Question

The electrons in a metal that produce electric currents behave approximately as molecules of an ideal gas. The mass of an electron is \(m_{\mathrm{e}} \doteq 9.109 \cdot 10^{-31} \mathrm{~kg} .\) 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 is approximately \(1.079 \times 10^4~m/s\).

Step-by-step solution

Write down the given values

We have been given the mass of the electron (m) and the temperature (T) of the metal. Let's write them down: \(m_e = 9.109 \times 10^{-31}~kg\) \(T= 300.0~K\)

Find the Boltzmann constant k

The Boltzmann constant (k) is a physical constant that relates the average kinetic energy of particles in a gas with the temperature of the gas. Its value is: \(k = 1.381 \times 10^{-23} \frac{J}{K}\)

Use the formula of root-mean-square speed

We can now use the formula for the root-mean-square speed: \(v_{rms} = \sqrt{\frac{3kT}{m}}\) Plug in the given values: \(v_{rms} = \sqrt{\frac{3(1.381 \times 10^{-23} \frac{J}{K})(300.0 K)}{(9.109 \times 10^{-31} kg)}}\)

Calculate v_rms

Perform the calculations to find the root-mean-square speed of the electrons: \(v_{rms} = \sqrt{\frac{3(1.381 \times 10^{-23})(300.0)}{(9.109 \times 10^{-31})}}\) \(v_{rms} = \sqrt{1.164 \times 10^{8}}\) \(v_{rms} = 1.079 \times 10^4 m/s\) Therefore, the root-mean-square speed of the electrons in the metal is approximately \(1.079 \times 10^4~m/s\).

Key concepts

Boltzmann Constant

The Boltzmann constant (symbol: k or k_B) is a fundamental constant in physics that plays a key role in the kinetic theory of gases. It essentially bridges the gap between the macroscopic and microscopic worlds by connecting temperature, a macroscopic property, with the average kinetic energy of particles, a microscopic property.

Expressed in joules per kelvin (J/K), its value is approximately \(1.381 \times 10^{-23} J/K\). When studying the behavior of gases, this constant is used to calculate the average kinetic energy per particle with the formula \(\langle E_{\text{kin}} \rangle = \frac{3}{2} k_B T\), where \(T\) is the temperature in kelvins. In the context of root-mean-square speed for electrons, as in our exercise, the Boltzmann constant directly informs us of the energy associated with their thermal motion at a given temperature.

Electron Mass

Electrons are subatomic particles with a specific mass denoted by \(m_e\), which is essential for calculations in both atomic physics and solid state physics. The mass of an electron, \(9.109 \times 10^{-31} kg\), is a constant that allows us to compute various properties of electrons when coupled with other constants and equations.

Understanding electron mass is crucial when it comes to discussing the motion of electrons, such as their speed in a metal conductor. Here, it is important to note that the mass of an electron remains constant, regardless of its speed. This trait greatly simplifies many physics calculations, including our example where we determine the root-mean-square speed of electrons within the metal.

Kinetic Theory of Gases

The kinetic theory of gases is a theoretical framework that describes a gas as a collection of particles in random motion, colliding elastically with each other and with the walls of any container. One of the principal outcomes of this theory is that the temperature of a gas is directly proportional to the average kinetic energy of its molecules.

One key aspect of this theory is the concept of root-mean-square speed (\(v_{rms}\)). It is the measure of the speed of particles in a gas, which when squared and averaged, gives an indication of the kinetic energy in the system. It is calculated using the formula \(v_{rms} = \sqrt{\frac{3kT}{m}}\), where \(k\) is the Boltzmann constant, \(T\) is the temperature, and \(m\) is the mass of a single particle. In essence, the kinetic theory provides a molecular-level insight into the macroscopic properties of gases, which in turn can be applied to electrons in metals as in our exercise, considering them as particles in a 'gas' of electrons.

Thermal Physics

Thermal physics is the branch of physics that deals with heat, temperature, and their relation to energy, work, radiation, and the properties of matter. It encompasses the study of thermodynamics, kinetic theory, and statistical mechanics. A fundamental aspect of thermal physics is understanding how energy is transferred within a system and how it affects the system's properties.

If we consider the electrons in a metal as per our example, thermal physics would examine how the electrons' thermal energy (dictated by temperature) translates into their kinetic energy and subsequently their motion. By understanding these relationships, thermal physics allows us to predict how the electrons will behave under different thermal conditions, such as how fast they will move at a certain temperature, which is fundamentally related to the concept of root-mean-square speed we encountered in the exercise.