  Motivated by the recent discoveries of some compounds such as the
$\mathrm{Bi}_{2}\mathrm{Fe}_{4}\mathrm{O}_{9}$ which crystallizes in an
orthorhombic crystal structure with the $\mathrm{Fe}^{3+}$ ions, and iron-based
oxyfluoride ${\rm Bi}_{4}{\rm Fe}_{5}{\rm O}_{13}{\rm F}$ compounds following
the pattern of Cairo pentagonal structure, among some other compounds. We
propose a model for one stripe of the Cairo pentagonal Ising-Heisenberg
lattice, one of the edges of a pentagon is different, and this edge will be
associated with a Heisenberg exchange interaction, while the Ising exchange
interactions will associate the other edges. We study the phase transition at
zero temperature, illustrating five phases: a ferromagnetic phase (FM), a dimer
antiferromagnetic (DAF), a plaquette antiferromagnetic (PAF), a typical
antiferromagnetic (AFM) and a peculiar frustrated phase (FRU) where two types
of frustrated states with the same energy coexist. To obtain the partition
function of this model, we use the transfer matrix approach and following the
eight vertex model notation. Using this result we discuss the specific heat,
internal energy and entropy as a function of the temperature, and we can
observe some unexpected behavior in the low-temperature limit, such as
anomalous double peak in specific heat due to the existence of three phase
(FRU, PAF(AFM) and FM) transitions occurring in a close region to each other.
Consequently, the low-lying energy thermal excitation generates this double
anomalous peak, and we also discuss the internal energy at the low temperature
limit, where this double peak curve occurs. Some properties of our result were
compared with two dimensional Cairo pentagonal lattices, as well as orthogonal
dimer plaquette Ising-Heisenberg chain.
