Q: If a molecule in water has a speed $$v = 320 m/s $$and a step size $$d = 0.1 nm$$, what is the estimated diffusion coefficient $$D$$ using $$D = \frac{1}{3} v d$$?

$$D = 8 \times 10$$^{-9}$$ m^2$/s

Question 1

Which of the following best describes the role of models in physics, especially when applied to biological systems?

Accepted Answer

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

Question 2

In a random walk model of molecular diffusion, a large number of molecules start at the origin and take 100 steps, each of length $$d in a $$random direction along the $$x-$$axis. What is the most likely final position for any given molecule?

Accepted Answer

$$x = 0$$

Question 3

After 100 steps of a random walk, each of length $$d$$, what is the diffusion distance from the origin for a large number of molecules?

Accepted Answer

$$60d$$

Question 4

Which of the following is NOT a major goal for students studying physics for the life sciences, according to the text?

Accepted Answer

Memorize all equations without understanding their application.

Question 5

The diffusion coefficient $$D$$ for small molecules in water is estimated using $$D = \frac{1}{3} v d$$, where $$v is $$molecular speed and $$d is $$the step size. If the same type of reasoning is applied to molecules diffusing through air, how does $$D$$ compare to its value in water?

Accepted Answer

Larger than in water

Question 6

If the electric force between two ions is $$1.00 nN $$and the distance between them doubles, what is the new force?

Accepted Answer

$$0.25 nN$$

Question 7

The mean-square (rms) distance for a random walk is given by $$x_{rms} = \sqrt{n} d. If a $$molecule takes $$10^6$$ steps of length $$d$$, what is the rms distance traveled?

Accepted Answer

$$10^3 d$

Question 8

In biology, scaling laws often relate two quantities by a power law: $$Y = C X^r. If $$you plot $$\log Y$$ versus $$\log X$$, what does the slope of the line represent?

Accepted Answer

The exponent $$r$$

Question 9

An animal's mass $$m is $$proportional to the cube of some linear dimension $$l$$ ($$m \propto l^3$), while bone strength is proportional to the square of the diameter (S$$ \propto $$l^2). As $$animals get larger, how must bone diameter scale to support their increased mass?

Accepted Answer

Bone diameter must increase faster than $$l$$

Question 10

A horse has a mass of $$400 kg $$and a resting heart rate of $$38$$ beats/min. According to the scaling law $$R \propto m$$^{-1/4}$$, what is the expected resting heart rate of an 80$ g hamster?

Accepted Answer

Much greater than $$38$$ beats/min

Question 11

Which of the following statements about proportional reasoning is correct?

Accepted Answer

Linear proportionality allows us to use ratios to draw conclusions without knowing the proportionality constant.

Question 12

If the basal metabolic rate (BMR) of mammals scales with mass as $$BMR \propto m$$^{3/4}$$, what happens to BMR when mass increases by a factor of 16$?

Accepted Answer

BMR increases by a factor of $$8$$

Question 13

Which property of logarithms allows you to convert a power law $$Y = C X^r$$ into a linear relationship on a log-log plot?

Accepted Answer

$$\log(a^n) = n \log(a)$$

Question 14

Why do physicists often use simple models and quantitative analysis when studying biological systems?

Accepted Answer

To capture essential features and make predictions about complex phenomena.

Question 15

If a molecule in water has a speed $$v = 320 m/s $$and a step size $$d = 0.1 nm$$, what is the estimated diffusion coefficient $$D$$ using $$D = \frac{1}{3} v d$$?

Accepted Answer

$$D = 8 	imes 10$$^{-9}$$ m^2$/s

Question 16

Which of the following is a correct statement about the spread of molecules due to diffusion?

Accepted Answer

The rms distance is the standard deviation of the histogram of positions.

Question 17

In the context of physics for life sciences, why is quantitative reasoning important?

Accepted Answer

It allows for precise predictions and understanding of biological phenomena.

Question 18

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

Accepted Answer

The ratio decreases.

Question 19

Which of the following is a real-world example of a scaling law in biology?

Accepted Answer

The relationship between the strength of bones and the cross-sectional area of the bone.

Physics for the Life Sciences: Foundations and Quantitative Reasoning

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