Heat Expansion

ΔLL=αΔT,ΔVV=βΔT,β=3α\frac{\Delta L}{L} = \alpha \Delta T, \quad \frac{\Delta V}{V} = \beta \Delta T, \quad \beta =3\alpha

Heat

Q=mcΔT,ls=Qsm,la˚=Qa˚mQ = mc \Delta T, \quad l_s = \frac{Q_s}{m}, \quad l_\textup{å} = \frac{Q_\textup{å}}{m}

Fluid Pressure

ptot=pfluid+pair=ρgh+pairp_{tot} = p_\textup{fluid} + p_\textup{air} = \rho gh + p_\textup{air}

Ideal Gas Law

pV=NkTorpV=nRTwheren=mtotM=NNAandR=kNA\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}

Gas Density and Particle Density

ρ=mtotV=pMRT,no=NV=pkT\rho = \frac{m_{tot}}{V} = \frac{pM}{RT}, \quad n_o = \frac{N}{V} = \frac{p}{kT}

Barometric Height Formula

p=p0eρ0gh/p0,h=p0ρ0glnp0pp = p_0e^{-\rho_0gh/p_0}, \quad h = \frac{p_0}{\rho_0g}\ln{\frac{p_0}{p}}

Relative Moisture

RM=pwaterpsaturationR_{M} = \frac{p_\textup{water}}{p_\textup{saturation}}

Van der Waal's Equation

(p+an2V2)(Vnb)=nRT\left(p + a\frac{n^2}{V^2}\right)(V - nb) = nRT

Critical Point

Vk=3nb,Tk=8a27Rb,pk=a27b2V_k = 3nb, \quad T_k = \frac{8a}{27Rb}, \quad p_k = \frac{a}{27b^2}

Molecule Radius

r=(3b16πNA)1/3r = \left(\frac{3b}{16\pi N_A}\right)^{1/3}

Vapor Pressure Curve

p=AeMla˚/(RT)p = Ae^{-Ml_\textup{å}/(RT)}

Reynolds Number

Re=ρvdη,Re<2300  laminarRe = \frac{\rho vd}{\eta}, \quad Re < 2300 \; \textup{laminar}

Volume Flow

Φ=dVdt=A1v1=A2v2\Phi = \frac{\textup{d}V}{\textup{d}t} = A_1v_1 = A_2v_2

Bernoullis Equation

p1+ρv122+ρgy1=p2+ρv222+ρgy2p_1 + \frac{\rho v_1^2}{2} + \rho gy_1 = p_2 + \frac{\rho v_2^2}{2} + \rho gy_2

Poiseuilles Law

Φ=πR48η(p1p2)L\Phi = \frac{\pi R^4}{8\eta}\frac{(p_1 - p_2)}{L}

Pressure (Microscopic)

p=23nomen2v2=23noWkinenp = \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}

Temperature (Microscopic)

Wkinen=32kT\left\langle W_\textup{kin} \right\rangle_\textup{en} = \frac{3}{2}kT

Inner Energy (change)

ΔU=f2NkΔT=f2nRΔT\Delta U = \frac{f}{2}Nk\Delta T = \frac{f}{2}nR\Delta T

First Theorem

Q=ΔU+WwithW=12pdVQ = \Delta U + W \quad \textup{with} \quad W = \int_1^2pdV

Isokor

W0W \equiv 0

Isobar

W=p(V2V1)W = p\left(V_2 - V_1\right)

Isotherm

W=nRTln(V2V1)W = nRT\ln\left(\frac{V_2}{V_1}\right)

Adiabat

W=ΔUW = -\Delta U

Molar Heat Capacity

C=Mc,CV=f2R,Cp=CV+RC = Mc, \quad C_V = \frac{f}{2}R, \quad C_p = C_V + R

Adiabat(Poissons Equations)

T1V1(γ1)=T2V2(γ1)p1V1γ=p2V2γ\begin{gathered} T_1V_1^{(\gamma - 1)} = T_2V_2^{(\gamma - 1)} \\ p_1V_1^\gamma = p_2V_2^\gamma \end{gathered}

Quotient

γCpCV=cpcV=1+2f\gamma \equiv \frac{C_p}{C_V} = \frac{c_p}{c_V} = 1 + \frac{2}{f}

Circuit Process

Qnet=Wnet=pdVQ_\textup{net} = W_\textup{net} = \oint pdV

Efficiency

η=WnetQin=QinQoutQin=1QoutQin\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}}

Ideal Efficiency

η=TwarmTcoldTwarm=1TcoldTwarm\eta = \frac{T_\textup{warm} - T_\textup{cold}}{T_\textup{warm}} = 1 - \frac{T_\textup{cold}}{T_\textup{warm}}

Cold Factor (def. and Ideal)

KfQinWnet,Kf=TcoldTwarmTcoldK_f \equiv \frac{Q_\textup{in}}{|W_\textup{net}|}, \quad K_f = \frac{T_\textup{cold}}{T_\textup{warm} - T_\textup{cold}}

Heat Factor (def. and Ideal)

VfQoutWnet,Vf=TwarmTwarmTcoldV_f \equiv \frac{Q_\textup{out}}{|W_\textup{net}|}, \quad V_f = \frac{T_\textup{warm}}{T_\textup{warm} - T_\textup{cold}}

Gauss Distribution

f(vz)=men2πkTemenvz2/(2kT)f(v_z) = \sqrt{\frac{m_\textup{en}}{2\pi kT}}e^{-m_\textup{en}v_z^2/(2kT)}

Maxwell--Boltzmann Distribution

f(v)=4πv2(men2πkT)3/2emenv2/(2kT)f(v) = 4\pi v^2 \left(\frac{m_\textup{en}}{2\pi kT}\right)^{3/2}e^{-m_\textup{en}v^2/(2kT)}

Averages

v=8kTπmen,v=2vxWkin=menv22=men2v2=32kT\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}

Collision Number (per second and square meter)

n=no4vn^* = \frac{n_o}{4}\langle v \rangle

Mean Free Path

l=1noπd22l = \frac{1}{n_o\pi d^2 \sqrt{2}}

Heat Conduction (General and Rod)

P=λAdTdx,P=λAT1T2LP = -\lambda A \frac{\textup{d}T}{\textup{d}x}, \quad P = \lambda A \frac{T_1 - T_2}{L}

Heat Transfer

P=αAΔTP = \alpha A \Delta T

Radiation

Pideal=σAT4,Preal=ePidealP_\textup{ideal} = \sigma A T^4, \quad P_\textup{real} = eP_\textup{ideal}