作者Frobenius (▽．(▽×▽φ)=0)
看板Physics
標題Re: [問題] 強簡併之 Fermi-Dirac 氣體：能量於低뜠…
時間Thu Mar 25 01:15:05 2010
3 1/3
Kf = 3π^2 N/V => Kf = ( 3π^2 N/V )
2 2 2
hbar Kf (h/2π) 1/3 2
Ef(0) = k Tf = ──── = ──── [ ( 3π^2 N/V ) ]
2m 2m
2 2/3
= h [ 3π^2 N/(8V) ] /2m
2 2/3 2 2/3 -2/3
=> Tf = h [ 3π^2 N/(8V) ] /2mk = h [ 3π^2 N/8 ] /2mk V
2 2/3 -5/3
dTf/dV = - 2/3 h [ 3π^2 N/8 ] /2mk V
2 2/3 -1
= - 2/3 h [ 3π^2 N/(8V) ] /2mk V
-2
= ─ Tf
3V
-1 -2 -2 -2 2 -1
d(Tf )/dV = - Tf dTf/dV = - Tf ( ─ Tf ) = ─ Tf
3V 3V
-3 -4 -4 -2 2 -3
d(Tf )/dV = - 3 Tf dTf/dV = - 3 Tf ( ─ Tf ) = - Tf
3V V
3 π^2 T 2 π^4 T 4
F(T) = N(0) k Tf [ - - ── ( ─ ) + ── ( ─ ) + … ]
5 4 Tf 80 Tf
3 Tf π^2 T π^4 T 3
= N(0) k T [ - ( ─ ) - ── ( ─ ) + ── ( ─ ) + … ]
5 T 4 Tf 80 Tf
The Fermion Gas Pressure P(T)：
因為 P = - ( ∂ F/∂ V ) T,N
3 1 -2 π^2 2 -1
= - N(0) k T [ - - ( ─ Tf ) - ── T ( ─ Tf )
5 T 3V 4 3V
π^4 3 2 -3
+ ── T ( - Tf ) + … ]
80 V
N(0) k T 2 Tf π^2 T π^4 T 3
= ──── [ - ( ─ ) + ── ( ─ ) - ── ( ─ ) + … ]
V 5 T 6 Tf 40 Tf
N(0) k Tf 2 π^2 T 2 π^4 T 4
= ───── [ - + ── ( ─ ) - ── ( ─ ) + … ]
V 5 6 Tf 40 Tf
2 N(0) k Tf 5 π^2 T 2 π^4 T 4
= - ───── [ 1 + ─── ( ─ ) - ── ( ─ ) + … ]
5 V 12 Tf 16 Tf
2 N(0) k Tf 5 π^2 T 2 π^4 T 4
P(T) = - ───── [ 1 + ─── ( ─ ) - ── ( ─ ) + … ]
5 V 12 Tf 16 Tf
2 N(0)Ef(0) 5 π^2 T 2 π^4 T 4
= - ───── [ 1 + ─── ( ─ ) - ── ( ─ ) + … ]
5 V 12 Tf 16 Tf
3 5 π^2 T 2 π^4 T 4
U(T) = - N(0) Ef(0) [ 1 + ─── ( ─ ) - ── ( ─ ) + … ]
5 12 Tf 16 Tf
P(T) 2 2U
── = ─ => P = ─
U(T) 3V 3V
--
Van Der Waals' equation： ( P + a/υ^2 )( υ - b ) = R T
--
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◆ From: 118.160.213.211
1F：推 feynman511:雖然我從沒好好看過你打的 可是真的打得很認真XD 03/25 02:03
2F：→ Frobenius:其實我本來以為很簡單，沒想到會用到這麼多近似的技巧XD 03/25 02:20
3F：推 ANUBISANKH:沒想到今天碰到同樣的習題 XDD 剛好在這板上看過! 04/02 12:06
4F：推 alfadick:從數學板跑過來。外行給推！ 05/28 09:47
5F：→ alfadick:可以用LaTeX打！打好再外連pdf網址 05/28 09:47