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Thermodynamics
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Svenska
Heat Expansion
Δ
L
L
=
α
Δ
T
,
Δ
V
V
=
β
Δ
T
,
β
=
3
α
\frac{\Delta L}{L} = \alpha \Delta T, \quad \frac{\Delta V}{V} = \beta \Delta T, \quad \beta =3\alpha
L
Δ
L
=
α
Δ
T
,
V
Δ
V
=
β
Δ
T
,
β
=
3
α
Heat
Q
=
m
c
Δ
T
,
l
s
=
Q
s
m
,
l
a
˚
=
Q
a
˚
m
Q = mc \Delta T, \quad l_s = \frac{Q_s}{m}, \quad l_\textup{å} = \frac{Q_\textup{å}}{m}
Q
=
m
c
Δ
T
,
l
s
=
m
Q
s
,
l
a
˚
=
m
Q
a
˚
Fluid Pressure
p
t
o
t
=
p
fluid
+
p
air
=
ρ
g
h
+
p
air
p_{tot} = p_\textup{fluid} + p_\textup{air} = \rho gh + p_\textup{air}
p
t
o
t
=
p
fluid
+
p
air
=
ρ
g
h
+
p
air
Ideal Gas Law
p
V
=
N
k
T
or
p
V
=
n
R
T
where
n
=
m
t
o
t
M
=
N
N
A
and
R
=
k
N
A
\begin{gathered} pV = NkT \quad \textup{or} \quad pV = nRT \\ \textup{where} \quad n = \frac{m_{tot}}{M} = \frac{N}{N_A} \quad \textup{and} \quad R = kN_A \end{gathered}
p
V
=
N
k
T
or
p
V
=
n
RT
where
n
=
M
m
t
o
t
=
N
A
N
and
R
=
k
N
A
Gas Density and Particle Density
ρ
=
m
t
o
t
V
=
p
M
R
T
,
n
o
=
N
V
=
p
k
T
\rho = \frac{m_{tot}}{V} = \frac{pM}{RT}, \quad n_o = \frac{N}{V} = \frac{p}{kT}
ρ
=
V
m
t
o
t
=
RT
pM
,
n
o
=
V
N
=
k
T
p
Barometric Height Formula
p
=
p
0
e
−
ρ
0
g
h
/
p
0
,
h
=
p
0
ρ
0
g
ln
p
0
p
p = p_0e^{-\rho_0gh/p_0}, \quad h = \frac{p_0}{\rho_0g}\ln{\frac{p_0}{p}}
p
=
p
0
e
−
ρ
0
g
h
/
p
0
,
h
=
ρ
0
g
p
0
ln
p
p
0
Relative Moisture
R
M
=
p
water
p
saturation
R_{M} = \frac{p_\textup{water}}{p_\textup{saturation}}
R
M
=
p
saturation
p
water
Van der Waal's Equation
(
p
+
a
n
2
V
2
)
(
V
−
n
b
)
=
n
R
T
\left(p + a\frac{n^2}{V^2}\right)(V - nb) = nRT
(
p
+
a
V
2
n
2
)
(
V
−
nb
)
=
n
RT
Molecule Radius
r
=
(
3
b
16
π
N
A
)
1
/
3
r = \left(\frac{3b}{16\pi N_A}\right)^{1/3}
r
=
(
16
π
N
A
3
b
)
1/3
Bernoullis Equation
p
1
+
ρ
v
1
2
2
+
ρ
g
y
1
=
p
2
+
ρ
v
2
2
2
+
ρ
g
y
2
p_1 + \frac{\rho v_1^2}{2} + \rho gy_1 = p_2 + \frac{\rho v_2^2}{2} + \rho gy_2
p
1
+
2
ρ
v
1
2
+
ρ
g
y
1
=
p
2
+
2
ρ
v
2
2
+
ρ
g
y
2
Pressure (Microscopic)
p
=
2
3
n
o
m
en
2
⟨
v
2
⟩
=
2
3
n
o
⟨
W
kin
⟩
en
p = \frac{2}{3}n_o\frac{m_\textup{en}}{2}\langle v^2 \rangle = \frac{2}{3}n_o\left\langle W_\textup{kin}\right\rangle_\textup{en}
p
=
3
2
n
o
2
m
en
⟨
v
2
⟩
=
3
2
n
o
⟨
W
kin
⟩
en
Temperature (Microscopic)
⟨
W
kin
⟩
en
=
3
2
k
T
\left\langle W_\textup{kin} \right\rangle_\textup{en} = \frac{3}{2}kT
⟨
W
kin
⟩
en
=
2
3
k
T
Inner Energy (change)
Δ
U
=
f
2
N
k
Δ
T
=
f
2
n
R
Δ
T
\Delta U = \frac{f}{2}Nk\Delta T = \frac{f}{2}nR\Delta T
Δ
U
=
2
f
N
k
Δ
T
=
2
f
n
R
Δ
T
First Theorem
Q
=
Δ
U
+
W
with
W
=
∫
1
2
p
d
V
Q = \Delta U + W \quad \textup{with} \quad W = \int_1^2pdV
Q
=
Δ
U
+
W
with
W
=
∫
1
2
p
d
V
Isokor
W
≡
0
W \equiv 0
W
≡
0
Isobar
W
=
p
(
V
2
−
V
1
)
W = p\left(V_2 - V_1\right)
W
=
p
(
V
2
−
V
1
)
Isotherm
W
=
n
R
T
ln
(
V
2
V
1
)
W = nRT\ln\left(\frac{V_2}{V_1}\right)
W
=
n
RT
ln
(
V
1
V
2
)
Adiabat
W
=
−
Δ
U
W = -\Delta U
W
=
−
Δ
U
Molar Heat Capacity
C
=
M
c
,
C
V
=
f
2
R
,
C
p
=
C
V
+
R
C = Mc, \quad C_V = \frac{f}{2}R, \quad C_p = C_V + R
C
=
M
c
,
C
V
=
2
f
R
,
C
p
=
C
V
+
R
Adiabat(Poissons Equations)
T
1
V
1
(
γ
−
1
)
=
T
2
V
2
(
γ
−
1
)
p
1
V
1
γ
=
p
2
V
2
γ
\begin{gathered} T_1V_1^{(\gamma - 1)} = T_2V_2^{(\gamma - 1)} \\ p_1V_1^\gamma = p_2V_2^\gamma \end{gathered}
T
1
V
1
(
γ
−
1
)
=
T
2
V
2
(
γ
−
1
)
p
1
V
1
γ
=
p
2
V
2
γ
Quotient
γ
≡
C
p
C
V
=
c
p
c
V
=
1
+
2
f
\gamma \equiv \frac{C_p}{C_V} = \frac{c_p}{c_V} = 1 + \frac{2}{f}
γ
≡
C
V
C
p
=
c
V
c
p
=
1
+
f
2
Circuit Process
Q
net
=
W
net
=
∮
p
d
V
Q_\textup{net} = W_\textup{net} = \oint pdV
Q
net
=
W
net
=
∮
p
d
V
Efficiency
η
=
W
net
Q
in
=
Q
in
−
∣
Q
out
∣
Q
in
=
1
−
∣
Q
out
∣
Q
in
\eta = \frac{W_\textup{net}}{Q_\textup{in}} = \frac{Q_\textup{in} -|Q_\textup{out}|}{Q_\textup{in}}= 1 - \frac{|Q_\textup{out}|}{Q_\textup{in}}
η
=
Q
in
W
net
=
Q
in
Q
in
−
∣
Q
out
∣
=
1
−
Q
in
∣
Q
out
∣
Ideal Efficiency
η
=
T
warm
−
T
cold
T
warm
=
1
−
T
cold
T
warm
\eta = \frac{T_\textup{warm} - T_\textup{cold}}{T_\textup{warm}} = 1 - \frac{T_\textup{cold}}{T_\textup{warm}}
η
=
T
warm
T
warm
−
T
cold
=
1
−
T
warm
T
cold
Cold Factor (def. and Ideal)
K
f
≡
Q
in
∣
W
net
∣
,
K
f
=
T
cold
T
warm
−
T
cold
K_f \equiv \frac{Q_\textup{in}}{|W_\textup{net}|}, \quad K_f = \frac{T_\textup{cold}}{T_\textup{warm} - T_\textup{cold}}
K
f
≡
∣
W
net
∣
Q
in
,
K
f
=
T
warm
−
T
cold
T
cold
Heat Factor (def. and Ideal)
V
f
≡
Q
out
∣
W
net
∣
,
V
f
=
T
warm
T
warm
−
T
cold
V_f \equiv \frac{Q_\textup{out}}{|W_\textup{net}|}, \quad V_f = \frac{T_\textup{warm}}{T_\textup{warm} - T_\textup{cold}}
V
f
≡
∣
W
net
∣
Q
out
,
V
f
=
T
warm
−
T
cold
T
warm
Gauss Distribution
f
(
v
z
)
=
m
en
2
π
k
T
e
−
m
en
v
z
2
/
(
2
k
T
)
f(v_z) = \sqrt{\frac{m_\textup{en}}{2\pi kT}}e^{-m_\textup{en}v_z^2/(2kT)}
f
(
v
z
)
=
2
πk
T
m
en
e
−
m
en
v
z
2
/
(
2
k
T
)
Maxwell--Boltzmann Distribution
f
(
v
)
=
4
π
v
2
(
m
en
2
π
k
T
)
3
/
2
e
−
m
en
v
2
/
(
2
k
T
)
f(v) = 4\pi v^2 \left(\frac{m_\textup{en}}{2\pi kT}\right)^{3/2}e^{-m_\textup{en}v^2/(2kT)}
f
(
v
)
=
4
π
v
2
(
2
πk
T
m
en
)
3/2
e
−
m
en
v
2
/
(
2
k
T
)
Average energy
⟨
W
kin
⟩
=
⟨
m
en
v
2
2
⟩
=
m
en
2
⟨
v
2
⟩
=
3
2
k
T
\begin{gathered} %\langle v \rangle = \sqrt{\frac{8kT}{\pi m_\textup{en}}}, \quad \langle %v\rangle = 2\langle |v_x| \rangle \\ \langle W_\textup{kin} \rangle = \left\langle \frac{m_\textup{en}v^2}{2}\right\rangle = \frac{m_\textup{en}}{2}\langle v^2 \rangle = \frac{3}{2}kT \end{gathered}
⟨
W
kin
⟩
=
⟨
2
m
en
v
2
⟩
=
2
m
en
⟨
v
2
⟩
=
2
3
k
T
Maxwell-Boltzmann velocities
v
r
m
s
=
⟨
v
2
⟩
=
3
k
T
m
v_{rms} = \sqrt{ \langle v^2 \rangle } = \sqrt{ \frac{3kT}{m} }
v
r
m
s
=
⟨
v
2
⟩
=
m
3
k
T
v
m
a
x
=
2
k
T
m
v_{max} = \sqrt{ \frac{2kT}{m} }
v
ma
x
=
m
2
k
T
⟨
v
⟩
=
∫
0
∞
f
M
B
⋅
v
⋅
d
v
=
8
k
T
π
m
\langle v \rangle = \int^\infty_0 f_{MB}\cdot v \cdot dv = \sqrt{ \frac{8kT}{\pi m} }
⟨
v
⟩
=
∫
0
∞
f
MB
⋅
v
⋅
d
v
=
πm
8
k
T
Mean Free Path
l
=
1
n
o
π
d
2
2
l = \frac{1}{n_o\pi d^2 \sqrt{2}}
l
=
n
o
π
d
2
2
1
Heat Conduction
P
=
k
⋅
A
⋅
∣
d
T
d
x
∣
,
R
=
Δ
x
k
A
P = k \cdot A \cdot \left| \frac{\textup{d}T}{\textup{d}x} \right|, R = \frac{\Delta x}{kA}
P
=
k
⋅
A
⋅
∣
∣
d
x
d
T
∣
∣
,
R
=
k
A
Δ
x
Thermal resistance
Δ
T
=
R
t
h
e
r
m
⋅
P
i
f
R
t
h
e
r
m
=
Δ
x
k
A
\Delta T = R_{therm} \cdot P \;\; if \;\; R_{therm} = \frac{\Delta x}{kA}
Δ
T
=
R
t
h
er
m
⋅
P
i
f
R
t
h
er
m
=
k
A
Δ
x
Heat Transfer
P
=
α
⋅
A
⋅
∣
Δ
T
∣
,
R
=
1
α
A
P = \alpha \cdot A \cdot \left| \Delta T \right|, \; R=\frac{1}{\alpha A}
P
=
α
⋅
A
⋅
∣
Δ
T
∣
,
R
=
α
A
1
Stefan-Boltzmann's law
P
=
A
σ
(
T
4
−
T
0
4
)
,
σ
=
5.67
⋅
1
0
−
8
W
/
m
2
K
4
P = A \sigma \left(T^4-T^4_0\right), \sigma = 5.67 \cdot 10^{-8} \textup{W}/\textup{m}^2 \textup{K}^4
P
=
A
σ
(
T
4
−
T
0
4
)
,
σ
=
5.67
⋅
1
0
−
8
W
/
m
2
K
4
P
r
e
a
l
=
ε
⋅
P
i
d
e
a
l
P_{real} = \varepsilon \cdot P_{ideal}
P
re
a
l
=
ε
⋅
P
i
d
e
a
l
Wien's law
λ
m
a
x
⋅
T
=
2.898
⋅
1
0
−
3
K
⋅
m
\lambda_{max}\cdot T = 2.898 \cdot 10^{-3} \textup{K} \cdot \textup{m}
λ
ma
x
⋅
T
=
2.898
⋅
1
0
−
3
K
⋅
m
Planck's law
ρ
(
f
)
d
f
=
8
π
h
f
3
c
3
⋅
1
e
h
f
/
k
T
−
1
d
f
\rho (f)df = \frac{8\pi hf^3}{c^3} \cdot \frac{1}{e^{hf/kT}-1} df
ρ
(
f
)
df
=
c
3
8
πh
f
3
⋅
e
h
f
/
k
T
−
1
1
df
The solar constant
Average value
≈
1380
W
/
m
2
\textup{Average value} \approx 1380 \; W/m^2
Average value
≈
1380
W
/
m
2