Solved thermodynamics exercise

Food calories and metabolic power

Exercise 5 · Lesson 2 — A History of Thermodynamics and Calorimetry

calorie
kilocalorie
food Calorie
mechanical equivalent
metabolic power
units

Statement

A nutrition label indicates that an adult consumes about $2000\,\text{kcal}$ per day on average. The kilocalorie is a non-SI unit still used in nutrition; its SI conversion is $1\,\text{kcal}=4{,}18\times10^3\,\text{J}$ .

Convert this daily intake into joules.
Deduce the adult's average metabolic power in watts, assuming the energy is consumed uniformly over 24 hours.
An energy bar is labelled “ $210\,\text{kcal}$ ”. What mass of water, initially at $20\,^\circ\text{C}$ , could be heated to boiling point, $100\,^\circ\text{C}$ , with the same energy if all of it were converted into heat? Take $c_{\text{water}}=4{,}18\times10^3\,\text{J}\!\cdot\!\text{kg}^{-1}\!\cdot\!\text{K}^{-1}$ .
In what forms is this energy actually used by the human body? Give a qualitative answer.

Hint

For Question 2, use

\mathcal P=E/\Delta t

. For Question 3, first convert the labelled energy into joules, then solve

Q=mc\Delta T

for

m

Detailed solution

Solution

1. In SI units,

E=2000\times4{,}18\times10^3 =8{,}36\times10^6\,\text{J}.

2. The average power is

\mathcal P=\frac{8{,}36\times10^6}{8{,}64\times10^4}\approx97\,\text{W}.

Over a whole day, a person therefore uses about as much average power as an incandescent light bulb. This order of magnitude is often surprising and illustrates the value of converting food calories into familiar physical units.

3. The energy of the bar is

Q=210\times4{,}18\times10^3\approx8{,}78\times10^5\,\text{J}.

With $\Delta T=80\,\text{K}$ ,

m=\frac{Q}{c\Delta T} =\frac{8{,}78\times10^5}{4{,}18\times10^3\times80} \approx2{,}63\,\text{kg}.

Thus, to order of magnitude, one energy bar contains enough energy to bring about $2{,}5\,\text{kg}$ of tap water to boiling point.

4. The body does not consume energy in the sense of making it disappear; it transforms chemical energy stored in nutrients. Some supports muscular work, organ function, cellular electrical gradients, molecular transport, and chemical synthesis. Some may be stored chemically as glycogen or fat. Ultimately, most of the energy used is dissipated as heat, helping maintain body temperature before being transferred to the environment. A small fraction also leaves the body as chemical energy in waste.