  We use a Monte Carlo simulation to calculate the spectra of mildly
relativistic thermal plasmas in pair balance. We use the exact integral
expression for the electron-positron thermal annihilation spectrum, and provide
accurate expressions for the Gaunt factors of electron-ion, electron-electron,
and electron-positron thermal bremsstrahlung in the transrelativistic
temperature regime. The particles are assumed to be uniformly distributed
throughout a sphere, and the pair opacity is self-consistently calculated from
the energy and angular distribution of scattered photons. The resultant photon
spectra are compared with the nonrelativistic diffusion treatment of Sunyaev
and Titarchuk, the bridging formulas of Zdziarski, and the relativistic
corrections proposed by Titarchuk. We calculate allowed pair-balanced states of
thermal plasmas with no pair escape which include bremsstrahlung and internal
soft photons. The results are presented in the spectral index/compactness
plane, and can be directly compared with observations of spectra from AGNs and
Galactic black hole candidates. By comparing with X-ray spectral indices of
Seyfert AGNs and compactnesses inferred from X-ray variability data, we find
that the allowed solutions for pair equilibrium plasma imply that the
temperatures of Seyfert galaxies are $\lesssim 300$ keV. This prediction can be
tested with more sensitive gamma-ray observations of Seyfert galaxies. We find
that if the X-ray variability time scale gives an accurate measure of the
compactness, pair-dominated solutions are inconsistent with the data.
