Home | | Latest | About | Random
# Look at that fluffy cloud!
An exercise of unit conversions. Look at that fluffy cloud!
How heavy is one fluffy cloud? Let us estimate it.
## Mass of a cubic cloud.
Suppose we have a cloud that's roughly a cube in shape, with a measure of 1 km on each side.
![[1 teaching/summer program 2023/puzzles-and-problems/---files/storm-and-cloud 2023-08-21 12.04.06.excalidraw.svg]]
%%[[1 teaching/summer program 2023/puzzles-and-problems/---files/storm-and-cloud 2023-08-21 12.04.06.excalidraw|🖋 Edit in Excalidraw]], and the [[summer program 2023/puzzles-and-problems/---files/storm-and-cloud 2023-08-21 12.04.06.excalidraw.dark.svg|dark exported image]]%%
![[---images/---assets/---icons/question-icon.svg]] What is its volume in $\text{km}^{3}$? What about its volume in $\text{m}^{3}$? (How many meters is in one kilometer?)
Now clouds are mostly made out of water droplets and crystals in the air, one estimate of a **typical cloud density** is about $0.5 \text{ g}/\text{m}^{3}$.
![[---images/---assets/---icons/question-icon.svg]] Use the density of a cloud above to find the **mass** of a typical cubical cloud that's $1\text{ km}$ on each side.
![[---images/---assets/---icons/question-icon.svg]] An average **blue whale** is about $300,000\text{ lb}$. In terms of mass, how many blue whales is a typical kilometer sized cubic cloud? Note: $1\text{ kg} \approx 2.2\text{ lb}$. ![[1 teaching/summer program 2023/puzzles-and-problems/---files/storm-and-cloud 2023-08-21 12.56.22.excalidraw.svg]]
%%[[1 teaching/summer program 2023/puzzles-and-problems/---files/storm-and-cloud 2023-08-21 12.56.22.excalidraw|🖋 Edit in Excalidraw]], and the [[summer program 2023/puzzles-and-problems/---files/storm-and-cloud 2023-08-21 12.56.22.excalidraw.dark.svg|dark exported image]]%%
## Density of air.
If cloud is to float, however, it should have a density less than the atmosphere it is floating in. **Let us estimate the density of air.**
The **ideal gas law** gives a relation between the number of molecules $N$, pressure $P$, volume $V$, and temperature $T$ as the following: $$
PV = kNT
$$where $k$ is **Boltzmann's constant**, approximately $k = 1.38 \times 10^{-23}\text{J/K}$, where $\text{J}$ is Joules, a unit for energy, and $\text{K}$ is Kelvin, a measure of temperature.
We also have the relation that $\text{J} =\text{Pa}\cdot\text{m}^{3}$, where $\text{Pa}$ is Pascal, a measure of pressure.
Near Los Angeles, the temperature on a nice summer day is about $75^{\circ}\text{F}$ by the beach, with a pressure of $1\text{ atm}$ near sea-level. ![[1 teaching/summer program 2023/puzzles-and-problems/---files/storm-and-cloud 2023-08-21 13.31.01.excalidraw.svg]]
%%[[1 teaching/summer program 2023/puzzles-and-problems/---files/storm-and-cloud 2023-08-21 13.31.01.excalidraw|🖋 Edit in Excalidraw]], and the [[summer program 2023/puzzles-and-problems/---files/storm-and-cloud 2023-08-21 13.31.01.excalidraw.dark.svg|dark exported image]]%%
The **relation between Fahrenheit and Celsius** is.....oh you forgot the conversion.
It's ok. We remember the following
(1) Water freezes at $0^{\circ}\text{C}$, as well as $32^{\circ}\text{F}$.
(2) Water boils at $100^{\circ}\text{C}$, as well as $212^{\circ}\text{F}$
(3) The relation between Celsius $\text{C}$ and Fahrenheit $\text{F}$ is **linear**, that is they obey the relation $$
C = aF + b
$$for some constants $a,b$.
![[1 teaching/summer program 2023/puzzles-and-problems/---files/storm-and-cloud 2023-08-21 13.46.17.excalidraw.svg]]
%%[[1 teaching/summer program 2023/puzzles-and-problems/---files/storm-and-cloud 2023-08-21 13.46.17.excalidraw|🖋 Edit in Excalidraw]], and the [[summer program 2023/puzzles-and-problems/---files/storm-and-cloud 2023-08-21 13.46.17.excalidraw.dark.svg|dark exported image]]%%
![[---images/---assets/---icons/question-icon.svg]] Use the information about water freezing and water boiling, figure out what $a,b$ are and solve for them, giving you a formula of the form $C=aF+b$.
Once you have done that, we note that it is more customary to have it in the form $$
C = a\left( F+ \frac{b}{a} \right)
$$Does this formula look familiar now?
![[---images/---assets/---icons/question-icon.svg]] Ok, now convert a nice Los Angeles summer beach day temperature of $75^{\circ}\text{F}$ to Celsius. We can also convert this into Kelvin scale, with the formula $K = C + 273.15$. What is this temperature in Kelvin?
![[---images/---assets/---icons/question-icon.svg]] Now that you have found the temperature of a nice summer Los Angeles beach day in Kelvins, let us find out how many air molecules total are in **one cubic meter box**, at $75^{\circ}\text{F}$ and $1\text{ atm}$. Use the ideal gas law $PV = kNT$, find how many molecules $N$ are in a box of volume $1\text{ m}^{3}$. Note, we define $1\text{ atm} = 101325\text{ Pa}$.
To help you troubleshoot, your answer above should be on the order of magnitude of $10^{25}$ molecules....
![[---images/---assets/---icons/question-icon.svg]] Now, typical atmospheric air is composed about $78\%$ nitrogen gas ($\text{N}_{2}$), $21\%$ oxygen gas ($\text{O}_{2}$), and about $1\%$ Argon ($\text{Ar}$) and other gases, by molecule counts. Estimate the number of nitrogen gas molecule, the number of oxygen gas molecule, and the number of argon gas molecule **in one cubic meter box**, at $75^{\circ}\text{F}$ and $1\text{ atm}$.
![[summer program 2023/puzzles-and-problems/---files/storm-and-cloud 2023-08-21 13.52.14.excalidraw.svg]]
%%[[summer program 2023/puzzles-and-problems/---files/storm-and-cloud 2023-08-21 13.52.14.excalidraw.md|🖋 Edit in Excalidraw]], and the [[summer program 2023/puzzles-and-problems/---files/storm-and-cloud 2023-08-21 13.52.14.excalidraw.dark.svg|dark exported image]]%%
![[---images/---assets/---icons/question-icon.svg]] Now, nitrogen gas molecules have a **molar mass** of $\displaystyle28.02\frac{\text{g}}{\text{mol}}$, that is to say, for every $1\text{ mol} = 6.022 \times 10^{23}$ many molecules of nitrogen gas, they will have a mass of $28.02\text{ g}$. This number $6.022\times 10^{23}$ is also known as **Avogadro's number**, approximately. The molar mass of oxygen gas is $\displaystyle32 \frac{\text{ g}}{\text{mol}}$, and the molar mass of argon gas is $\displaystyle 39.95 \frac{\text{ g}}{\text{mol}}$.
Using these information, find out the mass in grams of one cubic meter box, at a typical sunny day by the Los Angeles beach with temperature $75^{\circ}\text{F}$ and a pressure of $1\text{ atm}$.
To help you troubleshoot, your answer should be on the order of magnitude of $10^{3} \text{ g}/\text{m}^{3}$.
![[---images/---assets/---icons/question-icon.svg]] Finally, does our cubic kilometer cloud float (compare the densities) in the atmosphere on a nice Los Angeles beach day? How do you know?