References
211
sites occupied by O 2 molecules (N = 2). Let us choose the energy associated
with the unoccupied state as our zero of energy, and let the energy associated
with single occupancy of either site to be −0.55 eV, and let the energy for a
doubly occupied site be −1.3 eV (which means that the change in energy upon
binding the second O 2 molecule is −0.75 eV, rather than simply −0.55 eV).
To facilitate your calculations, write the energy 2 for double occupation as
2 = 2 1 + δ, with δ = −0.2. Calculate and plot the fraction of occupied sites
as a function of the effective partial pressure of O 2 in blood that is in contact
with a cell. Why might this behaviour for the fraction of sites occupied by O 2
be preferable (i.e., provide a more realistic representation of how haemoglobin
actually functions in our bodies) to the behaviour obtained using the single-site
model for haemoglobin?
34. The internal energy U is given in terms of the canonical partition function
Z(T , V , N) by Eq. (4.1.3b) and in terms of the isobaric–isothermal partition
function by Eq. (4.3.22), while the enthalpy H is given similarly in terms of the
canonical partition function by
H (T , V , N) = k B T
T
∂ ln Z
∂T
V ,N
+ V
∂ ln Z
∂V
T ,N
and in terms of the isobaric–isothermal partition function by Eq. (4.3.20). Of
course, the final expressions for U and H in terms of T , V , N or T , P , N
must be equivalent. Show the equality/equivalence of the expressions for U, H
obtained from the two partition functions in the case of an ideal gas.
References
1. P.T. Landsberg, J. Dunning-Davies, D. Pollard, Am. J. Phys. 62, 712 (1994)
2. W. Greiner, L. Neise, H. Stöcker, Thermodynamics and Statistical Mechanics (Springer, New
York, 1995), p. 138 (see also pp. 186–190)
211
sites occupied by O 2 molecules (N = 2). Let us choose the energy associated
with the unoccupied state as our zero of energy, and let the energy associated
with single occupancy of either site to be −0.55 eV, and let the energy for a
doubly occupied site be −1.3 eV (which means that the change in energy upon
binding the second O 2 molecule is −0.75 eV, rather than simply −0.55 eV).
To facilitate your calculations, write the energy 2 for double occupation as
2 = 2 1 + δ, with δ = −0.2. Calculate and plot the fraction of occupied sites
as a function of the effective partial pressure of O 2 in blood that is in contact
with a cell. Why might this behaviour for the fraction of sites occupied by O 2
be preferable (i.e., provide a more realistic representation of how haemoglobin
actually functions in our bodies) to the behaviour obtained using the single-site
model for haemoglobin?
34. The internal energy U is given in terms of the canonical partition function
Z(T , V , N) by Eq. (4.1.3b) and in terms of the isobaric–isothermal partition
function by Eq. (4.3.22), while the enthalpy H is given similarly in terms of the
canonical partition function by
H (T , V , N) = k B T
T
∂ ln Z
∂T
V ,N
+ V
∂ ln Z
∂V
T ,N
and in terms of the isobaric–isothermal partition function by Eq. (4.3.20). Of
course, the final expressions for U and H in terms of T , V , N or T , P , N
must be equivalent. Show the equality/equivalence of the expressions for U, H
obtained from the two partition functions in the case of an ideal gas.
References
1. P.T. Landsberg, J. Dunning-Davies, D. Pollard, Am. J. Phys. 62, 712 (1994)
2. W. Greiner, L. Neise, H. Stöcker, Thermodynamics and Statistical Mechanics (Springer, New
York, 1995), p. 138 (see also pp. 186–190)