- #1
Crosshash
- 50
- 0
Hello everyone
The equivalent of the Maxwell-Boltzman distribution for a two-dimensional
gas is
[itex]P(v) = Cv e^-\frac {mv^2}{kt}[/itex]
Determine [itex]C[/itex] so that
[itex]\int_0^\infty P(v)dv = N[/itex]
Not really sure
I wasn't really sure how to tackle this question so I figured i'd integrate [itex]P(v)[/itex] since the question says that'll equal N.
[itex]\int_0^\infty P(v)dv[/itex]
[itex]\int_0^\infty Cv e^-\frac {mv^2}{kt} dv[/itex]
[itex]C\int_0^\infty v e^-\frac {mv^2}{kt} dv[/itex]
[itex] u = \frac {mv^2}{kt}[/itex]
[itex]\frac {du}{dv} = \frac {2mv}{kt}[/itex]
[itex]dv = \frac {du kt}{2mv}[/itex]
[itex]C\int_0^\infty v e^{-u} \frac {du kt}{2mv}[/itex]
[itex]C\int_0^\infty e^{-u} \frac {du kt}{2m}[/itex]
[itex]\frac {Ckt}{2m} \int_0^\infty e^{-u} du[/itex]
[itex] = \frac {Ckt}{2m} \bigg[{-e^{-u}\bigg]_0^\infty[/itex]
[itex] = \frac {Ckt}{2m} \bigg[{-e^{-\frac {mv^2}{kt}}\bigg]_0^\infty[/itex]
I'm not really sure where to go from here. How would I evaluate this between infinity and zero?
Thanks
Homework Statement
The equivalent of the Maxwell-Boltzman distribution for a two-dimensional
gas is
[itex]P(v) = Cv e^-\frac {mv^2}{kt}[/itex]
Determine [itex]C[/itex] so that
[itex]\int_0^\infty P(v)dv = N[/itex]
Homework Equations
Not really sure
The Attempt at a Solution
I wasn't really sure how to tackle this question so I figured i'd integrate [itex]P(v)[/itex] since the question says that'll equal N.
[itex]\int_0^\infty P(v)dv[/itex]
[itex]\int_0^\infty Cv e^-\frac {mv^2}{kt} dv[/itex]
[itex]C\int_0^\infty v e^-\frac {mv^2}{kt} dv[/itex]
[itex] u = \frac {mv^2}{kt}[/itex]
[itex]\frac {du}{dv} = \frac {2mv}{kt}[/itex]
[itex]dv = \frac {du kt}{2mv}[/itex]
[itex]C\int_0^\infty v e^{-u} \frac {du kt}{2mv}[/itex]
[itex]C\int_0^\infty e^{-u} \frac {du kt}{2m}[/itex]
[itex]\frac {Ckt}{2m} \int_0^\infty e^{-u} du[/itex]
[itex] = \frac {Ckt}{2m} \bigg[{-e^{-u}\bigg]_0^\infty[/itex]
[itex] = \frac {Ckt}{2m} \bigg[{-e^{-\frac {mv^2}{kt}}\bigg]_0^\infty[/itex]
I'm not really sure where to go from here. How would I evaluate this between infinity and zero?
Thanks