With fixed n v t and h compute the molar energy entropy and

Question

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=1^N s_i.H

where μ is the magnetic moment of each spin and s_i = ±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=1^N s_i.H - j∑_i.jⁱ s_i.s_j,

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 < T_c < 3J.