Boltzmann constant: kB = 1.38 x 10-28 JK Thermal energy at 300 K: kBT = 0.0259 eV Energy unit conversion: 1eV = 1.60 x 10-19 J
The Correct Answer and Explanation is:
Correct Values
- Boltzmann constant kBk_B:
kB=1.38×10−23 J/Kk_B = 1.38 \times 10^{-23} \, \text{J/K}
(Note: Your original value 1.38×10−28 J/K1.38 \times 10^{-28} \, \text{J/K} is incorrect by 5 orders of magnitude.) - Thermal energy at 300 K:
kBT=1.38×10−23×300=4.14×10−21 Jk_B T = 1.38 \times 10^{-23} \times 300 = 4.14 \times 10^{-21} \, \text{J} - Energy conversion from Joules to eV:
1 eV=1.60×10−19 J1 \, \text{eV} = 1.60 \times 10^{-19} \, \text{J} - Thermal energy in eV at 300 K: kBT=4.14×10−21 J1.60×10−19 J/eV=0.0259 eVk_B T = \frac{4.14 \times 10^{-21} \, \text{J}}{1.60 \times 10^{-19} \, \text{J/eV}} = 0.0259 \, \text{eV}
Explanation
The Boltzmann constant kBk_B is a fundamental physical constant that relates the average kinetic energy of particles in a gas to the temperature of the gas. Its value is approximately 1.38×10−23 J/K1.38 \times 10^{-23} \, \text{J/K}, which means that for each degree Kelvin increase in temperature, the average energy per particle increases by this tiny amount of joules.
The thermal energy kBTk_B T at a given temperature TT is a useful measure in physics and chemistry because it sets the energy scale for many processes such as diffusion, reaction rates, and semiconductor behavior. At room temperature (about 300 K), thermal energy can be calculated by multiplying the Boltzmann constant by the temperature: kBT=1.38×10−23 J/K×300 K=4.14×10−21 J.k_B T = 1.38 \times 10^{-23} \, \text{J/K} \times 300 \, \text{K} = 4.14 \times 10^{-21} \, \text{J}.
Since many energies in atomic and solid-state physics are expressed in electron volts (eV) rather than joules, it is important to convert this value to eV using the conversion factor 1 eV=1.60×10−19 J1 \, \text{eV} = 1.60 \times 10^{-19} \, \text{J}: kBT=4.14×10−21 J1.60×10−19 J/eV=0.0259 eV.k_B T = \frac{4.14 \times 10^{-21} \, \text{J}}{1.60 \times 10^{-19} \, \text{J/eV}} = 0.0259 \, \text{eV}.
This energy scale (~0.026 eV) is often used in semiconductor physics to describe how electrons behave at room temperature, influencing carrier distributions and electronic properties. It also appears in Arrhenius equations describing temperature-dependent processes and rates of chemical reactions.
In summary, the Boltzmann constant bridges microscopic energy scales to macroscopic temperatures, and knowing kBTk_B T at room temperature helps scientists and engineers understand the thermal behavior of materials and particles.
