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PHXI13:KINETIC THEORY

360251 Five gas molecules chosen at random are found to have speeds of $500, 600, 700, 800$ and $900 m / s$ . Then:

1 The rms speed and the average speed are the same.

2 The average speed is

474 m / s

higher than the rms speed.

3 The rms speed is

474 m / s

lower than the average speed.

4 The rms speed is

14 m / s

higher than the average speed.

Explanation:

$v_{r m s} = \sqrt{\frac{\sum v^{2}}{n}}, v_{avg} = \frac{\sum v}{n}$
$v_{a v g} = \frac{500 + 600 + 700 + 800 + 900}{5}$
$= 20 \times 35 = 700 m / s$
$\sqrt{< v^{2} >} = 100 \sqrt{\frac{25 + 36 + 49 + 64 + 81}{5}}$
$= 100 \sqrt{51} = 714 m / s$
So rms speed is $14 m / s$ is higher than average speed.

PHXI13:KINETIC THEORY

360252 The root mean square speed of hydrogen molecule is $v_{1}$ at temperature $T K$ . The most probable speed of Oxygen molecule is $v_{2}$ at same temperature. Then $\frac{v_{1}}{v_{2}}$ is equal to:

1 4

2

2 \sqrt{6}

3

3 \sqrt{6}

4 2

Explanation:

For hydrogen $v_{H} = \sqrt{\frac{3 R T}{M_{H}}} (1)$
For oxygen $v_{0} = \sqrt{\frac{3 R T}{M_{0}}} (2)$
Given $v_{H} = v_{1}$ and $v_{0} = v_{2}$ at same temperature
$\begin{matrix} ∴ \frac{v_{1}}{v_{2}} = \sqrt{\frac{M_{0}}{M_{H}}} \\ M_{0} = 32, M_{H} = 2 \\ ∴ \frac{v_{1}}{v_{2}} = \sqrt{\frac{32}{2}} = \sqrt{16} = 4 . \end{matrix}$

PHXI13:KINETIC THEORY

360253 To what temperature should the hydrogen at $327^{\circ}$ be cooled at constant pressure, so that the root mean square velocity of its molecules become half of its previous value.

1

- 123^{\circ} C

2

23^{\circ} C

3

- 100^{\circ} C

4

0^{\circ} C

Explanation:

$V_{r m s} \propto \sqrt{T}$
$\frac{C_{1}}{C_{2}} = \sqrt{\frac{T_{1}}{T_{2}}} \Rightarrow 2 = \sqrt{\frac{T_{1}}{T_{2}}}$
$\Rightarrow T_{2} = \frac{T_{1}}{4} = \frac{600}{4} = 150 K$
$\Rightarrow T_{2} = - 123^{\circ} C .$

PHXI13:KINETIC THEORY

360254 The speed of 5 molecules of gas (in arbitrary units) are as follows :
$2, 3, 4, 5, 6$ The root mean square speed for these molecules is

1 2.91

2 4.00

3 3.52

4 4.24

Explanation:

Root mean square speed $v_{rms} = \sqrt{\frac{v_{1}^{2} + v_{2}^{2} + v_{3}^{2} + v_{4}^{2} + v_{5}^{2} + \dots}{n}}$
$v_{rms}$ for 5 molecules of gas is, $\begin{aligned} v_{r m s} & = \sqrt{\frac{2^{2} + 3^{2} + 4^{2} + 5^{2} + 6^{2}}{5}} \\ = \sqrt{\frac{90}{5}} = 4.24 units \end{aligned}$ .
So correct option is (4)

PHXI13:KINETIC THEORY

360251 Five gas molecules chosen at random are found to have speeds of $500, 600, 700, 800$ and $900 m / s$ . Then:

1 The rms speed and the average speed are the same.

2 The average speed is

474 m / s

higher than the rms speed.

3 The rms speed is

474 m / s

lower than the average speed.

4 The rms speed is

14 m / s

higher than the average speed.

Explanation:

$v_{r m s} = \sqrt{\frac{\sum v^{2}}{n}}, v_{avg} = \frac{\sum v}{n}$
$v_{a v g} = \frac{500 + 600 + 700 + 800 + 900}{5}$
$= 20 \times 35 = 700 m / s$
$\sqrt{< v^{2} >} = 100 \sqrt{\frac{25 + 36 + 49 + 64 + 81}{5}}$
$= 100 \sqrt{51} = 714 m / s$
So rms speed is $14 m / s$ is higher than average speed.

PHXI13:KINETIC THEORY

360252 The root mean square speed of hydrogen molecule is $v_{1}$ at temperature $T K$ . The most probable speed of Oxygen molecule is $v_{2}$ at same temperature. Then $\frac{v_{1}}{v_{2}}$ is equal to:

1 4

2

2 \sqrt{6}

3

3 \sqrt{6}

4 2

Explanation:

For hydrogen $v_{H} = \sqrt{\frac{3 R T}{M_{H}}} (1)$
For oxygen $v_{0} = \sqrt{\frac{3 R T}{M_{0}}} (2)$
Given $v_{H} = v_{1}$ and $v_{0} = v_{2}$ at same temperature
$\begin{matrix} ∴ \frac{v_{1}}{v_{2}} = \sqrt{\frac{M_{0}}{M_{H}}} \\ M_{0} = 32, M_{H} = 2 \\ ∴ \frac{v_{1}}{v_{2}} = \sqrt{\frac{32}{2}} = \sqrt{16} = 4 . \end{matrix}$

PHXI13:KINETIC THEORY

360253 To what temperature should the hydrogen at $327^{\circ}$ be cooled at constant pressure, so that the root mean square velocity of its molecules become half of its previous value.

1

- 123^{\circ} C

2

23^{\circ} C

3

- 100^{\circ} C

4

0^{\circ} C

Explanation:

$V_{r m s} \propto \sqrt{T}$
$\frac{C_{1}}{C_{2}} = \sqrt{\frac{T_{1}}{T_{2}}} \Rightarrow 2 = \sqrt{\frac{T_{1}}{T_{2}}}$
$\Rightarrow T_{2} = \frac{T_{1}}{4} = \frac{600}{4} = 150 K$
$\Rightarrow T_{2} = - 123^{\circ} C .$

PHXI13:KINETIC THEORY

360254 The speed of 5 molecules of gas (in arbitrary units) are as follows :
$2, 3, 4, 5, 6$ The root mean square speed for these molecules is

1 2.91

2 4.00

3 3.52

4 4.24

Explanation:

Root mean square speed $v_{rms} = \sqrt{\frac{v_{1}^{2} + v_{2}^{2} + v_{3}^{2} + v_{4}^{2} + v_{5}^{2} + \dots}{n}}$
$v_{rms}$ for 5 molecules of gas is, $\begin{aligned} v_{r m s} & = \sqrt{\frac{2^{2} + 3^{2} + 4^{2} + 5^{2} + 6^{2}}{5}} \\ = \sqrt{\frac{90}{5}} = 4.24 units \end{aligned}$ .
So correct option is (4)

PHXI13:KINETIC THEORY

360251 Five gas molecules chosen at random are found to have speeds of $500, 600, 700, 800$ and $900 m / s$ . Then:

1 The rms speed and the average speed are the same.

2 The average speed is

474 m / s

higher than the rms speed.

3 The rms speed is

474 m / s

lower than the average speed.

4 The rms speed is

14 m / s

higher than the average speed.

Explanation:

$v_{r m s} = \sqrt{\frac{\sum v^{2}}{n}}, v_{avg} = \frac{\sum v}{n}$
$v_{a v g} = \frac{500 + 600 + 700 + 800 + 900}{5}$
$= 20 \times 35 = 700 m / s$
$\sqrt{< v^{2} >} = 100 \sqrt{\frac{25 + 36 + 49 + 64 + 81}{5}}$
$= 100 \sqrt{51} = 714 m / s$
So rms speed is $14 m / s$ is higher than average speed.

PHXI13:KINETIC THEORY

360252 The root mean square speed of hydrogen molecule is $v_{1}$ at temperature $T K$ . The most probable speed of Oxygen molecule is $v_{2}$ at same temperature. Then $\frac{v_{1}}{v_{2}}$ is equal to:

1 4

2

2 \sqrt{6}

3

3 \sqrt{6}

4 2

Explanation:

For hydrogen $v_{H} = \sqrt{\frac{3 R T}{M_{H}}} (1)$
For oxygen $v_{0} = \sqrt{\frac{3 R T}{M_{0}}} (2)$
Given $v_{H} = v_{1}$ and $v_{0} = v_{2}$ at same temperature
$\begin{matrix} ∴ \frac{v_{1}}{v_{2}} = \sqrt{\frac{M_{0}}{M_{H}}} \\ M_{0} = 32, M_{H} = 2 \\ ∴ \frac{v_{1}}{v_{2}} = \sqrt{\frac{32}{2}} = \sqrt{16} = 4 . \end{matrix}$

PHXI13:KINETIC THEORY

360253 To what temperature should the hydrogen at $327^{\circ}$ be cooled at constant pressure, so that the root mean square velocity of its molecules become half of its previous value.

1

- 123^{\circ} C

2

23^{\circ} C

3

- 100^{\circ} C

4

0^{\circ} C

Explanation:

$V_{r m s} \propto \sqrt{T}$
$\frac{C_{1}}{C_{2}} = \sqrt{\frac{T_{1}}{T_{2}}} \Rightarrow 2 = \sqrt{\frac{T_{1}}{T_{2}}}$
$\Rightarrow T_{2} = \frac{T_{1}}{4} = \frac{600}{4} = 150 K$
$\Rightarrow T_{2} = - 123^{\circ} C .$

PHXI13:KINETIC THEORY

360254 The speed of 5 molecules of gas (in arbitrary units) are as follows :
$2, 3, 4, 5, 6$ The root mean square speed for these molecules is

1 2.91

2 4.00

3 3.52

4 4.24

Explanation:

Root mean square speed $v_{rms} = \sqrt{\frac{v_{1}^{2} + v_{2}^{2} + v_{3}^{2} + v_{4}^{2} + v_{5}^{2} + \dots}{n}}$
$v_{rms}$ for 5 molecules of gas is, $\begin{aligned} v_{r m s} & = \sqrt{\frac{2^{2} + 3^{2} + 4^{2} + 5^{2} + 6^{2}}{5}} \\ = \sqrt{\frac{90}{5}} = 4.24 units \end{aligned}$ .
So correct option is (4)

PHXI13:KINETIC THEORY

360251 Five gas molecules chosen at random are found to have speeds of $500, 600, 700, 800$ and $900 m / s$ . Then:

1 The rms speed and the average speed are the same.

2 The average speed is

474 m / s

higher than the rms speed.

3 The rms speed is

474 m / s

lower than the average speed.

4 The rms speed is

14 m / s

higher than the average speed.

Explanation:

$v_{r m s} = \sqrt{\frac{\sum v^{2}}{n}}, v_{avg} = \frac{\sum v}{n}$
$v_{a v g} = \frac{500 + 600 + 700 + 800 + 900}{5}$
$= 20 \times 35 = 700 m / s$
$\sqrt{< v^{2} >} = 100 \sqrt{\frac{25 + 36 + 49 + 64 + 81}{5}}$
$= 100 \sqrt{51} = 714 m / s$
So rms speed is $14 m / s$ is higher than average speed.

PHXI13:KINETIC THEORY

360252 The root mean square speed of hydrogen molecule is $v_{1}$ at temperature $T K$ . The most probable speed of Oxygen molecule is $v_{2}$ at same temperature. Then $\frac{v_{1}}{v_{2}}$ is equal to:

1 4

2

2 \sqrt{6}

3

3 \sqrt{6}

4 2

Explanation:

For hydrogen $v_{H} = \sqrt{\frac{3 R T}{M_{H}}} (1)$
For oxygen $v_{0} = \sqrt{\frac{3 R T}{M_{0}}} (2)$
Given $v_{H} = v_{1}$ and $v_{0} = v_{2}$ at same temperature
$\begin{matrix} ∴ \frac{v_{1}}{v_{2}} = \sqrt{\frac{M_{0}}{M_{H}}} \\ M_{0} = 32, M_{H} = 2 \\ ∴ \frac{v_{1}}{v_{2}} = \sqrt{\frac{32}{2}} = \sqrt{16} = 4 . \end{matrix}$

PHXI13:KINETIC THEORY

360253 To what temperature should the hydrogen at $327^{\circ}$ be cooled at constant pressure, so that the root mean square velocity of its molecules become half of its previous value.

1

- 123^{\circ} C

2

23^{\circ} C

3

- 100^{\circ} C

4

0^{\circ} C

Explanation:

$V_{r m s} \propto \sqrt{T}$
$\frac{C_{1}}{C_{2}} = \sqrt{\frac{T_{1}}{T_{2}}} \Rightarrow 2 = \sqrt{\frac{T_{1}}{T_{2}}}$
$\Rightarrow T_{2} = \frac{T_{1}}{4} = \frac{600}{4} = 150 K$
$\Rightarrow T_{2} = - 123^{\circ} C .$

PHXI13:KINETIC THEORY

360254 The speed of 5 molecules of gas (in arbitrary units) are as follows :
$2, 3, 4, 5, 6$ The root mean square speed for these molecules is

1 2.91

2 4.00

3 3.52

4 4.24

Explanation:

Root mean square speed $v_{rms} = \sqrt{\frac{v_{1}^{2} + v_{2}^{2} + v_{3}^{2} + v_{4}^{2} + v_{5}^{2} + \dots}{n}}$
$v_{rms}$ for 5 molecules of gas is, $\begin{aligned} v_{r m s} & = \sqrt{\frac{2^{2} + 3^{2} + 4^{2} + 5^{2} + 6^{2}}{5}} \\ = \sqrt{\frac{90}{5}} = 4.24 units \end{aligned}$ .
So correct option is (4)