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The ratio of specific heats \(\left (\dfrac{C_p}{C_v}\right)\) in terms of degree of freedom \((f)\) is given by:
1. \(\left(1+\dfrac{f}{3}\right) \) 2. \(\left(1+\dfrac{2}{f}\right)\)
3. \(\left(1+\dfrac{f}{2}\right) \) 4. \(\left(1+\dfrac{1}{f}\right)\)
Subtopic:  Law of Equipartition of Energy |
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Molecules of an ideal gas are known to have three translational degrees of freedom and two rotational degrees of freedom. The gas is maintained at a temperature of \(T.\) The total internal energy, \(U\) of a mole of this gas, and the value of \(\gamma~\left(=\dfrac{C_P}{C_V}\right )\) are, respectively:
1. \( U=5 R T \text { and } \gamma=\dfrac{7}{5} \)

2. \( U=\dfrac{5}{2} R T \text { and } \gamma=\dfrac{6}{5} \)

3. \(U=5 R T \text { and } \gamma=\dfrac{6}{5} \)

4. \( U=\dfrac{5}{2} R T \text { and } \gamma=\dfrac{7}{5}\)

Subtopic:  Law of Equipartition of Energy |
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Given below are two statements: 

Statement I: For a monatomic gas atom, the number of degrees of freedom is \(3.\) 
Statement II: For a monatomic gas, the ratio \(\dfrac{C_P}{C_V}=\gamma=\dfrac{5}{3}.\)
 
1. Statement I is False but Statement II is True.
2. Both Statement I and Statement II are True.
3. Both Statement I and Statement II are False.
4. Statement I is True but Statement II is False.
Subtopic:  Law of Equipartition of Energy |
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A gas mixture consists of \(3\) moles of oxygen and \(5\) moles of argon at temperature \(T.\) Assuming the gases to be ideal and the oxygen bond to be rigid, the total internal energy (in units of \(RT\)) of the mixture is:
1. \(11\)
2. \(15\)
3. \(20\)
4. \(13\)

Subtopic:  Law of Equipartition of Energy |
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The molar specific heat capacity at constant volume for a rigid diatomic molecule is:
(\(R\) is the universal gas constant)
1.  \(\frac97R\)
2.  \(\frac72R\)
3.  \(\frac52R\)
4.  \(\frac32R\)
Subtopic:  Law of Equipartition of Energy |
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The translational degrees of freedom \(\left(f_t\right)\) and rotational degrees of freedom \(\left(f_r\right)\) of \(\mathrm{CH}_4\) molecule are:
1. \(f_t=3\) and \(f_r=3\)
2. \(f_t=2\) and \(f_r=3\)
3. \(f_t=2\) and \(f_r=2\)
4. \(f_t=3\) and \(f_r=2\)
Subtopic:  Law of Equipartition of Energy |
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In a mixture, \(0.5\) moles of \(\mathrm{O_2}\) and \(4\) moles of \(\mathrm{Ne}\) gas are taken at temperature \(T\). The internal energy of the system is equal to:
1. \(\left ( \dfrac{13}{2} \right )RT\)

2. \(\left ( \dfrac{11}{4} \right )RT\)

3. \(\left ( \dfrac{29}{4} \right )RT\)

4. \(\left ( \dfrac{13}{4} \right )RT\)
Subtopic:  Law of Equipartition of Energy |
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Three moles of oxygen and four moles of helium are mixed at room temperature. What is the degree of freedom of the mixture?
1. \(\dfrac{15}{7}\) 2. \(\dfrac{12}{7}\)
3. \(\dfrac{27}{7}\) 4. \(\dfrac{3}{2}\)
Subtopic:  Law of Equipartition of Energy |
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At room temperature, how many degrees of freedom are associated with the motion of an oxygen molecule \((\mathrm{O_2})\text{?}\)
1. \(5\)
2. \(4\)
3. \(3\)
4. \(2\)
Subtopic:  Law of Equipartition of Energy |
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Match the \(C_p/C_V\)  ratio for ideal gases with different types of molecules:

Column I Column II
(A) Monatomic (I) \(7/5\)
(B) Diatomic rigid molecules (II) \(9/7\)
(C) Diatomic non-rigid molecules (III) \(4/3\)
(D) Triatomic rigid molecules (IV) \(5/3\)
 
1. (A)-(III), (B)-(IV), (C)-(II), (D)-(I)
2. (A)-(II), (B)-(III), (C)-( I), (D)-(IV)
3. (A)-(IV), (B)-(II), (C)-(I), (D)-(III)
4. (A)-(IV), (B)-(I), (C)-(II), (D)-(III)
 

Subtopic:  Law of Equipartition of Energy |
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