Chap 4 熱與熱力學第一定律:

4-1 Work and Heat

When two objects have different temperatures and were placed together, (Generally we say one is hot, the other is cold) they will reach the same temperature.

During the processes to reach the final temperature, people may think there exists a substance or a form of matter termed caloric ( now we call that is heat ) in

every body(which is wrong), and this substance transferred from hot to cold object. Though we know that heat is not a substance and heat is the something to have

two objects to reach thermal equilibrium.

Definition of Heat: Heat is transferred between a system and its surroundings by virtue of a temperature difference only.

Adiabatic wall : no heat can transfer through this wall, heat insulator.

Adiabatic process: During adiabatic expansion and adiabatic compression, no heat will transfer between system and its surrounding.

Diathermic wall : heat can transfer through this wall completely, heat conductor.

Figure 4-1 a), it is adiabatic work, ( from right to left is adiabatic compression, while from left to right is adiabatic expansion) since the system is surrounded

by adiabatic wall, note the temperature of system is irrelevant to its of the surroundings.

b), the system is heated up by its surroundings and heat is transfered into the system, but no work is done, since dV = 0.

c), Work and heat are done or transferred between system and its surroundings, this is the general case.

Work and heat have the same unit, so, during a state changed, does it perform a work or just the heat transferred. The answer of this question is " Find the system and its surroundings".

Figure 4-2: can clearly specify this answer.

Weight (as a surroundings of resistance) do work to the Resistance system. The resistance ( as a system and water as surroundings) transferred heat to the water surroundings.

4-2 Adiabatic Work

Adiabatic System: A system is enclosed by an adiabatic boundary (or walls) and its temperature is independent of that of the surroundings, and this system

may still be coupled to the surroundings so that work (not heat transferred) may be done. See Figure 4-3, and the work done between system and its surroundings is call adiabatic work.

Figure 4-4a is

Irreversible adiabatic process: Adiabatic free expansion, if the piston is moving faster than the velocity of liquid molecules contained inside the adiabatic walls, then the fluid will do no work on the piston.

Reversible adiabatic process: Move the piston slowly and it is a quasi-static processes, the fluid system is in quasi-equilibrium.

In figure 4-4b:

iaf path: ia is a reversible adiabatic process due to slowly moving pistons( can go both directions ia or ai), while the af is irreversible adiabatic and isothermal process.

During af process (can only go one direction), the resisters dissipate electrical energy in conjunction with piston movements that keep the system at constant temperature.

ibf path: ib is an irreversible adiabatic and isothermal process, while the bf is a reversible process due to slowly moving pistons. During ib process, the resisters dissipate

electrical energy in conjunction with piston movements that keep the system at constant temperature.

iedf path: ie is a reversible adiabatic process due to slowly moving pistons, and the ed is an irreversible process (free expansion), and df is an isochoric process (constant volume V and V') because

the resisters dissipate electrical energy in conjunction with piston movements that keep the system at constant volume.

iebf path: ie is an irreversible adiabatic process (free expansion), and the eb is an isochoric process (constant volume) because the resisters dissipate electrical energy in conjunction with piston

movements that keep the system at constant volume., and bf is a reversible process due to slowly moving pistons.

【Although accurate measurements of adiabatic work along different paths between the same two states have never been made, indirect experiments indicate that the adiabatic work is the same along all such paths.】 This implies that all the paths mentioned above have done the same amount of work, and the generalization of this result is known as the first law of thermodynamics:

→【If a system is caused to change from an initial state to a final state by adiabatic means only , the work done is the same for all adiabatic paths connecting the two states.】

The first law gives the following conclusion: the work done adiabatically is independent of path between initial and final states, and there exists a function of the coordinates of thermodynamic system. The adiabatic work done between initial and final states are equal to the difference of the function between initial and final states. The function is known as internal-energy function, U, which is

If the adiabatic work is done on the system, thus it is a positive work, the internal energy increases. From the discussion above, the adiabatic work done on the system is equal to increase of internal energy,→this simply expresses the principle of the conservation of energy.

4-3 Internal Energy Function

is the energy change of the system, and U is a function of thermodynamic coordinates, also it can specify the state of a system. General speaking, any function, which is the function of thermodynamic coordinates, can be expressed as function of any two thermodynamic coordinates because the third coordinate can be linked through the equation of state. We can write dU as function of θ and V,

if dU as a function of θ and P, we have

Note: .

4-4 Mathematical Formulation of the First Law of Thermodynamic

Nonadiabatic process: The work done during this process is not equal to the difference of internal energy between initial and final states(or simply say that the system is not surrounded by adiabatic wall, the temperature of system and its surroundings is related).

Question: What is the nonadiabatic process? For examples? (Figure 4-5)

Figure 4-5 a),the system is heated up by the heat source, through the diathermic wall.

b), Paramagnetic solid is surrounded by the liquid heliun, and they have temperature different, also the magnetic field is acting on the solid.

We know that the adiabatic work is equal to the change of internal energy, thus energy of the system is conserved. If the work is nonadiabatic, then it is not equal to . Due to the concept of energy conservation, mathematically we can rewrite the first law of thermodynamic as:

where Q is heat and W is the general work (not adiabatic work only), we are forced to conclude that energy has been transferred by means of heat other than the performance of work. Heat is transferred between system and its surroundings and this process has taken place only by virtue of the temperature difference between the system and its surroundings.

The first law contains three related ideas:

1. The existence of an internal-energy function, whose infinitesimal change is an exact differentials.

2. The principle of energy conservation.

3. The definition of heat as energy in transit by virtue of a temperature difference.

Re-define heat: 『When a system whose surroundings are at a different temperature and on which work may be done undergoes a process, the energy transferred by nonmechanical means, equal to the difference between the internal-energy change and the work done, is called HEAT.』

This means heat is a form of energy.

4-5 Concept of Heat

【Heat is internal energy in transit. It flows from one part of a system to another, or from one system to another, by virtue of only a temperature difference.】 The performance of work and the flow of heat are methods whereby the internal energy of a system is chanaged.

Since work W generally depend on the path and U is path independ function, so Q must be path-dependent. The first law is a path-independent function equal to two path dependent functions W and Q.

4-6 Differential Form of the First Law of Thermodynamics

For an infinitesimal quasi-static process, the first law can be express as

where only dU and can be represented by the thermodynamic coordinates. For hydrostatic system, and the other systems can be check in Table 4-1. For example: the paramagnetic gas system:

This is known as a Pfaffian differential form which is inexact. This system has two independent thermodynamic coordinates, (θ, P,V) and (θ,H,M). To integrate , generally we offen mutiply an intrgrating function to make it as an exact differential ( this is purely mathematical property. For simple system that undergoes an infinitesimal quasi-static process, e.g. , we can always find an integrating factor because there only two independent thermodynamic coordinates. (the third one can be eliminated by the equation of states). In general, a Pfaffian differential from containning three differentials does not admit of an integrating factor. But for having three or more independent coordinates, it still can find an integrating factor due to the second law of thermodynamics and the empirical temperature θ. The intrgrating factor for with any number of independent variables is an arbitrary function of the empirical temperature only and it is the same fumction for all systems.→this gives the definition of absolute thermodynamic temperature.

4-7 Heat Capacity and Its Measurement

If a system undergoes a change of temperature from qi to qf during the transfer of Q units of heat, the average heat capacity of the system is defined as the ratio

Average heat capacity =

The instantaneous value of heat capacity C is

→ C =

Specific heat: means the heat capacity per unit mass. Specific → Per Unit Mass.

Molar heat capacity c: The heat capacity per mole. Molar → Per Mole

Heat capacity can be positive, zero, negative or infinite (phase transition), depending on the process the system undergoes during the heat transfer.

Heat capacity at constant pressure: ; a function of P and θ.

Heat capacity at constant volume:; a function of V and θ.

The measurement of the heat capacity of solid is very important, because the numerical values of heat capacity provide one of the most direct means of verifying the calculations of theoretical physicists and of deciding on the validity of the assumptions constituting some of the modern theories.

測量熱容量現今多採用 electrical method, 如下圖

將樣品置於真空中並用導熱非常差的物質將其圍繞以避免熱的散失。利用熱電偶或電阻溫度計來測系統的溫度。

c 為莫耳熱容量，Dq和Dt的測量如下圖:
Dq為溫度差，Dt為加熱時間。AB段溫度計與樣品接觸但加熱電源並為未啟動。BD段加熱電源啟動，DEAB段加熱電源關閉。F和G圍外插得到的。
4-8: 
18世紀用水作為量取熱容量的方法。純水由升14.5℃到15.5℃所需之熱為一卡。
1cal = 4.1860 J

4-9 Equation for a Hydrostatic System

Cp、Cv的實驗值對理論的驗證十分重要。

代入第一定律


上式對靜流體而言是一般式。
a) 若V是常數→dV=0，則

U可以用理論模型計算，Cv可以由實驗求出。兩相驗證則可以知理論的假設是否正確。
b) 若P是常數→dP=0，則


實驗上可以測量則可以推算驗證理論。

4-10 熱庫

在準靜過程中系統與外界的溫差是非常小，且溫度的變化也是非常慢，一般假設系統的溫度是均勻的。這樣情況下熱的傳遞也是非常慢，如此熱的傳遞可以用熱力學坐標進行簡單的運算。要如何有慢且均勻的熱傳遞呢?熱庫是一方法。
Heat Reservior: (熱庫)一物體質量很大，可以吸收或放出無限量的熱量而本身溫度不會改變，且其他熱力學坐標亦不會改變。(或說有改變但很小而無法測知)
一個準靜過程，可以視為與一系列之不同溫度(其溫差很小)的熱庫相接觸，而所考慮之系統之溫度由θi變化到θf，則熱流量可以被計算。例如等壓準靜過程，
        
若是等壓準靜過程，則
        
對於其它系統的準靜過程，也是有類似的考慮與計算方法。

4-11 Heat Conduction (熱傳導)

當一物質的兩個部分各別保持在不同的溫度(或說與兩個不同溫度的熱庫接觸)，則能量在這兩部分之間的物質傳遞。實驗上證實在這兩者之間溫度是連續分部的。由於溫度不相同而在臨近區域傳遞能量的現象是為熱傳導。下圖的結果是

，

→

dθ/dx是溫度梯度，K為熱傳導係數(當溫差不大時可以被是為常數)。負號是因為熱傳導的方向與溫度由高而低的方向相反，及溫度梯度是小於零。

K基本上為溫差的函數，熱傳導係數在當熱庫溫差不大時可以被視為常數。

4-12 熱傳導係數

樣品為金屬時，製成圓柱型，熱傳導係數，K，的測量方法如下圖所示。一端用電熱絲加熱保持定溫，另一端以水流保持定溫。熱由高溫端流向低溫時，且絕大多數的能量將由水帶走以保持定溫，對金屬而言，表面散失的能量相較之少許多。

若為非金屬，則製成薄盤型並用銅板夾住樣品，同金屬一樣，一端用電熱絲加熱保持定溫，另一端以水流保持定溫。

問題: 氣體和液體的熱傳導係數，K，如何測?

影響熱傳導係數，K，的因素

一) 熱傳導係數，K，一般為溫度的函數，

二) 也因材料之結構改變而改變(如加溫或加壓使之相變)，

三) 金屬熱傳導係數對雜質的存在也極端的敏感(如銅中攙有少許砷則K減小一千倍)。

四) 對固體和液體而言，其熱傳導係數對適中的壓力的變化不敏感。

五) 固體液化時，熱傳導係數減小。

六) 多數液體其熱傳導係數隨壓力的增加而變小，隨溫度的增加而增加。

七)非金屬之熱傳導係數在常溫時很小且特性與液體類似，也隨溫度之增加而增加。

八) 非金屬之熱傳導係數在常溫時，一般較小，但在低溫時情況就不同了。見圖4-8中藍寶石與固態氦的曲線。

九) 氣體是較差的熱導體，見4-8圖中的氦氣曲線。

4-13 對流(Convection)

流動的氣體或液體可以在某處吸收熱量，然後移到另處將熱傳遞出去，這樣的現象是對流。如果是因為流體因不同密度而造成溫差，所產生對流的氣體或液體是為自然對流。例如？？？如果是因邦浦或風扇的作用所產生對流的氣體或液體是為強迫對流。例如？？？

由上圖知

，

h是對流係數，A是牆的面積，Dq是流體與牆的溫差。影響對流係數的因素有: 見課本

4-14 熱輻射(Thermal Radiation)

熱的固體或液體放出電磁波的現象，稱之為熱輻射。熱的物質所放出的電磁波經光譜分析知是連續光譜，同一種物質在不同的溫度的情況下其熱輻射之光譜的分佈(波長範圍)會不同。如在500℃時，一般物質其多光譜分佈在紅外線附近，當溫度增高時光譜的分部會趨進可見光區。總之溫度愈高，熱輻射的總能量愈高，光譜的分佈愈趨向短波長。物體可以由輻射傳出熱量，而溫度降低。也可以吸收輻射熱量而升高溫度。實驗上的觀察證明知熱輻射率與熱輻吸收率和物體的溫度及表面的特質有關。（一般而言由外界傳來之熱輻射是各向同性可以被物體反射、吸收和透射。）

Radiant exitance = R = total radiant power emitted per unit area.→輻射度例如鎢在2177℃是500 kW/m2。

absorptivity = a = fraction of the total energy of isotropic radiation that is absorbed.→吸收率。例如鎢在2477℃是0.25。

Blackbody (黑體): 黑體是指吸收率等於1的物質，也就是可以吸收所有的熱輻射的物體。→這是一種理想物質，並不真實存在，但實驗可以找到非常近似的物體。什麼樣的裝置是接近黑體呢？下圖所示

黑體的熱輻射與材料本身的性質無關，而，如下所述得知、其僅與溫度有關。被輻射度 H(Irradiance) 是指照射在空腔中任何單位面積上、單位時間內的輻射量。對黑體而言，單位面積所吸收之輻射功率等於，()。

當黑體的溫度保持一定時，也就是黑體本身與外界達成熱平衡(所吸收之熱輻射等於所發出之熱輻射，熱的流出與流入是相等的)，這也就是說黑體所發出之輻射，，與其所吸收之熱輻射是的一樣多；即

→

The irradiance within a cavity whose walls are at the temperature θis equal to the radiant exitance of a blackbody at the same temperature.→The radiation within a cavity is called Blackbody Radiation. 因為H與物質本身的特性無關，所以黑體輻射度只是溫度的函數。任何黑體若其溫度相等，則熱輻射率相同，輻射光譜也相同，這種輻射與物質無關。稱之為黑體輻射。

4-15 Kirchhoff's Law: Heat Radiation

克希何夫定律是指：物體之吸收率等於輻射度比同溫度之黑體輻射度即【The gain or loss of internal energy, equal to the difference between the energy of the thermal radiation which is absorbed and that which is radiated, is called heat.】當一系統或物體與外界有溫差時、且無作功、無熱傳導或對流的現象時；與外界達成熱平衡過程中，內能的增加或減少等於輻射出的能量與所吸收之輻射兩者之間的差值，此內能的改變稱之為熱。