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Ok, so I'm reading Schroeder's Thermal Physics and I've reached the part where he 'derives' the Maxwell speed distribution. Most of what he writes makes perfect sense, however there is one bit that is rather confusing to me.
Here's a quick breakdown of his derivation, up to the point where I become confused, just in case you don't have access to this book.
He says that the probability of a molecule having a speed between v_1 and v_2 is equal to
[tex]\int_{v_1}^{v_2} D(v)dv [/tex]
where D(v) is the distribution function that is proportional to the product of the probability of a molecule having velocity v and the number of vectors v corresponding to the speed v.
He then goes on to say the first factor mentioned above is proportional to the Boltzmann factor where the energy in the Boltzmann factors is just the molecule's kinetic energy.
Everything up to this point makes sense to me.
He then proceeds to talk about the second factor mentioned above, i.e., the number of vectors corresponding to the speed v. What follows is a quote from the book:
"To evaluate this factor, imagine a three-dimensional "velocity space" in which each point represents a velocity vector. The set of velocity vectors corresponding to any given speed v lives on the surface of a sphere with radius v."
Ok, this makes sense, although I wouldn't say that the set of velocity vectors corresponding to any given speed "lives on the surface a sphere", to me it looks like said set is just member of R^3. In fact, one may say that the set of velocity vectors corresponding to a given speed v could be expressed as
[tex] \{(v_x,v_y,v_x) \in R^3 : v_x^2 + v_y^2 +v_z^2 = v^2\} [/tex].
Anyway, he continues,
"The larger v is, the bigger the sphere, and the more possible velocity vectors there are."
The italics are mine; that is the statement that confuses me. Here is what I think he is saying: let [tex] v_1, v_2 [/tex] denote two speeds such that [tex] v_2 > v_1 [/tex]. Then
[tex]card \{(v_x,v_y,v_x) \in R^3:v_x^2 + v_y^2 +v_z^2 = v_2^2\} > card \{(v_x,v_y,v_x) \in R^3:v_x^2 + v_y^2 +v_z^2 = v_1^2\} [/tex],
where "card", of course, denotes the cardinality of the set. This is clearly a false statement; both sets are infinite.
And he continues on with the derivation after that. It's really just that above point that is distressing.
So... am I misinterpreting what he is trying to say? And if so, how should I read what he wrote? Or is this just sloppy language and/or logic? Or... help?!
Here's a quick breakdown of his derivation, up to the point where I become confused, just in case you don't have access to this book.
He says that the probability of a molecule having a speed between v_1 and v_2 is equal to
[tex]\int_{v_1}^{v_2} D(v)dv [/tex]
where D(v) is the distribution function that is proportional to the product of the probability of a molecule having velocity v and the number of vectors v corresponding to the speed v.
He then goes on to say the first factor mentioned above is proportional to the Boltzmann factor where the energy in the Boltzmann factors is just the molecule's kinetic energy.
Everything up to this point makes sense to me.
He then proceeds to talk about the second factor mentioned above, i.e., the number of vectors corresponding to the speed v. What follows is a quote from the book:
"To evaluate this factor, imagine a three-dimensional "velocity space" in which each point represents a velocity vector. The set of velocity vectors corresponding to any given speed v lives on the surface of a sphere with radius v."
Ok, this makes sense, although I wouldn't say that the set of velocity vectors corresponding to any given speed "lives on the surface a sphere", to me it looks like said set is just member of R^3. In fact, one may say that the set of velocity vectors corresponding to a given speed v could be expressed as
[tex] \{(v_x,v_y,v_x) \in R^3 : v_x^2 + v_y^2 +v_z^2 = v^2\} [/tex].
Anyway, he continues,
"The larger v is, the bigger the sphere, and the more possible velocity vectors there are."
The italics are mine; that is the statement that confuses me. Here is what I think he is saying: let [tex] v_1, v_2 [/tex] denote two speeds such that [tex] v_2 > v_1 [/tex]. Then
[tex]card \{(v_x,v_y,v_x) \in R^3:v_x^2 + v_y^2 +v_z^2 = v_2^2\} > card \{(v_x,v_y,v_x) \in R^3:v_x^2 + v_y^2 +v_z^2 = v_1^2\} [/tex],
where "card", of course, denotes the cardinality of the set. This is clearly a false statement; both sets are infinite.
And he continues on with the derivation after that. It's really just that above point that is distressing.
So... am I misinterpreting what he is trying to say? And if so, how should I read what he wrote? Or is this just sloppy language and/or logic? Or... help?!
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