Q: A mole of air weighs about 30 grams. What is the mass of a single air molecule?

$$m = \frac{30 \text{ g}}{6.02 \times 10$$^{23}$$} = 5.0 \times 10$$^{-23}$$ \text{ g}$$

Q: Estimate the height at which the gravitational potential energy of an air molecule equals its thermal energy at room temperature ($$T = 300 K). $$Use $$m = 5.0 \times 10$$^{-26}$$ kg, g = 9.8$ m/s^2$, and k_B = 1.38$$ \times $$10^{-23} J/K.$$

$$h = \frac{k_B T}{mg} = \frac{1.38 \times 10$$^{-23}$$ \times 300}{5.0 \times 10$$^{-26}$$ \times 9.8} \approx 8.5 \times 10^3$ m

Q: Water has a specific heat of $$4 \ \text{J}/(\text{g} \ ^\circ\text{C})$$ and a density of $$1 \ \text{g/cm}^3. If it $$takes about a minute to boil a glass of water (250 mL) in a microwave, estimate the power of the microwave in Watts.

Power $$= \frac{mc\Delta T}{t} = \frac{250 \times 4 \times 80}{60} \approx 1333 W$$

Q: A system can exist in two energy states, $$E_1 = 2$ meV and E_2$ = 30$$ meV. If the system is $$e$$ times as likely to be found in $$E_1$$ than in $$E_2$$, what is the temperature of the system in Kelvin? ($$1$$ meV $$= 1.6 \times 10$$^{-22}$$ J, k_B = 1.38$$ \times $$10^{-23} J/K)$$

$$T = \frac{E_2$ - E_1$}{k_B \ln e} = \frac{(30 - 2) \times 1.6 \times 10$$^{-22}$$}{1.38 \times 10$$^{-23}$$ \times 1} \approx 32 K$$

Q: Consider an ideal gas at $$27.0\ ^\circC $$and $$1.00$$ atm pressure. If each molecule is at the center of a small cube and adjacent cubes touch but do not overlap, what is the length $$L of an $$edge of each small cube? ($$k_B = 1.38 \times 10$$^{-23}$$ J/K, T = 300$ K, P = 1.01$$ \times $$10^5 Pa)$$

$$L = \left( \frac{k_B T}{P} \$$right)^{1/3}$$ \approx 3.7 \times 10$$^{-9}$$ m

Q: A 25% efficient electric power plant produces one gigajoule of electric energy per second and discharges waste heat into the ocean. If a typical American home requires 20 kW for heating, how many homes could be heated with the waste heat from this plant?

The waste heat is $$3 \times 10^9$ J/s, so $$\frac{3 \times $$10^9$$}{2 \times $$10^4$$} = 150,000$$ homes.

Question 1

Why do air molecules not all settle on the ground like most other massive objects on Earth?

Accepted Answer

Because the thermal energy of air molecules is comparable to their gravitational potential energy, causing them to move rapidly and remain suspended.

Question 2

The density of air decreases with elevation. What mathematical function best describes this decrease, and why?

Accepted Answer

An exponential function, because the probability of finding a molecule at height $$h is $$proportional to $$e^{-mgh/(k_BT)}.$$

Question 3

A mole of air weighs about 30 grams. What is the mass of a single air molecule?

Accepted Answer

$$m = \frac{30 	ext{ g}}{6.02 	imes 10$$^{23}$$} = 5.0 	imes 10$$^{-23}$$ 	ext{ g}$$

Question 4

Estimate the height at which the gravitational potential energy of an air molecule equals its thermal energy at room temperature ($$T = 300 K). $$Use $$m = 5.0 	imes 10$$^{-26}$$ kg, g = 9.8$ m/s^2$, and k_B = 1.38$$ 	imes $$10^{-23} J/K.$$

Accepted Answer

$$h = \frac{k_B T}{mg} = \frac{1.38 	imes 10$$^{-23}$$ 	imes 300}{5.0 	imes 10$$^{-26}$$ 	imes 9.8} \approx 8.5 	imes 10^3$ m

Question 5

Why is the height where gravitational potential energy equals thermal energy a good estimate for the thickness of the atmosphere?

Accepted Answer

Because the density of air falls off exponentially, and most molecules are found below this height.

Question 6

Water has a specific heat of $$4 \ 	ext{J}/(	ext{g} \ ^\circ	ext{C})$$ and a density of $$1 \ 	ext{g/cm}^3. If it $$takes about a minute to boil a glass of water (250 mL) in a microwave, estimate the power of the microwave in Watts.

Accepted Answer

Power $$= \frac{mc\Delta T}{t} = \frac{250 	imes 4 	imes 80}{60} \approx 1333 W$$

Question 7

An isolated box is divided into two equal compartments. An ideal gas occupies half of the box and the other half is empty. The partition is quickly removed without friction. How does the internal energy of the gas at the end compare to its energy at the start?

Accepted Answer

The internal energy stays the same.

Question 8

An isolated box is divided into two equal compartments. An ideal gas occupies half of the box and the other half is empty. The gas pushes the partition to the right, with friction. How does the energy of the gas at the end compare to its energy at the start?

Accepted Answer

The internal energy increases.

Question 9

An isolated box is divided into two equal compartments. An ideal gas occupies half of the box and the other half is empty. The partition is quickly removed without friction. How does the pressure of the gas at the end compare to its pressure at the start?

Accepted Answer

The pressure decreases.

Question 10

Consider two different weakly interacting gases, each with $$N$$ molecules, initially separated and then allowed to mix. What is the ratio of the total number of microstates after mixing $$W_f to $$before mixing $$W_i$$?

Accepted Answer

$$\frac{W_f}{W_i} = 2^N$$

Question 11

What is the total entropy change $$\Delta S$$ for the mixing of two different weakly interacting gases, each with $$N$$ molecules, at temperature $$T$$?

Accepted Answer

$$\Delta S = 2N k_B \ln 2$$

Question 12

A system can exist in two energy states, $$E_1 = 2$ meV and E_2$ = 30$$ meV. If the system is $$e$$ times as likely to be found in $$E_1$$ than in $$E_2$$, what is the temperature of the system in Kelvin? ($$1$$ meV $$= 1.6 	imes 10$$^{-22}$$ J, k_B = 1.38$$ 	imes $$10^{-23} J/K)$$

Accepted Answer

$$T = \frac{E_2$ - E_1$}{k_B \ln e} = \frac{(30 - 2) 	imes 1.6 	imes 10$$^{-22}$$}{1.38 	imes 10$$^{-23}$$ 	imes 1} \approx 32 K$$

Question 13

Consider an ideal gas at $$27.0\ ^\circC $$and $$1.00$$ atm pressure. If each molecule is at the center of a small cube and adjacent cubes touch but do not overlap, what is the length $$L of an $$edge of each small cube? ($$k_B = 1.38 	imes 10$$^{-23}$$ J/K, T = 300$ K, P = 1.01$$ 	imes $$10^5 Pa)$$

Accepted Answer

$$L = \left( \frac{k_B T}{P} \$$right)^{1/3}$$ \approx 3.7 	imes 10$$^{-9}$$ m

Question 14

A 25% efficient electric power plant produces one gigajoule of electric energy per second and discharges waste heat into the ocean. If a typical American home requires 20 kW for heating, how many homes could be heated with the waste heat from this plant?

Accepted Answer

The waste heat is $$3 	imes 10^9$ J/s, so $$\frac{3 	imes $$10^9$$}{2 	imes $$10^4$$} = 150,000$$ homes.

Question 15

A molecule undergoes a one-dimensional random walk, starting at $$x=0. $$After 4 steps, what is the probability that the molecule is at $$x=0$$?

Accepted Answer

There are $$\binom{4}{2} = 6$$ ways to return to $$x=0$$, so $$P = \frac{6}{16} = 0.375$$

Question 16

In the random walk model, after 4 steps, what is the most likely position for the molecule?

Accepted Answer

$$x = 0$$, because there are the most possible paths to return to the origin.

Question 17

In the mixing scenario shown (two gases separated, then allowed to mix), what happens to the entropy of the system?

Accepted Answer

The entropy increases because the number of accessible microstates increases.

Random Walk Model for Diffusing Molecules

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