Equipartition Theorem

Q.

A gas mixture consists of $2$ moles of oxygen and $4$ moles of argon at temperature$"\mathrm{t}"$. Neglecting all vibrational modes, the total internal energy of the system is?

Q. A mixture of $10\text{g}$ of hydrogen and $40\text{g}$ of helium are kept in a closed vessel. To change the temperature of the mixture by ${50}^{\circ }\text{C}$, the amount of heat energy should be added to the mixture is
(Given $R=2{\text{cal mol}}^{-1}{}^{\circ }{\text{C}}^{-1}$)

1. $2500\text{cal}$
2. $2000\text{cal}$
3. $2750\text{cal}$
4. None of these

Q. A gas mixture consists of 2 moles of oxygen and 4 moles of Argon at temperature $T$. Neglecting all vibrational degrees of freedom, the total internal energy of the system is

1. $4RT$
2. $15RT$
3. $9RT$
4. $11RT$

Q. A proton and an alpha particle are accelerating by the same potential difference. Find the ratio of their de-Broglie wavelength? $\left(\text{charge}\right(q_{\alpha })=+2e,q_{proton}=+e\text{and}{m}_{\alpha }=4m_{proton})$

1. $\sqrt{16}$
2. $\sqrt{6}$
3. $\sqrt{8}$
4. $\sqrt{2}$

Q. One $\text{mole}$ of an ideal monoatomic gas is taken at a temperature of $300\text{K}$. If the volume is doubled keeping its pressure constant, change in internal energy is

1. $450R$
2. $600R$
3. $500R$
4. $400R$

Q. One mole of an ideal monatomic gas requires 210 J heat to raise the temperature by 10 K, when heated at constant pressure. If the same gas is heated at constant volume to raise the temperature by 10 K then heat required is

1. 238 J
2. 126 J
3. 210 J
4. 350 J

Q. One $\text{mole}$ of an ideal monoatomic gas is taken at a temperature of $300\text{K}$. If the volume is doubled keeping its pressure constant, change in internal energy is

1. $450R$
2. $600R$
3. $500R$
4. $400R$

Q. A mixture of $10\text{g}$ of hydrogen and $40\text{g}$ of helium are kept in a closed vessel. To change the temperature of the mixture by ${50}^{\circ }\text{C}$, the amount of heat energy should be added to the mixture is
(Given $R=2{\text{cal mol}}^{-1}{}^{\circ }{\text{C}}^{-1}$)

1. $2500\text{cal}$
2. $2000\text{cal}$
3. $2750\text{cal}$
4. None of these

Q. If 2 moles of an ideal monoatomic gas at temperature $T_0$ is mixed with 4 moles of another ideal monoatomic gas at temperature $2T_0$ in a closed vessel, then the temperature of the mixture is

1. $\frac{5}{3}{T}_0$
2. $\frac{3}{2}{T}_0$
3. $\frac{4}{3}{T}_0$
4. $\frac{5}{4}{T}_0$

Q. A proton and an alpha particle are accelerating by the same potential difference. Find the ratio of their de-Broglie wavelength? $\left(\text{charge}\right(q_{\alpha })=+2e,q_{proton}=+e\text{and}{m}_{\alpha }=4m_{proton})$

1. $\sqrt{2}$
2. $\sqrt{6}$
3. $\sqrt{16}$
4. $\sqrt{8}$

Q. If 2 moles of an ideal monoatomic gas at temperature $T_0$ is mixed with 4 moles of another ideal monoatomic gas at temperature $2T_0$ in a closed vessel, then the temperature of the mixture is

1. $\frac{5}{3}{T}_0$
2. $\frac{3}{2}{T}_0$
3. $\frac{4}{3}{T}_0$
4. $\frac{5}{4}{T}_0$

Q. One mole of a mono-atomic ideal gas is contained in a closed vessel. The vessel is moving with a constant velocity $v$ and the molecular mass of the gas is $M$. The rise in temperature of the gas when the vessel is suddenly stopped is given by:
$[\gamma =\frac{C_p}{C_v}]$. Assume that the walls of the container are adiabatic.

1. $\frac{M{v}^2}{R}$
2. $\frac{M{v}^2}{2R}$
3. $\frac{2M{v}^2}{R}$
4. $\frac{M{v}^2}{3R}$