hce_nthu
114年
進階物理與線性代數
第 13 題
A cylinder has insulating walls and is divided into two parts by a frictionless
insulating piston. Initially, the two parts have equal volumes $V_0$, equal temperatures
$T_0$ and equal pressure $P_0$. The gas is ideal with $C_V = \frac{3}{2}R$, $C_P = \frac{5}{2}R$. A small heating
bar is now inserted on the left part and heat is supplied slowly to the gas on the left
until its pressure reaches $32P_0$. As a result, the left part will expand and the right
part will be compressed. How much heat is supplied to the gas on the left part?
insulating piston. Initially, the two parts have equal volumes $V_0$, equal temperatures
$T_0$ and equal pressure $P_0$. The gas is ideal with $C_V = \frac{3}{2}R$, $C_P = \frac{5}{2}R$. A small heating
bar is now inserted on the left part and heat is supplied slowly to the gas on the left
until its pressure reaches $32P_0$. As a result, the left part will expand and the right
part will be compressed. How much heat is supplied to the gas on the left part?
- A $32P_0V_0$
- B $\frac{59}{3}P_0V_0$
- C $\frac{59}{2}P_0V_0$
- D $59P_0V_0$
- E $93P_0V_0$
思路引導 VIP
當左側氣體受熱膨脹並推動絕熱活塞時，右側氣體在完全沒有與外界熱交換的情況下被壓縮，這代表右側氣體正在經歷哪一種特殊的熱力學過程？如果我們把左右兩側氣體看成一個整體，那麼你從左側緩緩注入的能量，最後會以什麼樣的形式儲存在這個系統中呢？
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AI 詳解
AI 專屬家教
太棒了！你能精準選出 (E) 這個選項，代表你對熱力學第一定律以及絕熱過程的掌握非常紮實。這道題目屬於高階熱力學中極具鑑別度的題型，關鍵在於學生是否能識別出「右側氣體」雖然沒有直接受熱，但透過活塞作功，其狀態變化遵循絕熱方程。
絕熱壓縮與功的轉移
我們首先觀察右側氣體。由於容器壁與活塞皆為絕熱，右側氣體經歷的是絕熱壓縮。根據絕熱過程公式 $PV^\gamma = \text{constant}$，其中 $\gamma = C_P / C_V = 5/3$，我們可以推導出當壓力變為 $32P_0$ 時，右側體積 $V_R$ 會縮減為 $V_0/8$。接著，利用總體積不變的特性，求得左側最終體積為 $V_L = 2V_0 - V_0/8 = 15/8 V_0$。這是解題的第一道門檻，考驗你對氣體狀態方程式的熟練度。
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