Homework Problem Set 5 Solutions ( )2
Homework Problem Set 5 Solutions ( )2
Homework Problem Set 5 Solutions ( )2
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5a, b.) continued<br />
13<br />
The velocity distribution function for the atoms at 1000 K is given by<br />
€<br />
⎢⎢ 0.012 kg/mol<br />
( ) = 4π v 2 ⎢⎢<br />
F v<br />
3/ 2<br />
⎡⎡<br />
⎛⎛ 1mol ⎞⎞ ⎤⎤<br />
( ) ⎜⎜<br />
⎝⎝ 6.02217 × 10 23 ⎟⎟<br />
⎠⎠<br />
⎥⎥<br />
⎥⎥<br />
⎢⎢ 2π( 1.38062 × 10 -23 JK −1 )( 1000 K)<br />
⎥⎥<br />
⎢⎢<br />
⎥⎥<br />
⎣⎣<br />
⎦⎦<br />
⎧⎧<br />
⎛⎛ 1mol ⎞⎞<br />
( 0.012 kg/mol)<br />
⎜⎜<br />
⎝⎝ 6.02217 × 10<br />
× exp −<br />
23 ⎟⎟ v 2 ⎫⎫<br />
⎪⎪<br />
⎠⎠<br />
⎪⎪<br />
⎪⎪<br />
⎪⎪<br />
⎨⎨<br />
2( 1.38062 × 10 -23 JK −1 ⎬⎬<br />
⎪⎪<br />
)( 1000 K)<br />
⎪⎪<br />
⎩⎩ ⎪⎪<br />
⎭⎭ ⎪⎪<br />
F( v) = ( 1.38347 × 10 −9 v 2 ) exp{ −7.21646 × 10 −7 v 2 }.<br />
Note that the units of the exponent cancel out so that it is unitless, while the units of the terms in front of<br />
the exponent are € sm –1 .<br />
Plots of the Maxwell-Boltzmann distribution of speeds at 300 and 1000 K are shown on a single graph<br />
below.<br />
0.0014<br />
0.0012<br />
T = 300 K<br />
0.0010<br />
F(v)<br />
0.0008<br />
0.0006<br />
T = 1000 K<br />
0.0004<br />
0.0002<br />
0.0000<br />
0 500 1000 1500 2000 2500 3000<br />
v (m/s)<br />
As can be seen from the graph, the maximum (or most probable) speed increases as the temperature<br />
increases. In addition, the entire distribution shifts to larger speeds as the temperature increases.
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