What are the dimensions of Boltzmann's constant?

Options

A. `MLT^(-2)K^(-1)`
B. `ML^(2)T^(-2)K^(-1)`
C. `M^(0)LT^(-2)`
D. `M^(0)L^(2)T^(-2)K^(-1)`

Answer

To find the dimensions of Boltzmann's constant (denoted as \( K_c \)), we can follow these steps:

### Step 1: Understand the definition of Boltzmann's constant
Boltzmann's constant relates energy to temperature. It is defined as the amount of energy per temperature increment. Mathematically, it can be expressed as:
\[ K_c = \frac{E}{T} \]
where \( E \) is energy and \( T \) is temperature.

### Step 2: Find the dimensions of energy
Energy can be defined in terms of work done, which is given by the formula:
\[ \text{Work} = \text{Force} \times \text{Displacement} \]
The dimension of force is given by Newton's second law:
\[ \text{Force} = \text{mass} \times \text{acceleration} = m \cdot a = m \cdot \frac{v}{t} = m \cdot \frac{L}{T^2} \]
Thus, the dimensions of force are:
\[ [F] = M L T^{-2} \]
Now, substituting this into the work formula:
\[ \text{Energy} = \text{Force} \times \text{Displacement} = (M L T^{-2}) \times L = M L^2 T^{-2} \]
So, the dimensions of energy \( [E] \) are:
\[ [E] = M L^2 T^{-2} \]

### Step 3: Find the dimensions of temperature
The dimension of temperature is represented as:
\[ [T] = K \]
where \( K \) stands for Kelvin.

### Step 4: Combine the dimensions to find the dimensions of Boltzmann's constant
Now, substituting the dimensions of energy and temperature into the expression for Boltzmann's constant:
\[ [K_c] = \frac{[E]}{[T]} = \frac{M L^2 T^{-2}}{K} \]
This simplifies to:
\[ [K_c] = M L^2 T^{-2} K^{-1} \]

### Conclusion
Thus, the dimensions of Boltzmann's constant \( K_c \) are:
\[ [K_c] = M L^2 T^{-2} K^{-1} \]