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 Definition of Boltzmann's Constant
Boltzmann's constant \( k \) is defined as:
\[ k = \frac{R}{N_A} \]
where \( R \) is the universal gas constant and \( N_A \) is Avogadro's number. Since \( N_A \) is a number of entities (molecules, atoms, etc.), it is dimensionless.

### Step 2: Find the Dimensions of the Universal Gas Constant \( R \)
The universal gas constant \( R \) can be derived from the ideal gas law:
\[ PV = nRT \]
Here, \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the gas constant, and \( T \) is temperature.

Rearranging gives:
\[ R = \frac{PV}{nT} \]

### Step 3: Determine the Dimensions of Each Variable
- **Pressure \( P \)** has dimensions of force per unit area:
\[ [P] = \frac{F}{A} = \frac{MLT^{-2}}{L^2} = ML^{-1}T^{-2} \]
- **Volume \( V \)** has dimensions:
\[ [V] = L^3 \]
- **Number of moles \( n \)** is dimensionless.
- **Temperature \( T \)** has dimensions:
\[ [T] = K \quad \text{(Kelvin)} \]

### Step 4: Substitute the Dimensions into the Equation for \( R \)
Substituting the dimensions of \( P \) and \( V \) into the equation for \( R \):
\[ [R] = \frac{[P][V]}{[n][T]} = \frac{(ML^{-1}T^{-2})(L^3)}{1 \cdot K} = \frac{ML^{2}T^{-2}}{K} \]

### Step 5: Conclude the Dimensions of Boltzmann's Constant \( k \)
Since \( k = \frac{R}{N_A} \) and \( N_A \) is dimensionless, the dimensions of \( k \) are the same as those of \( R \):
\[ [k] = \frac{ML^{2}T^{-2}}{K} \]

### Final Answer
The dimensions of Boltzmann's constant \( k \) are:
\[ [k] = ML^{2}T^{-2}K^{-1} \]

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