# Formelsammlung

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BFH/MNG, Dr. F. L&ouml;wenthal Version 4.2
𝜃𝜃
𝜔𝜔
𝛼𝛼
Formulas in physics
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
𝛼𝛼 =
𝑑𝑑𝑑𝑑
Oscillations and waves ………………………………………….4
Geometrical optics ………………………………………………..5
Thermodynamics …………………………………………………..7
Atomphysics ………………………………………………………….9
Constants ……………………………………………………………..10
𝛥𝛥𝛥𝛥
𝛥𝛥𝛥𝛥
𝑥𝑥 = 𝑣𝑣 ⋅ 𝑡𝑡
displacement [m]
Mean velocity [m/s]
Movement on a circuit with
constant acceleration
𝑣𝑣 2
𝑟𝑟
2𝜋𝜋𝜋𝜋 1 2𝜋𝜋
= =
𝑣𝑣
𝑓𝑓
𝜔𝜔
Angular acceleration
Centripetal acceleration
Vb: speed on the cicuit
Uniform movement on a
circuit
centripetal acceleration a
Periode T
Frequency f
1
𝐸𝐸𝑟𝑟𝑟𝑟𝑟𝑟 = 𝐼𝐼𝜔𝜔2
2
Energy of rotation
𝐼𝐼𝐴𝐴 = 𝐼𝐼𝑆𝑆 + 𝑚𝑚𝑚𝑚 2
Theorem of Steiner
𝐿𝐿�⃗ = 𝑟𝑟⃗ &times; 𝑝𝑝⃗
angular momentum
��⃗ = 𝑟𝑟⃗ &times; 𝐹𝐹⃗
𝑀𝑀
Moment of force
Orbital angular momentum
distance [m]
velocity [m/s]
time [s]
𝐿𝐿 = 𝐼𝐼 ⋅ 𝜔𝜔
distance x [m]
acceleration a [m/s^2]
𝑀𝑀 =
𝑑𝑑𝑑𝑑
= 𝐼𝐼 ⋅ 𝛼𝛼
𝑑𝑑𝑑𝑑
angular momentum of a
solid
Equation of circular motion
Force and Movement
Mean acceleration [m/s^2]
𝐹𝐹⃗ = 𝑚𝑚 ⋅ 𝑎𝑎⃗
Moment in two and three dimensions
𝑟𝑟⃗ = 𝑥𝑥𝑒𝑒���⃗1 + 𝑦𝑦𝑒𝑒����⃗
���⃗𝑧𝑧
𝑦𝑦 + 𝑧𝑧𝑒𝑒
1
𝜃𝜃 = 𝜃𝜃0 + 𝜔𝜔 ⋅ 𝑡𝑡 + 𝛼𝛼𝑡𝑡 2
2
𝑣𝑣 2
𝑎𝑎 =
𝑟𝑟
𝑇𝑇 =
Moment with constant acceleration
1
𝑥𝑥 = 𝑥𝑥0 + 𝑣𝑣0 ⋅ 𝑡𝑡 + 𝑎𝑎𝑡𝑡 2
2
𝑣𝑣(𝑡𝑡) = 𝑣𝑣0 + 𝑎𝑎 ⋅ 𝑡𝑡
𝑣𝑣 2 = 𝑣𝑣02 + 2a(𝑥𝑥 − 𝑥𝑥0 )
1
𝑥𝑥 − 𝑥𝑥0 = (𝑣𝑣0 + 𝑣𝑣)𝑡𝑡
2
𝛥𝛥𝛥𝛥
𝑎𝑎̄ =
𝛥𝛥𝛥𝛥
Angular speed
𝑎𝑎 =
Movement in one dimension
Angular speed and
tangential speed
𝜔𝜔 =
𝑣𝑣𝑏𝑏 = 𝜔𝜔 ⋅ 𝑟𝑟
Mechanics
𝑣𝑣̄ =
𝑣𝑣 = 𝜔𝜔 ⋅ 𝑟𝑟
Electronics …………………………………………………………….3
𝛥𝛥𝛥𝛥 = 𝑥𝑥2 − 𝑥𝑥1
𝑠𝑠
𝑟𝑟
𝜙𝜙 =
Mechanis ………………………………………………………………1
angular acceleration
Reibung
vector
𝑓𝑓𝑠𝑠,𝑚𝑚𝑎𝑎𝑥𝑥 = 𝜇𝜇𝑠𝑠 𝑁𝑁
displacement
𝛥𝛥𝑟𝑟⃗ = ���⃗
𝑟𝑟2 − ���⃗
𝑟𝑟1
throw
𝑥𝑥 − 𝑥𝑥0 = (𝑣𝑣0 𝑐𝑐𝑐𝑐𝑐𝑐𝛩𝛩0 )𝑡𝑡
1
𝑦𝑦 − 𝑦𝑦0 = (𝑣𝑣0 𝑠𝑠𝑠𝑠𝑠𝑠𝛩𝛩0 )𝑡𝑡 − 𝑔𝑔𝑡𝑡 2
2
𝑔𝑔𝑔𝑔 2
𝑦𝑦(𝑥𝑥) = (𝑡𝑡𝑡𝑡𝑡𝑡𝛩𝛩0 )𝑥𝑥 −
2(𝑣𝑣0 cos𝛩𝛩0 )2
2
𝑣𝑣0
𝑅𝑅 = sin2𝛩𝛩0
Horizontal reach
𝑔𝑔
𝑓𝑓𝑘𝑘 = 𝜇𝜇𝑘𝑘 𝑁𝑁
Resistance of fluid
1
𝐹𝐹 = 𝐶𝐶𝑤𝑤 𝜌𝜌𝜌𝜌𝑣𝑣 2
2
Movement on a circle
1
𝑣𝑣𝑡𝑡 = �
2𝐹𝐹𝑔𝑔
𝐶𝐶𝑤𝑤 𝜌𝜌𝜌𝜌
2. axiom of Newton
F [N]
Maximal friction
N: Normal force
Dynamic friction
𝐶𝐶𝑤𝑤 : constant
𝜌𝜌: density of the fluide
A: front surface
v: speed
𝑣𝑣𝑡𝑡 Maximal speed
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BFH/MNG, Dr. F. L&ouml;wenthal Version 4.2
𝐹𝐹 = 𝑚𝑚
𝑣𝑣 2
𝑟𝑟
momentum
Centripetal force
Equation of motion
Mechanical Energy
𝐸𝐸𝑘𝑘 =
1
𝑚𝑚𝑣𝑣 2
2
Kinetic energy
kinetic energy
𝑊𝑊 = 𝐹𝐹⃗ ⋅ 𝑟𝑟⃗
Golden rule of mechnanics
W: work
F: force
r: distance
𝐸𝐸𝑓𝑓 = 𝐸𝐸𝑖𝑖 + 𝑊𝑊
Change of energy
Ef: final energy
Ei: initial energy
W: work
𝐹𝐹 = −𝑘𝑘 ⋅ 𝑥𝑥
Hook's law
k: spring constant
𝑊𝑊 = 𝐹𝐹𝑟𝑟 ⋅ 𝑥𝑥 = 𝜇𝜇 ⋅ 𝑁𝑁 ⋅ 𝑥𝑥
Friction work
𝑊𝑊1,2 = ∫ 𝐹𝐹⃗ ⋅ ����⃗
𝑑𝑑𝑑𝑑
Generall math formulation
𝐸𝐸 =
1 2
𝑘𝑘𝑥𝑥
2
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
𝑊𝑊
𝑃𝑃̄ =
= 𝐹𝐹 ⋅ 𝑣𝑣
𝑡𝑡
𝑃𝑃 =
P: Momentum
M: mass
Total momentum
Spring energy
Instantaneous power [W]
Mean power
Potential energy
𝛥𝛥𝛥𝛥 = 𝑚𝑚𝑔𝑔(𝑦𝑦𝑓𝑓 − 𝑦𝑦𝑖𝑖 )𝐸𝐸𝑝𝑝𝑝𝑝𝑝𝑝
= 𝑚𝑚𝑚𝑚ℎ
1
𝑈𝑈(𝑥𝑥) = 𝑘𝑘𝑥𝑥 2
2
𝐹𝐹(𝑥𝑥) =
−𝑑𝑑𝑑𝑑(𝑥𝑥)
𝑑𝑑𝑑𝑑
Potential energy
Elastic energy
Force dermined from a
potiental
Collisions
J-Integral
Fm: mean force
J-Integral == change of
momentum
Impulse p or
𝐽𝐽 = 𝐹𝐹𝑚𝑚 ⋅ 𝛥𝛥𝛥𝛥
𝐽𝐽⃗ = 𝛥𝛥𝑝𝑝⃗
𝑝𝑝⃗ = 𝑚𝑚 ⋅ 𝑣𝑣⃗
2
𝐹𝐹 =
𝐸𝐸𝑘𝑘𝑘𝑘𝑘𝑘 =
�������⃗
𝑃𝑃𝑡𝑡𝑡𝑡𝑡𝑡 = ∑
𝑝𝑝
���⃗𝚤𝚤
𝛥𝛥𝛥𝛥
𝛥𝛥𝛥𝛥
𝑝𝑝2
2 ⋅ 𝑚𝑚
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BFH/MNG, Dr. F. L&ouml;wenthal Version 4.2
Elektrotechnics
Sources
𝑄𝑄𝑞𝑞 Open circuit voltage
𝑈𝑈𝐾𝐾 terminal voltage
𝑅𝑅𝑖𝑖 Internal resistance
Electrical charge
1 ∣𝑞𝑞1 ∣∣𝑞𝑞2 ∣
𝐹𝐹 =
4𝜋𝜋𝜀𝜀0 𝑟𝑟 2
Electrical field (static)
𝐸𝐸�⃗ =
𝐸𝐸 =
𝐸𝐸 =
Current
Coulombs law
q charges
r distance between charges
Coulomb-Gesetz
𝜀𝜀0 = 8.85 ⋅ 10−12 𝐶𝐶 2 ⁄𝑁𝑁 ⋅ 𝑚𝑚2
𝑃𝑃𝑉𝑉 Power of the consumer
RV: Resistance of the
consumer
Ri: internal resistance of
the source
F Force
𝐹𝐹⃗
𝑞𝑞0
1 ∣𝑞𝑞∣∣
4𝜋𝜋𝜀𝜀0 𝑟𝑟 2
E-Field of a point charge
1 𝑝𝑝
𝑝𝑝 = 𝑝𝑝𝑝𝑝
2𝜋𝜋𝜀𝜀0 𝑧𝑧 3
E-Field of a dipole
𝑈𝑈 = 𝑅𝑅 ∙ 𝐼𝐼
Ohms law
U tension [V]
R Ohm’s resistance
I Current
𝐹𝐹⃗ = 𝑞𝑞𝐸𝐸�⃗
𝑅𝑅 = 𝜌𝜌 ∙
𝐿𝐿
𝐴𝐴
Force of a charge q in the EField
Specific resistance 𝜌𝜌
L Length
A Area
1
𝑅𝑅𝑡𝑡 = �� �
𝑅𝑅𝑅𝑅
−1
𝑖𝑖
𝑅𝑅𝑡𝑡 = � 𝑅𝑅𝑖𝑖
Parallel resistances
Serial resistances
𝑖𝑖
Condensor
Capacity
𝑪𝑪𝒕𝒕 = � 𝑪𝑪𝒊𝒊
𝒊𝒊
Ct = ��
𝑖𝑖
1
�
𝐶𝐶𝑖𝑖
−1
Parallel condensors
Serial condensors
3
𝑈𝑈𝐾𝐾 = 𝑄𝑄𝑞𝑞 − 𝐼𝐼 ⋅ 𝑅𝑅𝑖𝑖
𝑈𝑈1 𝑅𝑅1
=
𝑈𝑈2 𝑅𝑅2
𝑃𝑃𝑉𝑉 = 𝑅𝑅𝑉𝑉 ⋅
𝑈𝑈𝑞𝑞2
(𝑅𝑅𝑖𝑖 + 𝑅𝑅𝑉𝑉 )2
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BFH/MNG, Dr. F. L&ouml;wenthal Version 4.2
Oscillations
𝛾𝛾 =
Oscillations without
friction
𝑥𝑥̈ + 𝜔𝜔02 𝑥𝑥 = 0
2𝜋𝜋
𝑇𝑇 =
𝜔𝜔
𝑓𝑓 =
1
𝑇𝑇
Harmonic oscillations
𝐸𝐸(𝑡𝑡) = 𝐸𝐸0 ⋅ 𝑒𝑒 −2𝛾𝛾𝛾𝛾
vmax = xm 𝝎𝝎
𝜔𝜔 = �
𝑘𝑘
𝑚𝑚
𝑚𝑚
𝑇𝑇 = 2𝜋𝜋�
𝑘𝑘
𝑇𝑇 = 2𝜋𝜋�
𝐿𝐿
𝑔𝑔
𝑇𝑇 = 2𝜋𝜋�
𝐿𝐿𝑟𝑟 =
𝐼𝐼𝑐𝑐𝑐𝑐 + 𝑚𝑚𝑚𝑚 2
𝑚𝑚𝑚𝑚𝑚𝑚
𝐼𝐼𝑐𝑐𝑐𝑐 + 𝑚𝑚𝑚𝑚 2
𝑚𝑚𝑚𝑚
𝑟𝑟𝑡𝑡 = �
Maximal speed
Maximal acceleration
Damping term
M: mass
b: Ns/m: Fr = - b v
Damped frequency
Energy loss
Doppler shift
Periode T
𝑓𝑓′ = 𝑓𝑓 ⋅
Frequence [Hz]
𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚 = 𝜔𝜔2 𝑥𝑥𝑚𝑚
t
𝜔𝜔2 = 𝜔𝜔02 − 𝛾𝛾 2
Equation of oscillations
𝑥𝑥(𝑡𝑡) = 𝑥𝑥𝑚𝑚 sin(𝜔𝜔𝜔𝜔 − 𝜙𝜙0 )
𝑏𝑏
2𝑚𝑚
𝑣𝑣
Resonance
=
1 &plusmn; 𝑣𝑣 ⁄𝑐𝑐
1 ∓ 𝑣𝑣 ⁄𝑐𝑐
f, f': Frequency
v: velocity Sender / recipient
c: velocity of sound
Resonance Equation
𝑥𝑥̈ + 2𝛾𝛾𝑥𝑥̇ + 𝜔𝜔02 = 𝑓𝑓𝑚𝑚 cos(𝜔𝜔𝜔𝜔)
Spring oscillations
𝜔𝜔 Angular velocity
Stationary solution
Periode of a spring oscillator
Periode
Mathematical pedulum
Periode
Physical pendulum
Amplitude 𝐴𝐴(𝜔𝜔) =
Reduced length of pendulum
Lr
Phase
𝛿𝛿(𝜔𝜔)
𝐹𝐹𝑚𝑚 ⁄𝑚𝑚
�(𝜔𝜔02 −𝜔𝜔2 )2 +𝑏𝑏 2 𝜔𝜔2 �𝑚𝑚2
= arctan �
𝐼𝐼𝑐𝑐𝑐𝑐
𝑚𝑚
𝑎𝑎
=
Damped oscillations
Equation of damped harmonic oscillations
𝑥𝑥̈
+
+
=
̈ + 2 𝛾𝛾 ẋ + 𝜔𝜔02 x = 0
x(t)
4
𝑏𝑏𝑏𝑏
�
𝑚𝑚(𝜔𝜔02 − 𝜔𝜔 2 )
𝜔𝜔𝑅𝑅 = �𝜔𝜔02 −
𝑏𝑏 2
2m2
𝑏𝑏𝑘𝑘𝑘𝑘𝑘𝑘𝑡𝑡 = √4𝑘𝑘𝑘𝑘
𝑥𝑥(t) = 𝐴𝐴(𝜔𝜔)cos(𝜔𝜔𝜔𝜔 − 𝛿𝛿(𝜔𝜔))
𝑏𝑏
𝛾𝛾 =
2𝑚𝑚
𝜔𝜔2 = 𝜔𝜔02 − 𝛾𝛾 2
𝐸𝐸(𝑡𝑡) = 𝐸𝐸0 ⋅ 𝑒𝑒 −2𝛾𝛾𝛾𝛾
Resonance frequency
Critical damping
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BFH/MNG, Dr. F. L&ouml;wenthal Version 4.2
Waves
𝑘𝑘 =
2𝜋𝜋
𝜆𝜆
Wave number
𝑇𝑇 =
2𝜋𝜋
𝜔𝜔
Periode
𝜔𝜔 = 2𝜋𝜋𝑓𝑓
Angular frequency
𝜏𝜏
𝜇𝜇
Speed in the rope
𝑣𝑣 = 𝜆𝜆 ⋅ 𝑓𝑓
𝑣𝑣 = �
𝑃𝑃𝑚𝑚 =
1
2
𝜇𝜇𝜇𝜇𝜔𝜔2 ⋅ 𝑦𝑦𝑚𝑚
2
Speed of phase
Power
Stationary waves
𝑘𝑘 =
2𝜋𝜋
𝜆𝜆
𝑓𝑓𝑛𝑛 = 𝑛𝑛 ⋅
𝑐𝑐
2L
wave number
Eigenfrequenzen
n = 1,2,3, …
5
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[P] = W
Geometrical Optics
Besselsche Methode
Mirrors
f&uuml;r Sammellinse
p: distance of object
i: distance of image
f: focal length
a: Abstand Objekt –
Bild
e: Abstand der beiden
Linsenpositionen
f: Brennweite
Refraction
n =
F&uuml;r Streulinse
Refraction index n
c: speed of light in
vacuum
cm: speed of light in
medium
c
cm
f1: Brennweiter Streulinse
a: Distanz ObjektStreulinse
d: Distanz StreulinseSammellinse
b: Distanz: SammellinseBild
Law of refaction
Sign convention
r &lt; 0 for concave
surface
r &gt; 0 f&uuml;r convex
surface
Dioptry D
Korrektur Kurzsichtig
smax: maximale
Sehdistanz
Lens maker formula
1
= (n-1)�r 1
1
r2
�
Abbildungsgleichung
d&uuml;nner Linsen
Intensit&auml;t bei
isotropem
Leuchtk&ouml;rper
[I] = W/m2
𝑛𝑛 ⋅ sin(𝛼𝛼) = 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐
𝑀𝑀 =
−𝐵𝐵 −𝑖𝑖
=
𝐺𝐺
𝑝𝑝
1 1 1
+ =
𝑝𝑝 𝑖𝑖 𝑓𝑓
Korrektur Weitsichtig 𝐼𝐼(𝑟𝑟) = 𝑃𝑃
2
ri &lt; 0 for convave
ri &gt; 0 for convex
f
𝑐𝑐0
𝑐𝑐𝑚𝑚
Linse in Serie
D&uuml;nne Linsen in Serie
𝑛𝑛1 𝑛𝑛2 𝑛𝑛2 − 𝑛𝑛1
+
=
𝑝𝑝
𝑖𝑖
𝑟𝑟
Refraction on a
spherical surface
1
𝑛𝑛𝑚𝑚 =
s0: normale
Sehdistanz
smin: minimale
Sehdistanz
4𝜋𝜋𝑟𝑟
smin &gt; s0 f&uuml;r
Weitsichtigkeit
1 1 1
+ =
𝑝𝑝 𝑖𝑖 𝑓𝑓
𝑅𝑅 = 2 ⋅ 𝑓𝑓
Optische
Instrumente
Konvex f &lt; 0
konkav f &gt; 0
6
𝑛𝑛𝑚𝑚 =
𝑐𝑐0
𝑐𝑐𝑚𝑚
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Lupe
s0 = 25 cm
f Brennweite der Lupe
Beamexpander
D:
Ausgangsdurchmesse
r
d:
Eingangsdurchmesser
f1: Eingangslinse
f2: Ausgangslinse
𝑒𝑒 = 2�
𝑎𝑎2
− 𝑎𝑎 ⋅ 𝑓𝑓
4
𝑎𝑎2 − 𝑒𝑒 2
4 ⋅ 𝑎𝑎
1 1
1
= +
𝑓𝑓1 𝑎𝑎 𝑑𝑑 − 1
1 1
−
𝑓𝑓2 𝑏𝑏
𝑓𝑓 =
𝑓𝑓𝑜𝑜𝑜𝑜𝑜𝑜
Fernrohr
𝑚𝑚 =
𝑓𝑓𝑜𝑜𝑜𝑜
m:
Winkelvergr&ouml;sserung
fok: Okularlinse
fobj: Objektivlinse
Mikroskop
s0 = 25 cm
s = Tubusl&auml;nge
fok = Okularbrennweite
fobj = Objektivbrennweite
𝑀𝑀 =
𝛥𝛥𝛥𝛥 =
−𝐵𝐵 −𝑖𝑖
=
𝐺𝐺
𝑝𝑝
𝐷𝐷 =
1
1
𝑓𝑓
=
−1
𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚
𝑓𝑓𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑒𝑒
𝛥𝛥𝛥𝛥 &lt; 0
1
1
𝛥𝛥𝛥𝛥 = −
𝑠𝑠0 𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚
𝛥𝛥𝛥𝛥 &gt; 0
𝑒𝑒 = 2�
𝑓𝑓 =
𝑎𝑎2
− 𝑎𝑎 ⋅ 𝑓𝑓
4
𝑎𝑎2 − 𝑒𝑒 2
4 ⋅ 𝑎𝑎
7
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BFH/MNG, Dr. F. L&ouml;wenthal Version 4.2
Thermodynamics
𝛥𝛥𝐿𝐿 = 𝛼𝛼 ⋅ 𝐿𝐿 ⋅ 𝛥𝛥𝑇𝑇
𝛥𝛥𝐴𝐴 = 𝛽𝛽 ⋅ 𝐴𝐴 ⋅ 𝛥𝛥𝛥𝛥
𝛽𝛽 = 2 ⋅ 𝛼𝛼
𝛥𝛥𝛥𝛥 = 𝛾𝛾 ⋅ 𝑉𝑉 ⋅ 𝛥𝛥𝛥𝛥
𝛾𝛾 = 3 ⋅ 𝛼𝛼
[Rs] = J/kg/K
[v] = m3/kg
M: mol volume
v: spezific volume
vm: molar volume
1.Dimensions
2 dimensions
Equation of real gas
3 dimensions
𝑝𝑝 ⋅ 𝑉𝑉 = 𝑍𝑍 ⋅ 𝑅𝑅𝑠𝑠 ⋅ 𝑇𝑇
𝑣𝑣𝑚𝑚 = 𝑀𝑀 ⋅ 𝑣𝑣
𝑅𝑅 = 𝑅𝑅𝑠𝑠 ⋅ 𝑀𝑀
Q = m C ∆T
Specific heat
capacity
[C] = J kg/K
NA = 6.022&middot;1023/mol
𝑄𝑄 = 𝑛𝑛 ⋅ 𝑐𝑐𝑚𝑚 ⋅ 𝛥𝛥𝑇𝑇
Molar heat
[cm] = J/mol/K
[n] mol-1
Work Wx
𝐶𝐶̄ =
𝑇𝑇2
∫𝑇𝑇1
𝐶𝐶(𝑇𝑇)𝑑𝑑𝑑𝑑
𝑇𝑇2 − 𝑇𝑇1
𝑄𝑄 = 𝐿𝐿 ⋅ 𝑚𝑚
𝑑𝑑𝑑𝑑𝑣𝑣 = 𝑝𝑝(𝑉𝑉) ⋅ 𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑𝑣𝑣 = 𝑝𝑝(𝑣𝑣)𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑𝑝𝑝 = −𝑉𝑉(𝑝𝑝) ⋅ 𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑𝑝𝑝 = −𝑣𝑣(𝑝𝑝) ⋅ 𝑑𝑑𝑑𝑑
Mean value
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
Melting enthalpy
[L] = J/kg
𝑝𝑝 ⋅ 𝑣𝑣 = 𝑅𝑅𝑠𝑠 𝑇𝑇
𝑝𝑝 ⋅ 𝑣𝑣𝑚𝑚 = 𝑅𝑅 ⋅ 𝑇𝑇
V-work Wv
p-work
Heat
[Q] = J
[q] = J/kg
(Spec.) Inner Energy
𝑓𝑓
[U] = J
⋅ 𝑁𝑁𝐴𝐴 ⋅ 𝑘𝑘𝐵𝐵 ⋅ 𝑇𝑇
[u] = J/kg
2
𝑅𝑅 = 𝑁𝑁𝐴𝐴 ⋅ 𝑘𝑘𝐵𝐵
Perfect gas: f = 3
𝑈𝑈 =
Perfect gas
𝑝𝑝 ⋅ 𝑉𝑉 = 𝑚𝑚 ⋅ 𝑅𝑅𝑠𝑠 ⋅ 𝑇𝑇
Real gas factor Z
𝛿𝛿𝛿𝛿
𝑇𝑇
𝛿𝛿𝑞𝑞
𝑑𝑑𝑑𝑑 =
𝑇𝑇
𝑑𝑑𝑑𝑑 =
Equation of equilibrium
of the perfect gas.
[V] = m3
[p] = Pa
[m] = kg
R = universelle
R = 8.314 J/mol/K
(spez.) Entropy S, s
[S] = J/K
[s] = J/K/kg
4 Principles of thermodynamics
8
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BFH/MNG, Dr. F. L&ouml;wenthal Version 4.2
Four elementary Processes in
closed systems
Closed systems
𝑑𝑑𝑑𝑑𝑝𝑝 = −𝑉𝑉(𝑝𝑝) ⋅ 𝑑𝑑𝑑𝑑 Volume work
𝑑𝑑𝑑𝑑𝑝𝑝 = −𝑣𝑣(𝑝𝑝) ⋅ 𝑑𝑑𝑑𝑑
Isochor process
1a. closed systems
𝑑𝑑𝑑𝑑 = 𝛿𝛿𝛿𝛿 − 𝛿𝛿𝑊𝑊𝑉𝑉
𝑑𝑑𝑑𝑑 = 𝛿𝛿𝛿𝛿 − 𝛿𝛿𝑤𝑤𝑣𝑣
V = const
𝑑𝑑𝑑𝑑𝑣𝑣 = 𝑝𝑝(𝑉𝑉) ⋅ 𝑑𝑑𝑑𝑑
𝑑𝑑𝑤𝑤𝑣𝑣 = 𝑝𝑝(𝑣𝑣)𝑑𝑑𝑑𝑑
Work
Conversion of energy
𝑝𝑝
= 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐
𝑇𝑇
Equation of state
1b. open Systems
heat
𝑑𝑑𝑑𝑑 𝑥𝑥 = 𝛿𝛿𝛿𝛿 − 𝛿𝛿𝑊𝑊𝑝𝑝
𝑑𝑑ℎ 𝑥𝑥 = 𝛿𝛿𝛿𝛿 − 𝛿𝛿𝑤𝑤𝑝𝑝
v-work
isobar process
2. natural processes 𝑑𝑑𝑑𝑑 = 𝛿𝛿𝛿𝛿
𝑇𝑇
takes the direction to
𝛿𝛿𝛿𝛿
𝑑𝑑𝑑𝑑 =
increase the total
𝑇𝑇
entropy.
𝑞𝑞12 = 𝑢𝑢2 − 𝑢𝑢1
𝑤𝑤𝑣𝑣 = 0
p = const
𝑣𝑣
= 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐
𝑇𝑇
Equation of state
heat
𝑑𝑑𝑑𝑑 &gt; 0
𝑑𝑑𝑑𝑑 &gt; 0
v-work
𝑞𝑞12 = ℎ2 − ℎ1
𝑤𝑤𝑣𝑣12 = 𝑅𝑅 ⋅ (𝑇𝑇2 − 𝑇𝑇1 )
isotherm process T = const
lim 𝛥𝛥𝛥𝛥 = 0
3. It is not possible to 𝑇𝑇→0
lim 𝛥𝛥𝛥𝛥 = 0
hit the absolut zero
𝑇𝑇→0
temperature by a
natural process.
Equation of state
heat
v-work
0. Two systems in a
thermodynamical
process have the
same temperature.
𝑝𝑝
= 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐
𝑇𝑇
𝑝𝑝 ⋅ 𝑣𝑣 = 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑡𝑡
𝑤𝑤𝑣𝑣12 = 𝑝𝑝1 ⋅ 𝑣𝑣1 ⋅ ln
𝑞𝑞12 = 𝑅𝑅 ⋅ 𝑇𝑇 ⋅ ln
𝑣𝑣2
𝑣𝑣1
exponent
k = cp/cv = 1.4
Equation of state
𝑝𝑝 ⋅ 𝑣𝑣 𝑘𝑘 = 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐
𝑝𝑝 ⋅ 𝑣𝑣 𝑘𝑘 = 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑡𝑡
𝑝𝑝 ⋅ 𝑣𝑣 𝑘𝑘 = 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑡𝑡
open systems
9
𝑣𝑣2
𝑣𝑣1
𝑝𝑝 ⋅ 𝑣𝑣 𝑘𝑘 = 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑡𝑡
𝐻𝐻 = 𝑈𝑈 + 𝑝𝑝 ⋅ 𝑉𝑉
ℎ = 𝑢𝑢 + 𝑝𝑝 ⋅ 𝑣𝑣
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BFH/MNG, Dr. F. L&ouml;wenthal Version 4.2
Enthalpy H, h
[H] = J
[h] = J/kg
Totale Enthalpy H*
[H*] = J
[h*] = J/kg
1
𝐻𝐻 𝑥𝑥 = 𝐻𝐻 + 𝑚𝑚𝑐𝑐 2
2
1
𝐻𝐻 𝑥𝑥 = 𝑈𝑈 + 𝑝𝑝𝑝𝑝 + 𝑚𝑚𝑚𝑚 2
2
1
ℎ 𝑥𝑥 = ℎ + 𝑐𝑐 2
2
1
ℎ 𝑥𝑥 = 𝑢𝑢 + 𝑝𝑝 ⋅ 𝑣𝑣 + 𝑐𝑐 2
2
10
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BFH/MNG, Dr. F. L&ouml;wenthal Version 4.2
Quantum
mechanics
Quantisierung der
Energieniveaus
h: Planckkonstante
f: Photonenfrequenz
Eu: oberes
Energieniveau
El: unteres
Energieniveau
Projektion des
Spindipolmoment
Photoelektrischer
Effekt
ℎ ⋅ 𝑓𝑓 = 𝐸𝐸𝑢𝑢 − 𝐸𝐸𝑙𝑙
Photonenenergie
f: Frequenz
Wellenl&auml;nge Frequenz
Photonenimpuls
Photonenmasse
Atomarer Drehimpuls
(Bahndrehimpuls)
Klassischer Drehimpuls
Atomarer
Bahndrehimpuls
L Bahndrehimpuls
l Quantenzahl, 𝑙𝑙𝑙𝑙𝑙𝑙, 𝑙𝑙 &lt;
𝑛𝑛
Photelektrische
Gleichung
𝐿𝐿 = 𝑚𝑚 ⋅ 𝑟𝑟 ⋅ 𝑣𝑣
𝐿𝐿 = �𝑙𝑙(𝑙𝑙 + 1) ⋅ ℏ
K: Kinetische Energie
der
Austrittselektronen
𝜙𝜙𝑎𝑎 Austrittsarbeit
V: Stoppotential
n: Hauptquantenzahl
Z-Projektion des
Drehimpulses,
∣𝑚𝑚𝑧𝑧 ∣ ≤ 𝑙𝑙, 𝑚𝑚𝑧𝑧 𝜖𝜖𝜖𝜖
Bohrsches
Magneton𝜇𝜇𝐵𝐵
Magnetisches
Dipolmoment
Z-Projektion des
magnetischen
Dipolmoments
𝐿𝐿𝑧𝑧 = 𝑚𝑚𝑙𝑙 ⋅ ℏ
𝜇𝜇𝐵𝐵 =
𝜇𝜇𝐵𝐵 =
����⃗
Materiewellen –
Wahrscheinlichkeit
𝑒𝑒
ℏ
2𝑚𝑚𝑒𝑒
De Broglie
Wellenl&auml;nge
−𝑒𝑒
�⃗
𝐿𝐿
2𝑚𝑚
𝜇𝜇𝐵𝐵,𝑧𝑧 = −𝑚𝑚𝑙𝑙 ⋅ 𝜇𝜇𝐵𝐵
𝑚𝑚𝑙𝑙 𝜖𝜖𝜖𝜖
Elektronenspin
(Eigendrehimpuls)
Spin des Elektrons
Z-Projektion des
Spins
Magnetisches
Dipolmoment des
Spins
𝑆𝑆 = �𝑠𝑠(𝑠𝑠 + 1) ⋅ ℏ
1
𝑠𝑠 =
2
𝑆𝑆𝑧𝑧 = 𝑚𝑚𝑠𝑠 ⋅ ℏ
1
𝑚𝑚𝑠𝑠 = &plusmn; � �
2
𝑒𝑒ℎ
𝜇𝜇𝑠𝑠 = − � � 𝑆𝑆⃗
���⃗
𝑚𝑚𝑒𝑒
���⃗
𝜇𝜇𝑠𝑠 = −2𝜇𝜇𝐵𝐵 𝑆𝑆⃗
11
𝜇𝜇𝑠𝑠,𝑧𝑧 = &plusmn;𝜇𝜇𝐵𝐵
𝜇𝜇𝐵𝐵 =
����⃗
−𝑒𝑒
𝐿𝐿�⃗
2𝑚𝑚
𝐸𝐸𝑝𝑝ℎ = ℎ𝑓𝑓
𝜆𝜆 ⋅ 𝑓𝑓 = 𝑐𝑐
ℎ ⋅ 𝑓𝑓 ℎ
=
𝑐𝑐
𝜆𝜆
ℎ ⋅ 𝑓𝑓
𝑚𝑚𝑝𝑝ℎ = 2
𝑐𝑐
𝑝𝑝 =
𝐸𝐸𝑝𝑝ℎ = 𝐾𝐾 + 𝜙𝜙𝑎𝑎
ℎ ⋅ 𝑓𝑓 = 𝑉𝑉𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ⋅ 𝑒𝑒 + 𝜙𝜙𝑎𝑎
ℎ
𝜙𝜙𝑎𝑎
𝑉𝑉𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = � � ⋅ 𝑓𝑓 −
𝑒𝑒
𝑒𝑒
𝜇𝜇𝑠𝑠,𝑧𝑧 = &plusmn;𝜇𝜇𝐵𝐵
𝜆𝜆 =
ℎ
𝑝𝑝
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BFH/MNG, Dr. F. L&ouml;wenthal Version 4.2
Constants
Speed of light
Elementary charge
Electron mass
Proton mass
Boltzmann constant
Planck constant h,
ℎ
oder ℏ = 2𝜋𝜋
Bohr magneton
𝑐𝑐 = 2.998 ⋅ 108 𝑚𝑚⁄𝑠𝑠
𝑒𝑒 = 1.60210 ⋅ 10−19 𝐶𝐶
𝑚𝑚𝑒𝑒 = 9.11 ⋅ 10−31 𝑘𝑘𝑘𝑘
𝑚𝑚𝑝𝑝 = 1.67 ⋅ 10−27 𝑘𝑘𝑘𝑘
𝑘𝑘 = 1.38 ⋅ 10−23 𝐽𝐽⁄𝐾𝐾
ℎ = 6.626 ⋅ 10−34 𝐽𝐽𝐽𝐽
ℏ = 1.054 ⋅ 10−34 𝐽𝐽𝐽𝐽
𝑚𝑚
=
⋅
𝜇𝜇𝐵𝐵 = 9.274 ⋅ 10−24 𝐽𝐽⁄𝑇𝑇
𝑁𝑁𝐴𝐴
= 6.0225
⋅ 1023 𝑚𝑚𝑚𝑚𝑚𝑚 −1
12
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BFH/MNG, Dr. F. L&ouml;wenthal Version 4.2
Version
updated
2.3
- Formatting, some minor corrections
2.4
2.5
2.6
- update: mechanics formula for air
resistance
2.7
Added: formulas for physics of atoms
Added: stern – triangle – formula
minor corrections
2.8
Minor corrections
3.0
- added: some formulas for oscillations
3.2
- added: some formulas of atom physics
- added: some formulas of electronics
22.2.18
3.3
- added: some formulas standing waves
11.6.18
4.0
- Change to english
- minor corrections
4.1