Electron concentration in an intrinsic semiconductor at room temperature
Thermal voltage (\$\sim 25\,meV\$) is the average kinetic energy of particles in gas. It's not the total energy of each and every electron.

The actual energy distribution among the electrons is described by Fermi energy and temperature via Fermi Dirac distribution. At zero temperature, all electrons have energy less than Fermi energy. At non-zero temperature, some electrons have energy greater than Fermi energy. 

So at any any temperature, the probability that an energy level is occupied is given by Fermi Dirac distribution, \$f(E)\$. The density of available states in an energy interval \$dE\$ at any energy is \$D(E)dE\$. Where \$D(E)\$ is the electron density of states function. Then the total electron concentration can be calculated by,

$$n=\int\limits_{E_C}^{\infty}f(E)D(E)dE$$

\$E_C\$ is the bottom of conduction band and \$D(E)\propto\sqrt{E}\$. Few approximations are used to do this integration. After integration,
$$n=\sqrt{N_CN_V}\exp\left(-\frac{E_g}{2k_BT}\right)$$

where \$N_C\$ and \$N_V\$ are material and temperature dependent parameters called as the effective density of states and \$E_g\$ is the bandgap of the material.

For silicon at room temperature, the bandgap is \$E_g=1.1\,eV\$ and the effective density of states values are in the order of \$10^{19}\,cm^{-3}\$. This will result in an intrinsic carrier concentration \$n_i\sim 10^{10}\$. The exact values can be obtained from books.