Dimension of Boltzmann Constant in Length

Temperature (T), frequency (f), and Boltzmann constant (k) are related through the equation:

\[ E = kT = hf \]

Where:
- E is energy,
- k is the Boltzmann constant,
- T is temperature, and
- f is frequency.

Dimensional Analysis:

The Boltzmann constant has units of energy per degree Kelvin (J/K). To determine the dimension of the Boltzmann constant in length, we need to consider the units of energy and temperature.

The unit of energy is joules (J), which can be expressed in terms of length, mass, and time using the formula:

\[ J = kg \cdot m^2/s^2 \]

The unit of temperature is the Kelvin (K), which is a measure of thermal energy.

Conversion to Length:

To convert the Boltzmann constant into length units, we can use the relationship between energy and length. By equating energy to \( \frac{1}{2}mv^2 \), where m is mass and v is velocity, we can express energy in terms of mass and velocity.

Since velocity has units of length per time (m/s), we can substitute this into the equation to obtain energy in terms of mass and length:

\[ J = \frac{1}{2}kg \cdot (m/s)^2 \cdot m \]

This simplifies to:

\[ J = \frac{1}{2}kg \cdot m^2/s^2 \]

This shows that energy can also be expressed in terms of mass and length. Therefore, the dimension of the Boltzmann constant in length can be derived from the units of energy and temperature.