Measuring Entropy Flashcards
What is the equation for the change in entropy ΔS?
ΔS = integral from T1 to T2 of Cv/T dT, where du = Cv dT and u is the internal energy
What is Gibbs’ entropy formula?
S = -k(B)sum over i of Piln(Pi), where Pi is the probability of being in state i
What is the equation for S in an isolated system in equilibrium?
S = -k(B)*sum from i = 1 to Ω of 1/Ω ln(1/Ω) = k(B)lnΩ
What is the equation for thr Boltzmann distribution?
P(x) = exp(-βαx^2)/(integral from -inf to inf of exp(-βαx^2) dx)
What is the equipartition theorem equation?
If a system has n independent quadratic modes, then = 1/2 k(B)T + 1/2k(B)T + ….. = n/2 *k(B)T
What is the equipartition theorem in words?
For a system in contact with a reservoir at temperature T, each independent quadratic energy mode contributes 1/2k(B)T to the average energy.
What is the equation for Cv, the heat capacity, in terms of u and E?
Cv = du/dT at const V = d/dT at const V = n/2 *k(B)
What is the first example of the equipartition theorem?
Translational motion of monatomic gas: E = 1/2 mv(x)^2 + 1/2 mv(y)^2 + 1/2 mv(z)^2 = 3 DoFs, so = 3/2*k(B)T -> Cv = 3/2 k(B) per atom = 3/2 N(A)k(B) per mole
What is the second example of the equipartition theorem?
Mass on a spring: E = 1/2kx^2 = 1/2mx’^2 -> 2 DoFs, so = k(B)*T
What is the third example of the equipartition theorem?
2 masses connected by a spring: E = 1/2k(|r1-r2|)^2 + 1/2μ(r1’-r2’)^2, so = k(B)*T
What is the fourth example of the equipartition theorem?
Lattice vibrations (a 3D arrangement of masses and spring - for a cubic lattice -> 3 springs per ion) therefore = 3k(B)T, Cv = 3k(B) per ion = 3N(A)k(B) per mole
What is the fifth example of the equipartition theorem?
Diatomic molecules with translational motion = 3/2 *k(B)T, vibrational motion = k(B)T and rotational motion Erot = 1/2 L1^2/I1 + 1/2L2^2/I2 -> = k(B)T
What is the total for a diatomic molecule?
= + + = 7/2 *k(B)T, so Cv = 7/2 *k(B)
