Assignment 6

Due Friday November 22 2024 (on paper at start of class, unless alternate arrangements have been made in advance)

Point values are indicated for each question.

Total points: 35

Question 1¶

4 points

(Problem 7.1 of the text)

By consulting Fig. 7.2 and Fig. 1.7, estimate the altitude at which the boiling point of water drops to room temperature (20ºC).

Question 2¶

6 points

(Problem 7.3 of the text)

A pot contains one kg of ice at 0ºC and is placed on a stove with a burner that provides 900 W of power to the pot. Ignoring the heat capacity of the pot itself as well as other heat losses, compute the following, using a specific heat capacity $c_{water} = 4186$ J kg^-1 K^-1 for water.

a) (2) The time required to melt the ice

b) (2) The time required to raise the water temperature to boiling

c) (2) The time required to boil the water away. Hint: use a value of latent heat of vaporization valid at the boiling point.

Question 3¶

7 points

(Based on Problem 7.4 of the text)

A container of pure water is set outside on a cold night and supercools to a temperature $T_{celsius}$ (expressed in ºC) without freezing. Once it is disturbed, however, ice crystals form and rapidly spread throughout the container until all that remains is a fairly uniform ice-water slush with temperature of 0ºC.

a) (5) Show that the fraction of ice $f$ in the resulting mixture depends on the initial temperature $T_{celsius}$ through this formula:

where $T_{celsius}$ is in ºC and is therefore a negative number.

Hint: First work out the energy required to warm the water up to 0ºC. Then work out the energy released by freezing the unknown fraction $f$ of ice at 0ºC. Set these two energies equal (which they must be in an isolated system) to determine the fraction of ice $f$.

b) (2) It is impossible to supercool pure water to less than about -40ºC without it spontaneously freezing. Therefore, what is the maximum fraction of ice that can be achieved from supercooled water without additional cooling?

Question 4¶

5 points

(Problem 7.5 of the text)

A chemist working in a laboratory requires a version of equation (7.18) that is valid in the vicinity of the boiling point of water. Rederive the coefficients $A$ and $B$ using $T_0 = 373$ K, noting that $e_{s0}$ at this temperature is equal to one standard atmosphere, and $L=2.26 \times 10^6$ J kg^-1.

Question 5¶

5 points

(Problem 7.6 of the text)

Use equation (7.18) to find the dewpoint temperature $T_d$ corresponding to a mixing ratio $w$ of 20 g kg^-1 at a pressure $p$ of 1000 hPa and a temperature of 35CºC.

Hint: recall that the dewpoint satisfies the equation $e = e_s(T_d)$, and vapor pressure can be calculated from mixing ratio and air pressure.

Question 6¶

6 points

(Problem 7.7 of the text)

On a humid day in the central of southeastern United States, one sometimes hears the casual comment that “it is 90 degrees (Fahrenheit) and 90% humidity.” What dewpoint (in ºF) would this combination represent?