Question

At room temperature, show that $\mathit{K}\mathit{B}\mathbf{}\mathit{T}\mathbf{\approx }\mathbf{1}\mathbf{/}\mathbf{}\mathbf{40}\mathbf{}\mathit{e}\mathit{V}$. It is useful to memorize this result, because it tells a lot about what phenomena are likely to occur at room temperature.

Short answer

The result can be proved using the energy equation

Step-by-step solution

Identification of given data

• Boltzmann’s constant is $k_b=8.62\times {10}^{-5}\frac{\text{eV}}{\text{K}}$.
• The room temperature is $T=20°\text{C}=293\text{K}$.

Concept of energy

The ability to do tasks is known as energy. Depending on the form it takes, it may be either potential or kinetic or thermal or electrical or chemical, or nuclear.

The energy equation is given by,

$E=k_{b}T$…… (i)

Here $k_b$is Boltzmann’s constant.

$T$is the room temperature.

Evaluation of energy equation

The energy equation can be proved using equation (i)

$$\begin{gathered} E=\left(8.62\times {10}^{-5}\frac{e^V}{K}\right)\times \left(293K\right) \\ E=\left(8.62\times {10}^{-5}\times 293\right).\left(\frac{1e^V}{1K}\times 1K\right) \\ E=0.025e^V \\ E\approx \frac{1}{40}{e}^V \end{gathered}$$

Thus, the temperature equation is proved.