Match the LIST-I with LIST-II
LIST-I LIST-II A. Boltzmann constant I. \( \text{ML}^2\text{T}^{-1} \) B. Coefficient of viscosity II. \( \text{MLT}^{-3}\text{K}^{-1} \) C. Planck's constant III. \( \text{ML}^2\text{T}^{-2}\text{K}^{-1} \) D. Thermal conductivity IV. \( \text{ML}^{-1}\text{T}^{-1} \)
Choose the correct answer from the options given below :
Match the LIST-I with LIST-II
| LIST-I | LIST-II | ||
| A. | Boltzmann constant | I. | \( \text{ML}^2\text{T}^{-1} \) |
| B. | Coefficient of viscosity | II. | \( \text{MLT}^{-3}\text{K}^{-1} \) |
| C. | Planck's constant | III. | \( \text{ML}^2\text{T}^{-2}\text{K}^{-1} \) |
| D. | Thermal conductivity | IV. | \( \text{ML}^{-1}\text{T}^{-1} \) |
Choose the correct answer from the options given below :
Show Hint
- A-III, B-IV, C-I, D-II
- A-II, B-III, C-IV, D-I
- A-III, B-II, C-I, D-IV
- A-III, B-IV, C-II, D-I
The Correct Option is A
Approach Solution - 1
Let's find the dimensions of each quantity in LIST-I.
A. Boltzmann constant (k): From the ideal gas law, \( PV = NkT \), where P is pressure
(\( \text{ML}^{-1}\text{T}^{-2} \)), V is volume (\( \text{L}^3 \)), N is the number of particles (dimensionless), k is the Boltzmann constant, and T is temperature (K).
So, \( k = \frac{PV}{NT} = \frac{(\text{ML}^{-1}\text{T}^{-2})(\text{L}^3)}{(1)(\text{K})} = \text{ML}^2\text{T}^{-2}\text{K}^{-1} \)
Thus, A matches with III.
B. Coefficient of viscosity (\( \eta \)): From viscous force \( F = 6\pi \eta r v \), where F is force (\( \text{MLT}^{-2} \)), r is radius (L), and v is velocity (\( \text{LT}^{-1} \)).
So, \( \eta = \frac{F}{6\pi r v} = \frac{\text{MLT}^{-2}}{(1)(\text{L})(\text{LT}^{-1})} = \frac{\text{MLT}^{-2}}{\text{L}^2\text{T}^{-1}} = \text{ML}^{-1}\text{T}^{-1} \)
Thus, B matches with IV.
C. Planck's constant (h): From the energy of a photon \( E = hf \), where E is energy (\( \text{ML}^2\text{T}^{-2} \)) and f is frequency (\( \text{T}^{-1} \)). So, \( h = \frac{E}{f} = \frac{\text{ML}^2\text{T}^{-2}}{\text{T}^{-1}} = \text{ML}^2\text{T}^{-1} \)
Thus, C matches with I.
D. Thermal conductivity (K): From the rate of heat flow \( \frac{dQ}{dt} = -KA \frac{dT}{dx} \), where \( \frac{dQ}{dt} \) is power (\( \text{ML}^2\text{T}^{-3} \)), A is area (\( \text{L}^2 \)), and \( \frac{dT}{dx} \) is temperature gradient (\( \text{KL}^{-1} \)).
So, \( K = \frac{(dQ/dt) dx}{A dT} = \frac{(\text{ML}^2\text{T}^{-3})(\text{L})}{(\text{L}^2)(\text{K})} = \frac{\text{ML}^3\text{T}^{-3}}{\text{L}^2\text{K}} = \text{MLT}^{-3}\text{K}^{-1} \)
Thus, D matches with II.
The correct matching is A-III, B-IV, C-I, D-II, which corresponds to option (A).
Approach Solution -2
To solve this problem, we need to match the physical quantities in LIST-I with their correct dimensional formulas in LIST-II. Let's analyze each option step-by-step:
- Boltzmann Constant: The dimensional formula for the Boltzmann constant (\( k \)) is given by \([k] = \text{ML}^2\text{T}^{-2}\text{K}^{-1}\). This dimensional formula corresponds to the option III in LIST-II. Therefore, Boltzmann constant is matched with III.
- Coefficient of Viscosity: The coefficient of viscosity (\( \eta \)) has the dimensional formula \([\eta] = \text{ML}^{-1}\text{T}^{-1}\). This matches with option IV in LIST-II. Therefore, the coefficient of viscosity is matched with IV.
- Planck's Constant: Planck's constant (\( h \)) has the dimensional formula \([h] = \text{ML}^2\text{T}^{-1}\). This corresponds to option I in LIST-II. Therefore, Planck's constant is matched with I.
- Thermal Conductivity: The dimensional formula for thermal conductivity (\( \kappa \)) is \([\kappa] = \text{MLT}^{-3}\text{K}^{-1}\). This matches with option II in LIST-II. Therefore, thermal conductivity is matched with II.
Based on the above analysis, the correct matching is:
- A - III
- B - IV
- C - I
- D - II
This corresponds to the correct answer: A-III, B-IV, C-I, D-II. Therefore, the solution is validated by matching the physical quantities with their correct dimensional formulas.
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