Q: A small molecule in water has a step size $$d = 0.1 nm $$and moves at a speed $$v = 320 m/s. $$What is the estimated diffusion coefficient $$D in $$one dimension?

$$D = v d = 8 \times 10$$^{-8}$$ m^2$/s

Q: On a log-log graph of basal metabolic rate (BMR) versus mass for mammals, the slope is approximately $$3/4. $$What does this imply about the relationship between BMR and mass?

BMR is proportional to the mass to the $$3/4$$ power: $$\text{BMR} \propto m$$^{3/4}$$.

Question 1

Which of the following best describes the purpose of a model in physics as applied to life sciences?

Accepted Answer

A model is a simplified picture of reality that captures the essence of what you want to study.

Question 2

In the context of diffusion, what does the root-mean-square (rms) distance represent?

Accepted Answer

The typical distance a molecule has moved from its starting point, calculated as the square root of the average of the squares of the positions.

Question 3

If a molecule undergoes a random walk with step size $$d$$ for $$n$$ steps, what is the formula for its rms distance from the origin?

Accepted Answer

$$x_{rms} = \sqrt{n}d$$

Question 4

Which statement about scaling laws in biology is correct?

Accepted Answer

Bone strength is proportional to the cross-sectional area, which scales with the square of a linear dimension.

Question 5

A small molecule in water has a step size $$d = 0.1 nm $$and moves at a speed $$v = 320 m/s. $$What is the estimated diffusion coefficient $$D in $$one dimension?

Accepted Answer

$$D = v d = 8 	imes 10$$^{-8}$$ m^2$/s

Question 6

Which of the following is a key difference between descriptive and explanatory models in physics?

Accepted Answer

Descriptive models examine essential characteristics, while explanatory models focus on causes and mechanisms.

Question 7

On a log-log graph of basal metabolic rate (BMR) versus mass for mammals, the slope is approximately $$3/4. $$What does this imply about the relationship between BMR and mass?

Accepted Answer

BMR is proportional to the mass to the $$3/4$$ power: $$	ext{BMR} \propto m$$^{3/4}$$.

Question 8

If the maximum possible travel distance for a molecule after $$n$$ steps is $$nd$$, and the diffusion distance is $$\sqrt{n}d$$, what happens to the ratio $$\frac{\sqrt{n}d}{nd} as$$ $$n$$ increases?

Accepted Answer

The ratio decreases as $$1/\sqrt{n}$$, meaning the diffusion distance becomes a smaller fraction of the maximum possible distance as $$n$$ increases.

Question 9

Which of the following best illustrates the use of quantitative reasoning in physics for life sciences?

Accepted Answer

Critically evaluating and solving problems using numerical information, mathematical concepts, and statistical methods.

Question 10

A researcher models the movement of molecules in a cell as a random walk. After 100 steps of size $$d$$, what is the expected average position $$x_{avg} of $$the molecules?

Accepted Answer

$$x_{avg} = 0$$

Question 11

Which mathematical property allows us to convert a power law relationship $$Y = C X^r$$ into a straight line on a log-log graph?

Accepted Answer

The properties of logarithms, specifically $$\log Y = r \log X + \log C.$$

Question 12

Why do the legs of an elephant need to be much thicker than those of a shrew, even if scaled proportionally?

Accepted Answer

Because bone strength scales with cross-sectional area, which increases slower than mass as size increases.

Question 13

Which of the following is NOT a common use of log-log graphs in biology?

Accepted Answer

Plotting the atomic structure of molecules.

Question 14

If the variance of a set of positions is $$\$$sigma^2$$, what is the standard deviation?

Accepted Answer

$$\sqrt{\$$sigma^2$$}$$

Question 15

Which of the following best describes allometry in biology?

Accepted Answer

The study of how characteristics of living creatures change with size, often following power-law scaling.

Question 16

A histogram of molecular positions after a random walk forms a bell curve. What does the width of this curve represent?

Accepted Answer

The rms distance, which quantifies the typical spread from the origin.

Question 17

Which equation correctly relates the diffusion distance $$x_{rms} to $$the diffusion coefficient $$D$$ and time $$t in $$one dimension?

Accepted Answer

$$x_{rms} = \sqrt{2Dt}$$

Question 18

Which of the following is a correct statement about the use of physics in life sciences?

Accepted Answer

Physics provides broad principles and quantitative analysis that help explain biological phenomena.

Physics for the Life Sciences: Chapter 1 Study Notes

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