[SOLVED] Phys 362 statistical and thermal physics: homework 11

A monoatomic ideal gas interacts with a reservoir, whose
temperature is T. The gas is allowed to expand, interacting only with the reservoir while it does this. Let Wby be
the work done by the gas during this process.The diagram illustrates two processes for the gas. Process 1 is an
isothermal expansion at the temperature of the reservoir,
represented by the curved line. Process 2 is a constant
pressure expansion followed by a constant volume cooling.
Pi
Vi
P
Va) How does the work done by gas in the process 2 compare (larger, smaller, same) to that
done by the gas in process 1? Explain your answer.
b) How does the change in Helmholtz free energy for process 2 compare (larger, smaller,
same) to that for process 1? Explain your answer.c) Is process 2 attainable given that the gas only interacts with the reservoir and its initial
and final temperatures are the same as that of the reservoir? Explain your answer.When methane (CH4) undergoes combustion according to
CH4(gas) + 2O2(gas) CO2(gas) + 2H2O(liquid),
at 298 K, the change in enthalpy is 890 kJ mol1.The entropy of the constituents is 186.3 JK1mol1
(CH4(gas)), 205.1 JK1mol1 (O2(gas)), 213.74 JK1mol1 (CO2(gas)) and 69.91 JK1mol1
(H2O(liquid)). Suppose that 1.00 mol of methane undergoes combustion and that this occurs
in such a way that the pressure at the beginning and end of the process is the same and that
the temperatures are also the same.a) Show that
H = E + P V
and, assuming that the gases involved are ideal, show that this gives
H = E + kT N.Determine the change in energy for the process. Note that the liquid that results from
the reaction no longer plays a role in the gas properties of the system.b) Considering the molecules involved in the reaction as the system, and S as the total
entropy of the system is S ! 0? Does this suggest that the process is possible or not?c) Determine the change in the Gibbs free energy and the maximum work that can be done
by the gas in this process. Is this smaller than or larger than the change in energy, E,
during the process? Provide a physical reason for your answer.d) If one generated energy by burning methane, what would be the minimum rate at which
methane would have to be burnt (in terms of mols/hour) in order to produce 1.0 kW of
power?Consider a system that is in contact with a reservoir, which has temperature T and pressure
P.
a) Suppose that the system undergoes a process in which it is initially in equilibrium with
the reservoir (i.e. has the same temperature and pressure as the reservoir) and is finally
in equilibrium with the reservoir.Use the argument involving the first and second law
to show that in this process
G P V ! W
where W is the work done on the system and the Gibbs free energy is
G = E T S + PV.b) Consider a quasistatic process in which the pressure of the system is constant and equal
to that of the reservoir. Assume that no non-mechanical work is done on the system.
Show that
G ! 0.c) Show that
G = H T S
d) Liquid water and carbon dioxide react as:
H2O(liquid) + CO2(gas) H2CO3(liquid).
For these the enthalpy (of formation from basic constituents) is
H(H2O(liquid)) = 285.83 kJmol1
H(CO2(gas)) = 393.51 kJmol1
H(H2CO3(liquid)) = 699.65 kJmol1
and the entropies are
S(H2O(liquid)) = 69.91 JK1mol1
S(CO2(gas)) = 213.74 JK1mol1
S(H2CO3(liquid)) = 187.4 JK1mol1Determine G for this reaction at 298 K. Will this occur spontaneously without any external non-mechanical work? If not, at what temperature could it occur spontaneously?