Boltzman's constant 'k' has dimensions of

Options

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

Answer

To determine the dimensions of Boltzmann's constant \( k \), we can follow these steps:

### Step 1: Understand the relationship involving Boltzmann's constant
We start with the ideal gas equation:
\[ PV = nRT \]
where:
- \( P \) is pressure,
- \( V \) is volume,
- \( n \) is the number of moles,
- \( R \) is the universal gas constant,
- \( T \) is temperature in Kelvin.

### Step 2: Rewrite the equation in terms of Boltzmann's constant
We can express the number of moles \( n \) in terms of the number of molecules \( N \) and Avogadro's number \( N_A \):
\[ n = \frac{N}{N_A} \]
Substituting this into the ideal gas equation gives:
\[ PV = \frac{N}{N_A}RT \]
Rearranging this, we can express \( R \) in terms of \( k \) (Boltzmann's constant):
\[ R = k N_A \]
Thus, we can rewrite the ideal gas equation as:
\[ PV = NkT \]

### Step 3: Identify the units of \( PV \)
The left side of the equation \( PV \) represents work done, which has the unit of Joules (J). Therefore:
\[ PV \text{ has units of Joules (J)} \]

### Step 4: Determine the units of \( k \)
From the equation \( PV = NkT \), we can isolate \( k \):
\[ k = \frac{PV}{N T} \]
Here, \( P \) has units of pressure (Pascals, \( \text{Pa} \)), \( V \) has units of volume (cubic meters, \( \text{m}^3 \)), \( N \) is the number of molecules (dimensionless), and \( T \) has units of Kelvin (K).

Thus, the units of \( k \) can be expressed as:
\[ \text{Units of } k = \frac{\text{J}}{\text{K}} = \frac{\text{N m}}{\text{K}} = \frac{\text{kg m}^2 \text{s}^{-2}}{\text{K}} \]

### Step 5: Determine the dimensions of \( k \)
We know that:
- Joule (J) has dimensions of \( [M L^2 T^{-2}] \)
- Kelvin (K) is a base unit.

Thus, the dimensions of Boltzmann's constant \( k \) can be expressed as:
\[ [k] = \frac{[M L^2 T^{-2}]}{[K]} = [M L^2 T^{-2} K^{-1}] \]

### Conclusion
The dimensions of Boltzmann's constant \( k \) are:
\[ [M L^2 T^{-2} K^{-1}] \]