Question

Match Column - I and Column - II and choose the correct match from the given choices. 

Column - I		Column - II
(A)	Root mean square speed of gas molecules	(P)	\(\frac{1}{3}\) nmv-2
(B)	The pressure exerted by the ideal gas	(Q)	\(\sqrt{\frac{3\,RT}{M}}\)
(C)	The average kinetic energy of a molecule	(R)	\(\frac{5}{2}RT\)
(D)	The total internal energy of 1 mole of a diatomic gas	(S)	\(\frac{3}{2}kBT\)

• (A) - (R), (B) - (Q), (C) - (P), (D) - (S)
• (A) - (R), (B) - (P), (C) - (S), (D) - (Q)
• (A) - (Q), (B) - (R), (C) - (S), (D) - (P)
• (A) - (Q), (B) - (P), (C) - (S), (D) - (R)

Answer 1

The Correct Option is D

To solve this matching question, we need to correctly pair the concepts in Column-I with their corresponding expressions or values in Column-II based on known gas laws and thermodynamic principles.

1. (A) Root mean square speed of gas molecules: The root mean square (rms) speed of gas molecules is given by the formula: \(\sqrt{\frac{3\,RT}{M}}\). This formula is derived from the kinetic theory of gases, where \(R\) is the universal gas constant, \(T\) is the absolute temperature, and \(M\) is the molar mass of the gas. Therefore, (A) matches with (Q).
2. (B) The pressure exerted by the ideal gas: For an ideal gas, the pressure is related to the number of molecules, mass, and their mean square speed by \(\frac{1}{3}\, nmv^2\), where \(n\) is the number of molecules per unit volume, \(m\) is the mass of a molecule, and \(v^2\) is the mean square speed. Thus, (B) matches with (P).
3. (C) The average kinetic energy of a molecule: The average kinetic energy (\(KE\)) of a single molecule of a gas is given by \(\frac{3}{2}\,k_B\,T\), where \(k_B\) is the Boltzmann constant and \(T\) is the temperature. This matches with option (S).
4. (D) The total internal energy of 1 mole of a diatomic gas: For a diatomic gas, the total internal energy is calculated as \(\frac{5}{2}\) per mole times the ideal gas constant times temperature, which is \(\frac{5}{2}\,RT\). Hence, for a diatomic gas, the internal energy of 1 mole is as given by (R).

Based on the above analysis, we have the correct matching:

• (A) - (Q)
• (B) - (P)
• (C) - (S)
• (D) - (R)

Thus, the correct answer is:

(A) - (Q), (B) - (P), (C) - (S), (D) - (R)