Homework Problem Set 5 Solutions ( )2

7. Given a collection of five atoms with speeds of 640, 552, 695, 533, and 697 m/s,
determine the temperature of the system. The mass of the atoms is 12.0 g/mol.
16
From the equipartition theorem for atoms,
U = 3 2 Nk B T ,
where N is the number of atoms and 3N is the number of degrees of freedom. Relating the internal energy to
the average kinetic energy, €
where the brackets indicate an average.
Solving for the temperature,
€
U = 3 2 Nk B T = 1
2 m v2 ,
T =
2
3Nk B
1
2 m v2 ,
or factoring out the 1 2 and m, €
€
T =
m
3Nk B
v 2 .
Determining
v 2 ,
€
v 2 = 1 ( 640 m/s ) 2 5 [ + ( 552 m/s)
2 + ( 695m/s)
2 + ( 533m/s)
2 + ( 697 m/s ) 2
]
v 2 = 3.934 × 10 5 m 2 s −2 .
€
The mass of one carbon atom is determined by dividing the molecular weight of one mole in kg by Avogadro’s
number,
€
m =
MW
Avogadro's Number
m = 1.9226 ×10 −26 kg.
=
0.01200 kg/mol
6.02214 ×10 23 /mol
Calculating the temperature,
€
T =
T = 189 K .
( 1.9926 ×10 −26 kg)
(
( ) 3.934 ×105 m 2 s −2
)
3 1.38065×10 −23 J K
€