Given below are two statements:

Statement (I) : Dimensions of specific heat is $[\mathrm{L}^2 \mathrm{~T}^{-2} \mathrm{~K}^{-1}]$.

Statement (II) : Dimensions of gas constant is $[\mathrm{M} \mathrm{L}^2 \mathrm{~T}^{-1} \mathrm{~K}^{-1}]$.

In the light of the above statements, choose the most appropriate answer from the options given below.

<p>To evaluate the veracity of the given statements, we need to understand the physical quantities involved and their dimensional formulas. Specifically, we're looking at the dimensions of specific heat and gas constant.</p> <p><strong>Specific Heat:</strong></p> <p>Specific heat (c) is the amount of heat required to raise the temperature of a unit mass of a substance by one degree Celsius (or one Kelvin, since the increment is the same in both scales). Its formula is $q = mc\Delta T$, where $q$ is the heat added, $m$ is the mass, $c$ is the specific heat, and $\Delta T$ is the change in temperature.</p> <p>From the formula, we can deduce the dimensions of specific heat as follows:</p> <p>$[q] = [m][c][\Delta T]$</p> <p>Knowing that the dimension of heat (q) is equivalent to energy, which is $[\mathrm{M} \mathrm{L}^2 \mathrm{T}^{-2}]$, the mass (m) is $[\mathrm{M}]$, and temperature ($\Delta T$) is $[\mathrm{K}]$, we can solve for $[c]$:</p> <p>$[\mathrm{M} \mathrm{L}^2 \mathrm{T}^{-2}] = [\mathrm{M}][c][\mathrm{K}]$</p> <p>So, $[c] = [\mathrm{L}^2 \mathrm{T}^{-2} \mathrm{K}^{-1}]$</p> <p>This reveals that Statement (I) is <strong>correct</strong>.</p> <p><strong>Gas Constant:</strong></p> <p>The gas constant (R) appears in the ideal gas law, represented as $PV = nRT$, where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is the temperature. The dimensions of the gas constant can be derived from this relation.</p> <p>Pressure (P) has dimensions $[\mathrm{M} \mathrm{L}^{-1} \mathrm{T}^{-2}]$, volume (V) has dimensions $[\mathrm{L}^3]$, and temperature (T) has dimensions $[\mathrm{K}]$.</p> <p>Therefore, $[\mathrm{M} \mathrm{L}^{-1} \mathrm{T}^{-2}][\mathrm{L}^3] = [n][R][\mathrm{K}]$</p> <p>Considering that the mole (n) is a dimensionless quantity, we can deduce the dimensions of R as:</p> <p>$$[R] = \frac{[\mathrm{M} \mathrm{L}^2 \mathrm{T}^{-2}]}{[\mathrm{K}]} = [\mathrm{M} \mathrm{L}^2 \mathrm{T}^{-2} \mathrm{K}^{-1}]$$</p> <p>This indicates that Statement (II) has stated the dimensions incorrectly, presenting them as $[\mathrm{M} \mathrm{L}^2 \mathrm{T}^{-1} \mathrm{K}^{-1}]$ when it should be $[\mathrm{M} \mathrm{L}^2 \mathrm{T}^{-2} \mathrm{K}^{-1}]$, making Statement (II) <strong>incorrect</strong>.</p> <p>Based on the analysis:</p> <p>Option A, "Statement (I) is incorrect but statement (II) is correct," is wrong because Statement (I) is correct.</p> <p>Option B, "Both statement (I) and statement (II) are incorrect," is wrong because Statement (I) is correct.</p> <p>Option C, "Both statement (I) and statement (II) are correct," is wrong because Statement (II) is incorrect.</p> <p>Option D, "Statement (I) is correct but statement (II) is incorrect," is the <strong>correct</strong> choice, reflecting the true nature of the statements provided.</p>