What is the defined value of \(k_B\) in the given equation?

\(k_B = \frac{PV}{TN}\)

\(1.38 × 10^{-23} JK^{-1}\)

The correct answer is option 1. \(1.38 × 10^{-23} JK^{-1}\).

Boltzmann's constant (kB) is a fundamental physical constant that relates the average kinetic energy of particles in a gas with the temperature of the gas. It is a crucial parameter in statistical mechanics and thermodynamics. The formula relating pressure (P), volume (V), temperature (T), and the number of particles (N) in an ideal gas is:

\[ PV = Nk_BT \]

Where:

\( P \) is the pressure of the gas

\( V \) is the volume of the gas

\( T \) is the temperature of the gas (in Kelvin)

\( N \) is the number of particles

\( k_B \) is Boltzmann's constant

By rearranging this formula, we can isolate Boltzmann's constant:

\[ k_B = \frac{PV}{NT} \]

In the context of your question, the value of Boltzmann's constant (kB) is being determined using this formula, with known values for pressure (P), volume (V), temperature (T), and the number of particles (N).

The correct value of Boltzmann's constant is approximately \(1.38 \times 10^{-23} \, \text{J K}^{-1}\). This constant relates the average kinetic energy of particles in a gas to the temperature of the gas, providing a fundamental link between microscopic behavior (individual particle energies) and macroscopic properties (temperature).