Question 1. Consider the combustion of methane which can be written as the following chemical reaction
CH4(g) + 2O2(g) → CO2(g) +2H2O(g)
The following data can be used to describe these molecules.
|
molecule
|
σ
|
Θ°rot(oK)
|
|
Θvib(oK)
|
Do (kcal/mol)
|
|
CO2
|
2
|
0.561
|
|
|
3360
|
954(2)
|
1890
|
|
381.5
|
|
H2O
|
2
|
40.1
|
20.9
|
13.4
|
5360
|
5160
|
2290
|
|
219.3
|
|
CH4
|
12
|
7.54
|
7.54
|
7.54
|
4170
|
2180(2)
|
4320(3)
|
1870(3)
|
392.1
|
|
O2
|
2
|
2.07
|
|
|
2256
|
|
|
|
118.0
|
where (2) indicates a two-fold degeneracy of that vir )ational temperature.
(a) Assuming that each component of this system is an ideal gas compute the equilibrium constant for the above reaction at T=300 K.
(b) gispts The standard free energy change for combustion of methane is reported as -800.8 kJ/mol. Provide both qualitative and quantitative comparisons between this value and the value computed in part (a). Is there a quantitative discrepancy? If so,
why?
Question 2. Consider a lattice of identical non-interacting two-state spins under an applied magnetic field H. The energy of the system is given as:
E = -μ∑i=1N si.H
where μ is the magnetic moment of each spin and si = ±1.
(a) With fixed N, V, T (and H) compute the molar energy, entropy and heat capacity of the system.
(b) How does increasing CHI change the heat capacity of the system?
Question 3. Consider the two-dimensional Ising model at a constant temperature 7' and external magnetic field H ≠ 0. The energy of this system is given as:
E= -μ∑i=1N si.H - j∑i.ji si.sj,
where the second sum is over nearest neighbor pairs i and j.
(a) What is the relative probability (P2/P1) of the following microstates in this ensemble? The gray squares are spin up and the white are spin down.
(b) Consider a simulation of this system using single site spin flips as trial moves. Write the Metropolis criteon for accepting a trial move where spin si is flipped as a function of the neighboring spins. Demonstrate that this scheme satisfies detailed balance.
(c) A code is provided that will run a simulation of this system with H = 0. Edit the code to study a H 0 sys-tem. This requires editing the code in two locations: 1) the subroutine TotalEnergy. 2) the line that calculates the variable deltaE for a trial move. The external mag¬netic field is found in the input file and is stored in the variable HO. Write out the two lines of edited code. [PUNT: The energy of a 20x20 completely aligned configuration with H = 1 J/ja is E = -1200.0 J.1
(d) Consider a 20 x 20 lattice at T = 2.01 and T = 3.01 with -0.2 J/ p, < H < 0.2 J/p. in 0.05 Jliz steps. Use your code to calculate (BIN) and (M/N) and plot the results as a function of external magnetic field for the two temperatures. It is sufficient to leave Nlter=107 and deltaWrite=105. Discuss how these plots are consistent with the fact that 2J < Tc < 3J.