Equation of continuity and Bernoulli's equation
v1A1 = v2A2, P1
+ ρgh1 + ½ρv12 = P2 + ρgh2
v2 = 4v1, h2 = h1, P2 - P1 = ½ρ(v22 - v12) = ½(1000 kg/m3)(15 m2/s2) = 7500 N/m2.
Maxwell speed distribution
dn(v)/dv = 0, 2Avexp(-mv2/(2kT)) - (mv/(kT))Av2exp(-mv2/(2kT)) = 0, v2 = 2kT/m.
The ideal gas law, P = nRT/V
W = ∫V1V2 PdV = ∫V1V2 (RT/V) dV = RT ln(V2/V1).
Latent heat of fusion, ΔQ = m*Lf
Heat of fusion = power *time/mass = (100 V)*(10 A)*(1020 s)/(3 kg) ~ 3.4*105 J/kg.
Stefan-Boltzmann law, radiated power = emissivity * σ * T4 * Area
Energy received is equal to energy emitted per unit time.
constant_1*πr2/(4πR2) = constant_2*4πr2*T4.
T is proportional to 1/R½.
Specific heat, ΔQ = cmΔT
ΔQ = ΔU at constant volume. U =
(3/2)kT, ΔQ/(mΔT) = (3/2)k/m.
Specific heat per atom: (3/2)k.
The PV diagram, dW = PdV
W = ½(VC - VA)(PB - PC) = area under the curve.
dS = dQ/T = c m dT/T, ΔS = c m ln(T2/T1).
dS = -dQ/T1 + dQ/T2.
The entropy of the gas in process (3) decreases and therefore the entropy of the reservoir (4) increases, since the total entropy for a reversible process does not change. For an isothermal process the internal energy of a gas is only a function of its temperature, ΔU = 0. For the isothermal compression of an ideal gas we have for the work done by the gas.
W = ∫V1V2 PdV = ∫V1V2 (nRT/V) dV = nRT ln(V2/V1).
W is negative if V2 < V1. Since ΔU = 0, the heat transferred to the gas is ΔQ = W. Therefore for one mole of gas ΔQ = RT ln(V2/V1) and ΔS = R ln(V2/V1).