is that we can use it even for systems whose microscopic details are very complicated, like living organisms. Consider a simple model of the body: energy enters, is stored and transformed internally, and leaves as thermal energy or as work done on the environment.

A schematic of the human body as an energy converter. A rounded rectangular system boundary labelled Body contains the internal stores and carriers ATP, glycogen, and fat. Two arrows point inward from the left: food, carrying chemical energy, and oxygen. Three arrows point outward to the right: mechanical work, thermal energy, and carbon dioxide plus water. The picture says energy and matter enter as food and oxygen and leave as work, heat, and waste products.

Chemical energy from food

The energy input to the body comes from food. In metabolism, carbohydrates, proteins, and fats are transformed into lower-energy products.

For example, consider oxidation of glucose. The net reaction is the same as combustion, although the body releases the energy through many controlled steps:

\[\mathrm{C_6H_{12}O_6 + 6\,O_2 \longrightarrow 6\,CO_2 + 6\,H_2O}, \qquad |\Delta E_\mathrm{chem}| \approx 2.8\ \mathrm{MJ/mol}.\]

The products $\mathrm{CO_2}$ and $\mathrm{H_2O}$ are lower-energy chemical states than glucose and oxygen, so the reaction releases energy.

The energy changes at the scale of individual molecules are tiny, but metabolism involves enormous numbers of molecules. Some useful scales for reference:

A vertical energy axis with an upward arrow labelled increasing energy and four tick marks, from a tiny molecular energy at the bottom to a large daily energy at the top. Bottom tick: one ATP hydrolysis, about 8 times 10 to the minus 20 joules. Next: one glucose molecule oxidized, about 5 times 10 to the minus 18 joules. Next: one food Calorie, 4184 joules. Top tick: one day at 100 watts, about 8.6 times 10 to the 6 joules, roughly 2000 Calories. The tick spacing is schematic, not a true logarithmic scale.

Macronutrient energy density

Different macronutrients release different amounts of chemical energy per gram:

Macronutrient Energy / g ($\mathrm{kJ}$) Energy / g ($\mathrm{kcal}$)
Protein $17$ $4$
Carbohydrate $17$ $4$
Fat $37$ $9$

Fat releases more energy per gram because it starts farther from the low-energy products $\mathrm{CO_2}$ and $\mathrm{H_2O}$. Fat molecules contain relatively little oxygen already (compare glucose $\mathrm{C_6H_{12}O_6}$ to, e.g., palmitic acid $\mathrm{C_{16}H_{32}O_2}$).

ATP and energy storage

At the cellular level, the body usually does not use the chemical energy in food directly. Instead, energy is transferred through ATP.

  • Hydrolyzing one ATP molecule to ADP releases about $8 \times 10^{-20}\ \mathrm{J}$, the energy scale of molecular processes such as ion pumps, protein synthesis, and muscle contraction.
  • A cell stores only a small amount of ATP at one time, so ATP is continually regenerated from ADP using energy from food. Larger energy reserves are stored as glycogen and as fat.

Metabolic power

How fast does the body use energy? As an order-of-magnitude estimate, the basal metabolic rate (BMR) is about $100\ \mathrm{W}$. This is the power required to maintain basic bodily functions at rest.

This means that, at rest, the body transforms roughly $100\ \mathrm{J}$ of chemical energy every second. Over one day,

\[\left(100\ \tfrac{\mathrm{J}}{\mathrm{s}}\right)\!\left(8.64 \times 10^{4}\ \tfrac{\mathrm{s}}{\mathrm{day}}\right) \approx 8.64\ \tfrac{\mathrm{MJ}}{\mathrm{day}} \approx 2000\ \tfrac{\mathrm{Cal}}{\mathrm{day}}.\]

Even at rest, maintaining ion gradients, running organs, and replacing molecules all require continuous power. Most of that energy eventually leaves as thermal energy.

Physical activity adds to the metabolic power. The additional energy expenditure depends on the type and intensity of the activity. You can find many values for various activities tabulated in your favorite exercise physiology textbook. For example, a $73\ \mathrm{kg}$ person walking on level ground at $3\ \mathrm{mph}$ expends about $4.4\ \mathrm{kcal/min}$ above their BMR.

To estimate metabolic power output (for aerobic metabolism), we can measure oxygen consumption as a proxy. About $20\ \mathrm{kJ}$ of chemical energy is released per liter of $\mathrm{O_2}$ consumed.

Mechanical efficiency

Earlier we saw that the body converts food chemical energy into mechanical work with an efficiency of about $e \approx 0.25$. The other ${\sim}75\%$ ends up mostly as thermal energy. If $e = 0.25$, then for every joule of mechanical work the body delivers,

\[\Delta E_\mathrm{th} \approx 3\,|\Delta E_\mathrm{mech}|.\]

This is why vigorous exercise warms the body. High metabolic power also means high thermal energy generation.

Choosing a system that includes the body

Now that we have a model of the body as a physical system with energy inputs and outputs, we can include the body in our energy bookkeeping.

For example, consider the process of lifting a book from rest and placing it on a shelf. The energy bookkeeping changes depending on our choice of system.

Book alone. The lifting work is balanced by the work done by gravity:

\[\Delta E_\mathrm{sys} = W_\mathrm{lift} + W_g = 0.\]

An energy-system diagram for the book alone. A rounded rectangular system boundary contains an empty kinetic-energy bar labeled K, showing that the kinetic energy does not change. A thick purple arrow labeled W sub lift points into the box from the left, and an equal thick purple arrow labeled W sub g points out of the box to the right: all the work done to lift the book is carried back out by gravity.

Book + Earth. The lifting work increases the gravitational potential energy:

\[\Delta U_g = W_\mathrm{lift}.\]

An energy-system diagram for the book plus the Earth. A rounded rectangular system boundary contains a gravitational potential-energy bar labeled U sub g, partly filled blue. A single thick purple arrow labeled W sub lift points into the box from the left; there is no outflow, so the work done by lifting is stored as increased gravitational potential energy.

Check your understanding: Now include the person lifting the book along with the book and Earth. What does the energy accounting look like in this case? Draw an energy bar chart for the system.

Answer

There is no external force on the system. The energy used to lift the book comes from chemical energy in the body, which is transformed into gravitational potential energy and thermal energy:

\[\Delta E_\mathrm{chem} + \Delta U_g + \Delta E_\mathrm{th} = 0.\]

An energy-system diagram for the book, the Earth, and the person together, with no external arrows because the system is isolated. Inside the rounded boundary, three energy-change bars stand on a dashed zero line. A small blue bar labeled U sub g rises slightly above the line (gravitational potential energy increases a little); a tall orange bar labeled Delta E sub th rises well above the line (thermal energy increases a lot); and a tall teal bar labeled Delta E sub chem hangs below the line (chemical energy decreases). The three changes sum to zero.

The big picture

What have we learned?

I hope you come away from this class with a better understanding of physics — but more importantly, with a clearer idea of what learning and doing physics is like. Going a step further, my ultimate goal is for you to leave this class with an expanded view of what you are capable of.

A few themes ran through everything we did:

  • We can understand many complicated phenomena using simplified mathematical models.
  • Even though the models are “simple” compared to the real world, the process of applying them may not be. Physics is not just about finding the right formula and plugging in!
  • Most of what we did in this class comes down to modeling interactions between systems — either through forces or through energy.

What does this process actually look like?

  • Represent the physical situation. Draw pictures, choose coordinates, define your system.
  • Plan your approach. What is your strategy? What do you know? What do you need?
  • Execute your plan. Write down equations, do the algebra.
  • Reflect on your result. (Don’t forget this step!) Does your answer make sense? Is there another way to think about the problem? What can you learn from your process?