Q = qᴺ
Nǃ
Where N is Avogadro number
= = 1 [(2πmkT)³/₂ ] ᴺ ..................... (1)
Nǃ h³
Using Stirling's approximation,
ln Nǃ = Nln N - N = ln Nᴺ − lne = ln Nᴺ − lneᴺ
Nǃ = (N ) ᴺ .............. (2)
e
Substituting for Nǃ from equation (2) into equation (1), we get
Q = (e)ᴺ [(2πmkT) ³/₂ V ] ᴺ ........... (3)
N h³
Q = [ (2πmkT)³/₂ Ve ] ᴺ
Nh³
lnQ = N ln [ (2πmkT)³/₂ Ve ]
Nh³
S = E + kln qᴺ = E + klnQ .............. (4)
T Nǃ T
Substituting for E = 3RT/2 ( trans. K.E) for one molecule.
S = 3 R + kNln [(2πmkT)³/₂ Ve ] = 3 R+ Rln
2 Nh³ 2
[(2πmkT)³/₂ Ve ]
Nh³
= 3 Rln e+ Rln [(2πmkT)³/₂ Ve ]
Nh³
= Rln [ (2πmkT)³/₂ Ve] = Rln
Nh³
[(2πmkT)³/₂ Ve 5/2 ]
Nh³
Since = M/N,
S=Rln [ e⁵/₂ (2πk)³/₂ V (TM)³/₂ ]...........(5)
N⁵/₂ h³
Eqution (5)/is known Sackur- Tetrode equation. Replacing V by RT/P in equation (5) we get
S = Rln [e⁵/₂ (2πk) ³/₂ RT⁵/₂ M³/₂ ........(6)
Equation (6) is another from of Sackir-Tetrode equation. From equation (5) separating the constants,
S=2.303 χ R [loɡ e⁵/₂ (2πk) ³/₂ + loɡ V
(TM)³/₂
Substitute the in c.g.s units and evaluate
S= 2.303×1.987 [loɡ V +3/2 loɡ T +3/2 loɡ M + loɡ(2.78)⁵/₂ (2χ3.14χ1.38χ10⁻¹⁶)³/₂ ]
(6.023χ10²³)⁵/₂ (6.626χ10⁻²⁷⁾³
=4.576 [loɡ V+3/2 loɡ T+3/2 loɡ M−2.43]
=−11.12+4.576 [loɡ V +3/2 loɡ T +3/2 loɡ M ] cal k⁻¹ mol⁻¹
Similarly from equation (6) in c.g.s units,
S=2.303xRln [e⁵/₂ (2πk)³/₂ R + Rln
N⁵/₂ h³
T⁵/₂ M³/₂ ]
P
S=2.303x1.987[5/2 loɡ T+3/2 loɡ M−
loɡ P− 0.5065]
S=-2.317+4.576 [5/2 log T+ 3/2 log M-log P] cal k⁻¹ mol ⁻¹
In S.I units, S= 2.303x8.314[5/2 log T+3/2 log M - log P + 8.9998]
= 172.32+19.147[5/2log T + 3/2 log M - log P] J K⁻¹ mol⁻¹
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