Question

The electrons that produce electric currents in a metal behave approximately like molecules of an ideal gas. The mass of an electron is \(m_{\mathrm{e}}=9.109 \cdot 10^{-31} \mathrm{~kg} .\) If the temperature of the metal is \(300.0 \mathrm{~K},\) what is the root-mean-square speed of the electrons?

Short answer

Answer: The root-mean-square speed of the electrons in the metal at a temperature of 300.0 K is approximately 1.16 x 10^6 m/s.

Step-by-step solution

Write down the given values

We are given the mass of an electron \(m_e=9.109 \times 10^{-31}\,\text{kg}\) and the temperature of the metal \(T=300.0\,\text{K}\).

Plug the values into the root-mean-square speed formula

Using the formula \(v_{rms}=\sqrt{\frac{3k_BT}{m}}\), we can plug the given values into the equation: $$v_{rms}=\sqrt{\frac{3(1.381\times 10^{-23}\,\text{J K}^{-1})(300.0\,\text{K})}{(9.109\times10^{-31}\,\text{kg})}}$$

Calculate the root-mean-square speed

Now, we just need to solve for \(v_{rms}\): $$v_{rms}=\sqrt{\frac{3(1.381\times 10^{-23}\,\text{J K}^{-1})(300.0\,\text{K})}{(9.109\times10^{-31}\,\text{kg})}}\approx1.16\times10^6\,\text{m/s}$$ So, the root-mean-square speed of the electrons in the metal at a temperature of \(300.0\,\text{K}\) is approximately \(1.16\times10^6\,\text{m/s}\).