Physics 410
Spring 2014
HW#5
Due Monday, 17 February 2014
1.
Carter, 11-7.
a) Show that the Maxwell speed distribution (number of particles, not fraction) can be written as
2
4𝑁
𝑁(𝑥) 𝑑𝑥 = 𝑥 2 𝑒 −𝑥 𝑑𝑥,
√𝜋
where 𝑥 ≡
𝑣
𝑣𝑚
2𝑘𝐵 𝑇
and 𝑣𝑚 = √
𝑚
.
b) Show that the number of molecules with speeds less than some specified speed vo is given by
2
2
𝑁0→𝑥𝑜 = 𝑁 [erf(𝑥𝑜 ) − 𝑥𝑜 𝑒 −𝑥𝑜 ],
√𝜋
2.
3.
4.
Where xo  vo/ vm. Here, erf(xo) is the error function, defined by
2 𝑥 −𝑢2
erf(𝑥) ≡
∫ 𝑒 𝑑𝑢.
√𝜋 0
c) Find the fractional number of molecules with speeds less than vm.
Carter, 11-12. The number density of the gas, mainly hydrogen, that fills interstellar space is one
molecule per cubic centimeter (106 m-3). If the molecular diameter is 10-10 m what is the mean free path
in interstellar space? What is the collision frequency of a hydrogen molecule in collisions per century if
the temperature of interstellar space is 10K?
You have a vacuum chamber that is roughly spherical in shape with a diameter of 1m. Calculate how
small the pressure inside must be before you can safely say that a molecule is more likely to hit the
wall of the container than it is another molecule.
Carter, Chapter 3 problem 3-1. Ten kilomoles of an ideal gas are compressed isothermally and
reversibly from a pressure of 1 atmosphere to 10 atmospheres at 300K. How much work is done?
Physics 410
Spring 2014
HW#5
Due Monday, 17 February 2014
1. Carter, 11-7.
a) Show that the Maxwell speed distribution (number of particles, not fraction) can be written as
2
4𝑁
𝑁(𝑥) 𝑑𝑥 = 𝑥 2 𝑒 −𝑥 𝑑𝑥,
√𝜋
where 𝑥 ≡
𝑣
𝑣𝑚
2𝑘𝐵 𝑇
and 𝑣𝑚 = √
𝑚
.
b) Show that the number of molecules with speeds less than some specified speed vo is given by
2
2
𝑁0→𝑥𝑜 = 𝑁 [erf(𝑥𝑜 ) − 𝑥𝑜 𝑒 −𝑥𝑜 ],
√𝜋
2.
3.
4.
Where xo  vo/ vm. Here, erf(xo) is the error function, defined by
2 𝑥 −𝑢2
erf(𝑥) ≡
∫ 𝑒 𝑑𝑢.
√𝜋 0
c) Find the fractional number of molecules with speeds less than vm.
Carter, 11-12. The number density of the gas, mainly hydrogen, that fills interstellar space is one
molecule per cubic centimeter (106 m-3). If the molecular diameter is 10-10 m what is the mean free path
in interstellar space? What is the collision frequency of a hydrogen molecule in collisions per century if
the temperature of interstellar space is 10K?
You have a vacuum chamber that is roughly spherical in shape with a diameter of 1m. Calculate how
small the pressure inside must be before you can safely say that a molecule is more likely to hit the
wall of the container than it is another molecule.
Carter, Chapter 3 problem 3-1. Ten kilomoles of an ideal gas are compressed isothermally and
reversibly from a pressure of 1 atmosphere to 10 atmospheres at 300K. How much work is done?