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Gas is found to obey the law $P^{2} V$ = constant. The initial temperature and volume are $T_{0}$ and $V_{0}$ . If the gas expands to a volume $3 V_{0}$ , its final temperature becomes:

$\frac{T_{0}}{3}$

$\frac{T_{0}}{\sqrt{3}}$

$3 T_{0}$

None of these

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Explanation

$P^{2} V$ = constant

$P = \frac{n R T}{V}$ or $P = \frac{C T}{V}$ (C=constant)

$∴ \frac{T^{2}}{V} = c o n s \tan t$

or $T \propto \sqrt{V}$

V is increased by 3 times.

Hence, T will increase by $\sqrt{3}$ times.

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For a gas, the r.m.s speed at 800 K is:

Four times the value at 200 K

Half the value at 200 K

Twice the value at 200 K

Same as at 200 K

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Explanation

$V_{r m s} = \sqrt{\frac{3 R T}{M}} ∴ V_{r m s} a t 800 K i s t w i c e t h a t a t 200 K .$

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The mean translational kinetic energy of a perfect gas molecule at the temperature T Kelvin is:

$\frac{1}{2} k T$

$\frac{3}{2} k T$

2 kT

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Explanation

Translational KE = $3 \times \frac{1}{2} k T$

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Choose the correct option related to the specific heat at constant volume $C_{V}$ -

$C_{V} = \frac{1}{2} f R$

$C_{V} = \frac{R}{γ - 1}$

$C_{V} = \frac{C_{P}}{γ}$

All of these

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Explanation

$C_{P} - C_{V} = R D i v i d i n g b y C_{V}, w e g e t γ - 1 = \frac{R}{C_{V}} ∴ C_{V} = \frac{R}{γ - 1} F u r t h e r U = \frac{f}{2} R T ∴ C_{V} = \frac{d U}{d T} = \frac{f}{2} R$

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Five molecules of a gas have speeds, 1, 2, 3, 4 and 5 km $s^{- 1}$ . The value of the root mean square speed of the gas molecules is:

3 km $s^{- 1}$

$\sqrt{11}$ km $s^{- 1}$

$\frac{1}{2} \sqrt{π}$ km $s^{- 1}$

3.5 km $s^{- 1}$

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Explanation

$V_{r m s} = \sqrt{\frac{1^{2} + 2^{2} + 3^{2} + 4^{2} + 5^{2}}{5}} = \sqrt{\frac{55}{5}} = \sqrt{11} k m s^{- 1}$

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The average velocity of an ideal gas molecule is: [OJEE 2010]

proportional to $\sqrt{T}$

proportional to $T^{2}$

proportional to $T^{3}$

zero

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Explanation

The average velocity of an ideal gas molecule is zero.

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For a gas, $\frac{R}{C_{V}} = 0.67$ . This gas is made up of molecules which are: [JIPMER 2001, 02]

Diatomic

Mixture of diatomic and polyatomic molecules

Monoatomic

Polyatomic

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Explanation

(c)

$C_{V} = \frac{R}{0.67} = 1.5 R = \frac{3}{2} R$

This is the value of $C_{V}$ for monoatomic gases.

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At constant temperature, on increasing the pressure of a gas by 5% its volume will decrease by: [MP PET 2002; Kerala 2006]

5.26 %

4.26 %

4.76 %

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Explanation

$P \propto \frac{1}{V} \Rightarrow \frac{V_{2}}{V 1} = \frac{P_{1}}{P_{2}} = \frac{100}{105} \Rightarrow V_{2} = \frac{100}{105} V_{1} = 0.953 V_{1} % c h a n g e i n v o l u m e = \frac{V_{1} - V_{2}}{V 1} \times 100 = \frac{V_{1} - 0.953 V_{1}}{V_{1}} \times 100 = 4.76 %$

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At what temperature, is the root mean square velocity of gaseous hydrogen molecules equal to that of oxygen molecules at 47 $°$ C:[MP PET 1997, 2000; RPET 1999; AIEEE 2002; J & K CET 2004; Kerala PET 2010]

20 K

80 K

- 73 K

3 K

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Explanation

$V_{r m s} = \sqrt{\frac{3 R T}{M}} \Rightarrow T \propto M [∵ V_{r m s}, R \to c o n s \tan t] \Rightarrow \frac{T_{H_{2}}}{T_{O_{2}}} = \frac{M_{H_{2}}}{M_{O_{2}}} \Rightarrow \frac{T_{H_{2}}}{(273 + 47)} = \frac{2}{32} \Rightarrow T_{H_{2}} = 20 K$

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The average kinetic energy of a gas molecule can be determined by knowing: [RPET 2000, MP PET 2010]

The number of molecules in the gas

The pressure of the gas only

The temperature of the gas only

None of the above is enough by itself

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Explanation

(c)

$K i n e t i c e n e r g y \propto T e m p e r a t u r e$

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