Thermal and Statistical Physics Essay Example

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    Physics
  • Document type:
    Math Problem
  • Level:
    Masters
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13THERMAL AND STATISTICAL PHYSICS

Thermal and Statistical Physics

Thermal and Statistical Physics

Question 1: Efficiency of a cannot cycle

The area under the curve is as below:

Thermal and Statistical Physics i

This equation gives the amount of energy transferred in the process

  1. If the process moves towards the less entropy, then heat will be removed from the system.

  2. If the process moves towards the great entropy, then heat will be absorbed in the system.

From the TS diagram for a reversible, the amount of work done over a cyclic process is:

Thermal and Statistical Physics 1 (ii)

Evaluating equation (ii) above:

Thermal and Statistical Physics 2

Thus the amount of energy transferred from the hot reservoir to the cold reservoir will be:

Thermal and Statistical Physics 3

and amount of energy transferred between the system and the cold reservoir will be:

Thermal and Statistical Physics 4

Therefore the efficiency, Thermal and Statistical Physics 5

Thermal and Statistical Physics 6 Is the maximum system entropy

Thermal and Statistical Physics 7 Is the minimum system entropy

Thermal and Statistical Physics 8 The amount of heat entering the system

Thermal and Statistical Physics 9 The amount of heat leaving the system

Thermal and Statistical Physics 10 The absolute temperature of the hot reservoir

Thermal and Statistical Physics 11 The absolute temperature of the cold reservoir

Question 2

  1. For a closed system

Thermal and Statistical Physics 12

Thermal and Statistical Physics 13

Thermal and Statistical Physics 14

Due to charge in temperature and volume, the internal energy charges as:

Thermal and Statistical Physics 15

Integrating the equation above:

Thermal and Statistical Physics 16

  1. The change in internal energy of a closed system

Thermal and Statistical Physics 17

Because of compression or expansion of the system, the equation becomes as below replacing the work in the system.

Thermal and Statistical Physics 18

Thermal and Statistical Physics 19

Taking F as a function of T and V, the heat capacity is as below:

Thermal and Statistical Physics 20

  1. From the law of thermodynamics, for a reversible process:

Thermal and Statistical Physics 21

Differentiating the internal energy equation

Thermal and Statistical Physics 22

Hence, Thermal and Statistical Physics 23

Thermal and Statistical Physics 24

Therefore Thermal and Statistical Physics 25

  1. Heat capacity with constant pressure

At constant pressure: Thermal and Statistical Physics 26

Simplifying Thermal and Statistical Physics 27

Thermal and Statistical Physics 28

Therefore at constant pressure:

Thermal and Statistical Physics 29

So Thermal and Statistical Physics 30

  1. For a constant temperature and volume

Change in heat capacity with constant volume is given as:

The change in internal energy of a closed system

Thermal and Statistical Physics 31

Because of compression or expansion of the system, the equation becomes as below replacing the work in the system:

Thermal and Statistical Physics 32

Thermal and Statistical Physics 33

Taking F as a function of T and V, the heat capacity is as below:

Thermal and Statistical Physics 34

Therefore Thermal and Statistical Physics 35

Question 3: Heat transfer

  1. Heat transferred, Thermal and Statistical Physics 36

Where: Q it the amount of heat transferred.

M is the mass

Thermal and Statistical Physics 37 is the heat capacity of the atmosphere

Thermal and Statistical Physics 38Is the temperature change

Thermal and Statistical Physics 39

Thermal and Statistical Physics 40

  1. Work doneThermal and Statistical Physics 41

Where: W = work done

R is the ideal gas constant

P1 is the pressure at the beginning

2 is the final pressure

T it is the temperature

Thermal and Statistical Physics 42

Thermal and Statistical Physics 43 -19703.72 J energy is lost in the system

  1. Change in internal energy

Thermal and Statistical Physics 44

Therefore Thermal and Statistical Physics 45

Thermal and Statistical Physics 46

Thermal and Statistical Physics 47

Question 4

  1. The change in internal energy of a closed system

Thermal and Statistical Physics 48

Because of compression or expansion of the system, the equation becomes as below replacing the work in the system:

Thermal and Statistical Physics 49

Thermal and Statistical Physics 50

Therefore at constant volume and entropy

Thermal and Statistical Physics 51

  1. At constant entropy

Thermal and Statistical Physics 52

Thermal and Statistical Physics 53

Thermal and Statistical Physics 54

Taking entropy at constant volume and temperature:

Thermal and Statistical Physics 55

This becomes, Thermal and Statistical Physics 56

  1. At constant entropy and pressure

Thermal and Statistical Physics 57

Thermal and Statistical Physics 58

Therefore diving the Thermal and Statistical Physics 59 and Thermal and Statistical Physics 60 at constant pressure and entropy

Thermal and Statistical Physics 61

Therefore Thermal and Statistical Physics 62

But Thermal and Statistical Physics 63

And Thermal and Statistical Physics 64

Therefore, Thermal and Statistical Physics 65

So, Thermal and Statistical Physics 66

Question 5

  1. Phase diagram of a fluid

Thermal and Statistical Physics 67

  1. Phase transitions

  1. Two properties of the system that change

During the phase transition, the properties that change are pressure and volume.

The two properties that remain constant are temperature and mass.

  1. Clapeyron’s Equation

The slope of the curve follows:

Thermal and Statistical Physics 68

During the phase change, the temperature is constant: Thermal and Statistical Physics 69

From the Maxwell’s relation:

Thermal and Statistical Physics 70

Taking an integral from one phase to another:

Thermal and Statistical Physics 71

Thermal and Statistical Physics 72

For a closed system and using the first law of thermodynamics:

Thermal and Statistical Physics 73

Using the fact that temperature and pressure are constant, we get:

Thermal and Statistical Physics 74

Therefore, Thermal and Statistical Physics 75

Thermal and Statistical Physics 76

Substituting in the equation of pressure change:

Thermal and Statistical Physics 77

Therefore, the equation becomes:

Thermal and Statistical Physics 78

  1. Clapeyron’s equation of sublimation

From the equation, Thermal and Statistical Physics 79

Thermal and Statistical Physics 80

Integrating the equation, Thermal and Statistical Physics 81

Therefore for a liquid vapor boundary:

Thermal and Statistical Physics 82

Given pressure and temperature are constant, therefore:

Thermal and Statistical Physics 83

Question 6

  1. Given the vapor pressure for liquid and solid and solving the equations simultaneously:

Thermal and Statistical Physics 84 i

Thermal and Statistical Physics 85 ii

Solving (i) and (ii) simultaneously

Thermal and Statistical Physics 86

Thermal and Statistical Physics 87

Thermal and Statistical Physics 88

Thermal and Statistical Physics 89

Thermal and Statistical Physics 90

Thermal and Statistical Physics 91

  1. Latent heat for triple points

At the triple points Thermal and Statistical Physics 92

Latent heat Thermal and Statistical Physics 93

Question 7

  1. Maximum inversion temperature of Helium

From the equation Thermal and Statistical Physics 94

Maximum inversion temperature is at Thermal and Statistical Physics 95

For Thermal and Statistical Physics 96

Thermal and Statistical Physics 97 0 = Thermal and Statistical Physics 98

Solving the quadratic equation

Thermal and Statistical Physics 99

Thermal and Statistical Physics 100

Thermal and Statistical Physics 101

Thermal and Statistical Physics 102

Thermal and Statistical Physics 103

Thermal and Statistical Physics 104

Thermal and Statistical Physics 105

Therefore, the maximum inversion temperature of Helium is Thermal and Statistical Physics 106

  1. Maximum inversion pressure is at maximum inversion temperature.

At 24 K, the inversion curve is at maximum therefore,

Thermal and Statistical Physics 107

Thermal and Statistical Physics 108

Thermal and Statistical Physics 109

References

Balmer, R. T. (2010). Modern engineering thermodynamics. New York: Elsivier.

Clausius, R. (1879). The mechanical theory of heat.London: Macmillan & Co.

Kittel, C. (1986). Introduction to solid state physics (6th edition). New York: John Wiley.

Stanley, H. E. (1971). Introduction to phase transitions and critical phenomena. New York:

Oxford University Press.