Atoms, Heat, and *Planetary atmospheres*
We saw that Conservation of Energy only worked once we could precisely account for transformations of other energies into ThermalE.
What is thermal energy?
How fast are atoms moving?
At room temperature, a typical speed is $v$ ~ 700 mi / h - close to the speed of sound.
This
is no accident: Sound is a pressure disturbance travelling through air.
It seems plausible that any disturbance should move with the same underlying
speed as the atoms are moving.
What is this in miles / sec? ...and in sec / mile? This time, see WolframAlpha
.Light travels at 3 X 10^8 m /sec = 186,000 miles/sec: much faster than sound.
Use this to figure out... How far away is this
airplane?
By the way... According to this USA Today article, civil aircraft are hit on average once per aircraft per year. But lightning hasn't caused any airplane to crash since 1981.
Temperature
The
metric system uses the Celsius degree to measure temperature.
0 C = water freezes = 32 F
100 C = water boils = 212 F
37 C = normal human body temperature = 98.6 F
40 C = bad fever = 104 F
1 C = 9/5 ${}^o$F~2${}^o$F
But what is temperature?
Thermal energy is the disorganized kinetic energy of atoms.
Why don't we have any hydrogen in our atmosphere?
Hydrogen is the most abundant atom in the universe. 70.6% of the atoms in our solar system are hydrogen:
Jupiter's atmosphere: 90% is $H_2$.
Saturn: 75% $H_2$.
Earth: 0.000055 % $H_2$.
??
The nature of temperature
Actually, Temperature is not proportional to the average speed of molecules, it is proportional to the average kinetic energy:
$$T\propto {\rm KineticE} = \frac{1}{2}mv^2.$$
If
two atoms have the same kinetic energy (that is, the same temperature),
but different masses, which one is moving faster?
Hydrogen gas ($H_2$) has an atomic weight of 2 grams / mole.
Nitrogen gas ($N_2$) has an atomic weight of 28 grams / mole. So, one nitrogen molecule has a mass of 14 times the mass of a hydrogen molecule.
If we have any hydrogen in the atmosphere along with the 78% of our atmosphere that's nitrogen, how much faster is the hydrogen moving?
$$\frac{1}{2}m_{H_2} v_{H_2}^2=\frac{1}{2}14*m_{H_2}(760 {\rm mph})^2.$$
With a little algebra...(and help from Google) $$v_{H_2} = \sqrt{14}*760 {\rm mph}= 2,844 mph = 1.3 km /sec.$$
This is about 10% of the escape velocity -- 11 km/sec -- from earth.
So occasionally, a hydrogen molecule will get going fast enough to reach escape velocity.
Remember this: at the same temperature, light atoms move faster than heavy atoms
We had this equation relating temperature and average speed: $$\frac 32 k_BT=\frac12 m \langle v^2 \rangle.\label{boltzy}$$
So, we looked at the effect of the mass of a gas using this equation: One reason that hydrogen should escape faster from Earth than Jupiter is that hydrogen molecules are moving fast ~ on average about 10% of Earth's escape velocity. Jupiter has a stronger gravitational pull (about 2.4 times Earth), and a higher escape velocity (about 5 times that of Earth).
But the other thing influencing how fast molecules are moving (on average) is the temperature. Look up the temperature on Jupiter's surface. Well, Jupiter doesn't have a surface exactly, so NASA uses the temperature in the atmosphere where the pressure is the same as at Earth's surface (1 atmosphere) for comparisons. Is Jupiter's temperature higher or lower? How would that change the speed of hydrogen molecules, and how fast they might "leak out" of the atmosphere?
Atmospheres of the planets
Gas X - assignment
We think the planets all formed from the same kind of dust / stuff--with roughly the same chemical composition, and yet their atmospheres are quite different--as we've seen with hydrogen. Let's see if we can account for another gas that's present on some planets and not on others:
- Jupiter is much further from the sun than Earth, and it's much larger and heavier, and therefore has higher gravity. Look up the atmospheric composition of two more earthlike planets--Venus and Mars. What is the most common gas in their atmosphere? (I'll call it "gas X").
- What is the percentage of Earth's atmosphere that gas X makes up? Wow, isn't that radically different from Mars and Venus? Maybe Earth once had a lot of that gas too?! One big difference between Earth and our neighbors is that weo have life here on Earth. How might life account for the fact that we have very little of gas X, but have a lot of oxygen--which is hardly present at all on any other planet?
- Now find the three most common gases in Jupiter's and Saturn's atmospheres. Any gas X to be seen?
- Look up the the boiling point of gas X, and the surface temperatures of Saturn and Jupiter. It may be that you find one in C and another in K: convert to the same temperature system so you can compare. E.g. Google "20 degrees C in degrees K". If one of these planets started with a lot of gas X at one point in time, what might have happened to it that it can't be found in the atmosphere? Where else (or in what form) might it still be present on Jupiter or on Saturn?
Absolute zero
So as you cool things down, the atoms are moving slower and slower. At a certain temperature they would stop completely:
-273 C = absolute zero
This is the basis of the Kelvin temperature scale: the degrees are the same size, but the zero point is absolute zero.
Room temperature = 20 C = 273 + 20 = 293 K.
There is a very simple formula relating kinetic energy per molecule to temperature (in Kelvin) it is...
$$K.E. = 2 \times 10^{-23} T_K.$$
So, the formula for thermal energy is related to the $N$--number of molecules-- and the temperature (in degrees Kelvin), like this
$$ThermalE = 2 \times 10^{-23} NT_K.$$
Which has more thermal energy? A mug of hot chocolate or a mug of ice water?
A mug of hot coffee or Lake Michigan at room temperature?
The Lake, even though it's at a lower temperature, still has lots more microscopic kinetic energy then a mere mug of hot coffee because the number of molecules $N$ in the lake is so much larger.
Thermal expansion or contraction?
In general... do things get bigger or smaller when you heat them?
The most familiar thermometers depend on a change in volume of alcohol (colored red). Years ago mercury was used. The glass enclosing the alcohol is also expanding. But not nearly as much as the alcohol.
Thermal expansion/contraction is the reason we put neat cracks into sidewalks, and complicated cracks ("expansions joints") into bridges.
If you have a lid that's too tight Should you run hot water on the jar or the lid??
Loosening lids by running under hot water uses thermal expansion.
Close fitting ball and ring... What happens if you
- heat the ball?
- heat the ring?
Typical thermal expansions are around... $$\frac{\Delta V}{V} =~ 10^{-4} / {}^o C$$ This means, if you increase the temperature by 1 degree C, the volume will increase fractionally by 1/10,000
Global warming and thermal expansion
Current predictions for average warming due to excess $CO_2$ are 2-4 C by 2100.
Even if all the ice (and glaciers) did *not* melt, the ocean could still rise due to thermal expansion. How much? Taking the higher value of $\Delta T = 4$ C,
Water has an expansion coefficient at room temperature of $ \frac{\Delta V}{V} = 2.0 \times 10^{-4} / {}^o C$ and the oceans have an average depth of 12,000 feet.
If the ocean is like a glass which is filled to a depth $h$,
$$\begin{align}\Delta h & = h \cdot 2.0 \times 10^{-4} \cdot \Delta T \\ &=(12,000 ft)(2.0 \times 10^{-4})(4 C) = 9.6 {\rm \ feet}\end{align}.$$ This is enough to be worrisome.
The IPCC has done a more detailed calculation. Luckily:
- If the water is between 0 and 4 C, heating water actually causes it to shrink instead of expand.
- Lots of the deep ocean is at temperatures below 4 C,
Their prediction is a maximum rise of 1.9 feet. If you add in the effect of polar ice melting, the maximum sea level is more like 2.5 feet.
You can explore the effect of a 3 ft rise with this Google Maps sea level simulator.
