Boltzman constant and Plank's constant differ in the dimensions of

Options

A. Mass and Time
B. Length and Time
C. Length and Mass
D. Time and temperature

Answer

To determine how Boltzmann's constant and Planck's constant differ in dimensions, we need to analyze their definitions and the formulas in which they appear.

### Step-by-Step Solution:

1. **Identify the Constants:**
- Boltzmann's constant (k) is used in thermodynamics, particularly in the ideal gas law.
- Planck's constant (h) is used in quantum mechanics, particularly in the energy of photons.

2. **Write Down the Relevant Formulas:**
- For Boltzmann's constant, we can use the ideal gas law:
\[ PV = nRT \]
Rearranging gives:
\[ k = \frac{PV}{nT} \]
- For Planck's constant, the energy of a photon is given by:
\[ E = h\nu \]
Rearranging gives:
\[ h = \frac{E}{\nu} \]

3. **Determine the Dimensions:**
- For Boltzmann's constant (k):
- Pressure (P) has dimensions of \([M L^{-1} T^{-2}]\).
- Volume (V) has dimensions of \([L^3]\).
- Number of moles (n) is dimensionless.
- Temperature (T) has dimensions of \([\Theta]\) (where \(\Theta\) is the dimension of temperature).
- Thus, the dimensions of k can be calculated as:
\[ [k] = \frac{[P][V]}{[n][T]} = \frac{[M L^{-1} T^{-2}][L^3]}{[T]} = [M L^2 T^{-2} \Theta^{-1}] \]

- For Planck's constant (h):
- Energy (E) has dimensions of \([M L^2 T^{-2}]\).
- Frequency (\(\nu\)) has dimensions of \([T^{-1}]\).
- Thus, the dimensions of h can be calculated as:
\[ [h] = \frac{[E]}{[\nu]} = \frac{[M L^2 T^{-2}]}{[T^{-1}]} = [M L^2 T^{-1}] \]

4. **Compare the Dimensions:**
- Boltzmann's constant: \([M L^2 T^{-2} \Theta^{-1}]\)
- Planck's constant: \([M L^2 T^{-1}]\)

5. **Identify the Differences:**
- The dimensions of mass (M) and length (L) are the same for both constants.
- The time dimension (T) differs: Boltzmann's constant has \(T^{-2}\) while Planck's constant has \(T^{-1}\).
- The temperature dimension (\(\Theta\)) is present in Boltzmann's constant but absent in Planck's constant.

6. **Conclusion:**
- The difference in dimensions between Boltzmann's constant and Planck's constant is due to the presence of temperature (\(\Theta\)) in Boltzmann's constant and the different powers of time (T).

### Final Answer:
Boltzmann's constant and Planck's constant differ in the dimensions of time and temperature.