Given below are two statements:
Statement (I): Dimensions of specific heat is $[L^{2}T^{-2}K^{-1}]$
Statement (II): Dimensions of gas constant is $[M L^{2}T^{-2}K^{-1}]$
Statement (I): Dimensions of specific heat is $[L^{2}T^{-2}K^{-1}]$
Statement (II): Dimensions of gas constant is $[M L^{2}T^{-2}K^{-1}]$
- Statement (I) is incorrect but statement (II) is correct
- Both statement (I) and statement (II) are incorrect
- Statement (I) is correct but statement (II) is incorrect
- Both statement (I) and statement (II) are correct
The Correct Option is C
Approach Solution - 1
\[\Delta Q = m S \Delta T\]
\[s = \frac{\Delta Q}{m \Delta T}\]
\[[s] = \frac{\left[ M L^2 T^{-2} \right]}{M \cdot K}\]
\[[s] = \left[ L^2 T^{-2} K^{-1} \right]\]
Statement-(I) is correct.
From \( PV = nRT \), we have:
\[R = \frac{PV}{nT}\]
Substitute dimensions:
\[[R] = \frac{\left[ M L^{-1} T^{-2} L^3 \right]}{\left[ \text{mol} \right] \cdot \left[ K \right]}\]
Simplify:
\[[R] = \left[ M L^2 T^{-2} \text{mol}^{-1} K^{-1} \right]\]
Statement-(II) is incorrect.
Approach Solution -2
To determine the correctness of the given statements about the dimensions of specific heat and gas constant, let's analyze each statement individually.
Statement (I): Dimensions of specific heat is \([L^{2}T^{-2}K^{-1}]\)
The specific heat (c) is defined as the amount of heat required to raise the temperature of a unit mass of a substance by one degree Kelvin. The formula for specific heat in terms of basic dimensions is:
\(c = \frac{Q}{m \Delta T}\)
- where \(Q\) is heat energy with dimension \([ML^{2}T^{-2}]\),
- \(m\) is mass with dimension \([M]\),
- \(\Delta T\) is temperature change with dimension \([K]\).
Therefore, the dimensions of specific heat are:
\([c] = \frac{[ML^{2}T^{-2}]}{[M][K]} = [L^{2}T^{-2}K^{-1}]\)
This matches the dimension provided in the statement. Thus, Statement (I) is correct.
Statement (II): Dimensions of gas constant is \([M L^{2}T^{-2}K^{-1}]\)
The universal gas constant (R) is defined as \(R = \frac{PV}{nT}\), where:
- \(P\) is pressure with dimension \([ML^{-1}T^{-2}]\),
- \(V\) is volume with dimension \([L^{3}]\),
- \(n\) is number of moles (dimensionless),
- \(T\) is temperature with dimension \([K]\).
Hence, the dimensions of the gas constant are:
\([R] = \frac{[ML^{-1}T^{-2}][L^{3}]}{[K]} = [ML^{2}T^{-2}K^{-1}]\)
Therefore, Statement (II) as given with \([M L^{2}T^{-2}K^{-1}]\) is correct.
However, there seems to be a discrepancy between the provided correct answer and our dimensional analysis. Upon review, the analysis shows that the dimensional formula for the gas constant is correct according to conventional understanding; thus, both statements are correct. The provided answer may have an error.
If adhering strictly to the provided correct answer, Statement (I) is correct but statement (II) is incorrect should be the conclusion, but analysis shows this evaluation might be misleading. Please verify with broader context or additional sources if this answer does not align with expected standards.
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