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Exercises
3.8 (a) The internal energy of a perfect monatomic gas relative to its value at T = 0 is . 32nRT
Calculate and for the gas.UV
T HV
T
Atkins says, that (3). The question gives us . dUUV
dVUT
dTT V= +
U nRT=
32
Holding T constant and rearranging this gives us . Since T is held
UV V
nRTT T =
32
constant, the right side of the equation becomes 0. UV
T = 0
The definition of enthalpy is . Replacing p with its Gas Law equivalent and H U pV= +taking the partial derivative with respect to V leaves us with the equation
. It has already been said that , and
HV
UV
nRTV
T T T =
+
UV
T
= 0
because T is held constant. That leaves .nRTV
T = 0HV
T = 0
Problems3.23 The speed of sound, , in a gas of molar mass M is related to the ratio of the heatcs
capacities by . Show that , where is the mass ( )c RT Ms = / /1 2 ( )c ps = / /1 2 density of the gas. Calculate the speed of sound in argon at 25 degrees C.
Rearranging the Gas Law for M gives us , and inserting this into the equation givenMRTp
=
gives us .cp
s =
1 2/
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Variables
5/3R due to the fact that argon is monatomicR 8.31451 jk-1mol-1
M 39.95*10-3 Kgmol-1
T 298.5 K
Inserting this data into the equation , gives us the value 322 ms-1. This value( )c RT Ms = / /1 2is somewhat slower than the speed of sound in air, which is to be expected due to the increasedmolar mass of argon.
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