Introduction to Biological Physics: Collection of physics equations for final exam kBT ≈ 1/40 eV ≈ 4.1 pN·nm = 4.1·10 –21 J; kB ≈ 1.38·10 –23 J/K; NA = 6.022·10 23 /mol; g = 9.81 m/s 2 ≈ 10 m/s 2 . Coulomb force: ! F = 1 4πεε 0 q 1 q 2 R 2 ˆ r , potential: U = 1 4πεε 0 q 1 q 2 R ; e = 1.6·10 –19 C; εo = 8.85·10 –12 As/Vm. Bjerrum length: l B = e 2 4πε 0 ε k B T ≈ 7 Å in water at T = 300 K. Born energy of an ion: U = q 2 8πε 0 ε r . dipole field: ! E = 3 ! μ ⋅ ˆ r ( ) ⋅ ˆ r − ! μ 4πε 0 ε r 3 ; interaction energy (IE) for static dipoles: U = − ! μ 1 ! μ 2 2πε 0 ε r 3 IE between charge and rotating dipole: U = − 1 4πε 0 ε ( ) 2 q 2 μ 2 6 k B T 1 r 4 ; polarizability (Bohr atom):α = 4πε 0 a 0 3 IE btw. charge and polarizable molecule: U = − α q 2 4πε 0 ε ( ) 2 1 r 4 ; Keesom energy: U = − 1 4πε 0 ε ( ) 2 μ 1 2 μ 2 2 3k B T 1 r 6 ; Debye energy: U = − 1 4πε 0 ε ( ) 2 μ 2 α r 6 ; London energy: U = − 3 24πε 0 ε ( ) 2 α 1 α 2 n 4 1 r 6 I 1 I 2 I 1 + I 2 ; Lennard-Jones: Ur () = 4ε rr 0 ( ) −12 − rr 0 ( ) −6 ⎡ ⎣ ⎤ ⎦ ; Gauss distribution: P( x ) = 1 2π σ ( ) ⋅ e − x− x 0 ( ) 2 2σ 2 statistical mean: f ( x ) = f x i ( ) i ∑ P( x i ) = f ( x )P( x )d x ∫ ; variance: var( x ) = x − x ( ) 2 = x 2 − x 2 ideal gas: E kin = 32 k B T ; speed distribution: P(v)dv = 4π v 2 2π k B Tm ( ) 3 e − mv 2 2 k B T dv ; v = 8 k B T π m generally: E = 12 k B T per degree of freedom; statistical weight/Boltzmann distribution: w i ∝ e −E i k B T partition function: Q = e −E i k B T i ∑ = g E e −Ek B T E ∑ thermodynamic energy/free energy: E = k B T 2 ∂ ∂T ln Q ; F = − k B T ln Q ; entropy: ∂F ∂T = − S end-to-end distances – freely jointed chain: R e = ! R N 2 = N l ; freely rotating chain: l eff = l ⋅ 1 + cos ϑ 1 − cos ϑ persistence length: l p = lim N →∞ ! r 1 l ! r n n ∑ ; Kuhn length: a = 2l p ; diffusion of spherical particle: D = k B T 6πηR end-to-end distance distribution: P N ( R)dR = 4π R 2 2π N l ( ) 3 e − R 2 2 Nl 2 dR ; radius of gyration: R G 2 = 1 6 R end-to-end 2 statistical entropy: S = k B ln W ; probability of knot formation: P N (0)dV = 1 2π N l ( ) 3 free energy of deformed polymer coil: F( x ) = k B T ⋅ x 2 2 Nl 2 ; resulting reactive force: f ( x ) = k B T ⋅ x Nl 2 entropy associated with statistical weight: S = k B ln W ; binomial distribution: P( N ) = 1 2 N N ! N 2 ( )! × N 2 ( )! Stirling's formula: ln N ! ≈ N ln N − N + 0.5ln 2π N ( ) ; displacement in n-dim. random walk: Δx 2 = 2nDt – 1 –