熱機Heat Engine

真實的熱機 A Real Heat Engine 完美的熱機A Perfect Heat Engine

熱幫浦Heat Pump 完美的熱幫浦A Perfect Heat Pump 真實的熱幫浦 A Real Heat Pump

Sadi Carnot 1796 – 1832 First to show the quantitative relationship between work and heat Book reviewed importance of the steam engine

Carnot Cycle

氣體運動論 Kinetic Theory of Gases Ludwid Boltzmann波茲曼1844 – 1906 貢獻 : 氣體運動論  統計力學 statistical mechanics 2. 電磁學Electromagnetism 3. 熱力學Thermodynamics

氣體運動論 以 理想氣體模型ideal gas model為基礎,只考慮單一物質的情況(每一個分子都一樣) 分子數目很大, 彼此距離很遠, 分子視為點粒子, 分子運動遵守牛頓力學, 分子運動方向是任意的 ,快慢也是任意的, 平均速率 分子間只有在彈性碰撞時,才有短距離的作用, 分子與牆壁間只有彈性碰撞, 用微觀量解釋ideal gas的壓力和溫度

Ideal Gas Notes 理想氣體  原子組成 理想氣體~分子氣體 分子的旋轉運動及 振動運動不影響以下考慮的運動情形

壓力與運動論 Pressure and Kinetic Energy 假設容器是邊長為d的正方體Assume a container is a cube Edges are length d 檢視分子運動的速度分量--- 例如x方向 Look at the motion of the molecule in terms of its velocity components 檢視它的動量及對牆壁產生的平均力Look at its momentum and the average force

Pressure and Kinetic Energy, 2 假設與牆壁產生的是完全彈性碰撞 Assume perfectly elastic collisions with the walls of the container 由動量及牛頓第二定律可得壓力與分子動能的關係 The relationship between the pressure and the molecular kinetic energy comes from momentum and Newton’s Laws

The relationship is 壓力與分子密度(N/V)成正比 壓力與平均的分子動能成正比 This tells us that pressure is proportional to the number of molecules per unit volume (N/V) and to the average translational kinetic energy of the molecules

如何增加壓力? 增加N/V One way to increase the pressure is to increase the number of molecules per unit volume 增加速率(動能) The pressure can also be increased by increasing the speed (kinetic energy) of the molecules

用分子(微觀)解釋溫度 A Molecular Interpretation of Temperature 由右式看出, 溫度是與分子的動能有關的 Therefore, the temperature is a direct measure of the average translational molecular kinetic energy

Simplifying the equation relating temperature and kinetic energy gives 整理得到 Simplifying the equation relating temperature and kinetic energy gives 把動能分成三個方向x,y及z, 因此x方向有 This can be applied to each direction, with similar expressions for vy and vz

結論 分子在每一個自由度方向的運動, 其貢獻的能量是一樣的 Each translational degree of freedom contributes an equal amount to the energy of the gas In general, a degree of freedom refers to an independent means by which a molecule can possess energy 推廣而言, 此即為能量均分原理 A generalization of this result is called the theorem of equipartition of energy

能量均分原理 Theorem of Equipartition of Energy 在熱平衡狀態下, 系統的能量平分到每一個自由度的運動上 The theorem states that the energy of a system in thermal equilibrium is equally divided among all degrees of freedom 每一個分子在每一個自由度上分到的能量是 ½ kBT Each degree of freedom contributes ½ kBT per molecule to the energy of the system

總動能Total Kinetic Energy The total translational kinetic energy is just N times the kinetic energy of each molecule This tells us that the total translational kinetic energy of a system of molecules is proportional to the absolute temperature of the system

單原子分子氣體Monatomic Gas For a monatomic gas, translational kinetic energy is the only type of energy the particles of the gas can have Therefore, the total energy is the internal energy: For 複原子分子polyatomic molecules, additional forms of energy storage are available, but the proportionality between Eint and T remains

均方根速率Root Mean Square Speed The root mean square (rms) speed is the square root of the average of the squares of the speeds Square, average, take the square root Solving for vrms we find M is the molar mass in kg/mole

Some Example vrms Values 同一溫度下, 輕的分子平均比較重的分子動的快 At a given temperature, lighter molecules move faster, on the average, than heavier molecules

Distribution of Molecular Speeds 熱平衡狀態下, 氣體的分子速率分佈如圖 The observed speed distribution of gas molecules in thermal equilibrium is shown NV is called the Maxwell-Boltzmann distribution function(速率分佈函數)

分佈函數Distribution Function The fundamental expression that describes the distribution of speeds in N gas molecules is mo is the分子質量 mass of a gas molecule, kB is Boltzmann’s constant and T is the absolute temperature

Average and Most Probable Speeds The average speed is somewhat lower than the rms speed The most probable speed, vmp is the speed at which the distribution curve reaches a peak

Speed Distribution T增加,速率分佈向右移, 寬度增加 The peak shifts to the right as T increases This shows that the average speed increases with increasing temperature The width of the curve increases with temperature The asymmetric shape occurs because the lowest possible speed is 0 and the upper classical limit is infinity

Speed Distribution, final The distribution of molecular speeds depends both on the mass and on temperature 液體裡的分子速率分佈與氣體的類似 The speed distribution for liquids is similar to that of gasses

液體的氣化Evaporation Some molecules in the liquid are more energetic than others Some of the faster moving molecules penetrate the surface and leave the liquid This occurs even before the boiling point is reached The molecules that escape are those that have enough energy to overcome the attractive forces of the molecules in the liquid phase較大動能的分子能脫離液體的吸引力 The molecules left behind have lower kinetic energies留下較少動能的分子, 液體分子的平均速率因而減小, 對應較低之溫度 因此Therefore, 氣化evaporation is a 冷卻過程cooling process

大氣Atmosphere For such a huge volume of gas as the atmosphere, the assumption of a uniform temperature throughout the gas is not valid There are variations in temperature Over the surface of the Earth At different heights in the atmosphere