Problem Set 2. Due Jan. 27 in class. 1. The Maxwell-Boltzmann Distribution· v is
ID: 979803 • Letter: P
Question
Problem Set 2. Due Jan. 27 in class.
1. The Maxwell-Boltzmann Distribution·v is:
4v2 m e v
2kT
Let’s rewrite the same as a function of energy, which is E = ½ mv2 . This is a very easy two-step process; first, wherever you see a “v” in the MB distribution, write it instead as a function of E. Note
2E 2E m m
that here this means replacing each instance of v with since v(E) = . Almost done!
There is one last part that is a small bit harder- you can’t just replace v with E , but instead you have
to change it as: v =v(E)E .
E
As such, this very long-worded question is actually asking you to write the Maxwell-Distribution function of velocity as a function of energy, which means that you just have to simplify the following:
4[v(E)2] m e E (7 pts) 2kT E
2. The Maxwell-Boltzmann distribution as function of energy is: 2E k1T 2 ekET
Can you calculate the most probable energy of a gas phase molecule from the above? (7 pts)
Is this the most probable kinetic energy, or the most probable potential energy, or the sum of the two
(i.e. the most probable total energy)? (2 pts)
1
3. The most probable velocity of a gas is vmp= 2kT 2 . Can you relate the most probable velocity
m
k T
to the most probable energy of a gas molecule, which is Emp = 2 ? (3 pts)
It is known that the most probable energy (the point at the top of the MB(E) distribution) is related to the most probable velocity via:
m v2mp
Emp =
4
and that the average energy <E> is related to the average square velocity <v2> as:
<E>= .
If v = mp, can you label, with a point or small circle,
where the most probable energy Emp and the average energy <E> on this graph of MB(E)?
Hint: you might want to do a little algebra on this one… (5 pts)
Let’s say I have krypton gas and I want to compress it so much that the mean free path is as big as the atom itself (0.176 nm diameter). At 30 °C, how much should I pressurize the gas? (5 pts)
Here is the Maxwell-Boltzmann probability density for the amount of money a person makes (“$”) as a function of number of credit cards (“credit_cards”) they have:
MB($) = 1 e4credit_$2cards credit_cards
Can you show that if you double your average income you can have 4 times as many credit cards? Hint: Determine <$> as a function of credit_cards. (10 pts)
Hint hint: there is such thing as negative income, and you can have an infinite amount of money. Also:
c 2 21
e ax x = 2e ax x = 12 a a
00
cc2 2 2 c2
0 x2eax2x = erf ca12 2ca ec2a c1xeax x = 21ae ax c1
x2 eax2 eax2x= a 0 x2 eax2x = 14 a
7. The correct equation that Einstein derived for the energy from the theory of relativity is not E=mc2, rather E2=p2·c2 + m2·c4 (m is mass, c is the speed of light).
What assumption allows you to say E=m·c2? (2 pts)
If energy of light is E=c·h/, can you derive what the momentum of light is, i.e. p = ? (3 pts)
pt. b. hint: photons have no mass, so E2 = p2·c2
Units! a. Write Plank’s constant (h=6.626×10-34 J·s) in units of kg, m, and s. b. What are the unit(s) of c·h/? c. What are the unit(s) of momentum squared over mass? d. What are the unit(s) of Plank’s
constant squared over mass over length squared? (8 pts)
a. Arrange the following forms of light in order of increasing energy (from lowest to highest): radiowave, ultraviolet, visible, infrared, microwave, and X-ray. b. What color is 620 nm light? What color is 525 nm light? What color is 400 nm light? c. How much energy does a 400 nm photon have?
How much energy does a 620 nm photon have? (12 pts) hint: yes you’re going to have to do some Googling to answer this one.
Explanation / Answer
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