Question 957: Thermodynamics & KTG

Question

The rms speed of oxygen molecule in a gas is $u$, If the temperature is doubled and the molecules dissociates into two atoms, the rms speed will be

Accepted Answer

To determine the new root mean square (rms) speed of the oxygen molecules when the temperature is doubled and they dissociate into atoms, we can use the formula for the rms speed:
$$ v_{	ext{rms}} = \sqrt{\frac{3kT}{m}} $$
where:
$ v_{	ext{rms}} $ is the root mean square speed,
$ k $ is the Boltzmann constant,
$ T $ is the temperature,
$ m $ is the mass of a molecule.
Initially, the oxygen molecule ($ O_2 $) has a mass $ m $, and its rms speed is $ u $. Therefore:
$$ u = \sqrt{\frac{3kT}{M}} $$
where $ M $ is the molar mass of the oxygen molecule $ O_2 $.
When the temperature is doubled ($ T' = 2T $), and oxygen molecules dissociate into atoms, the mass of an individual oxygen atom is $ \frac{M}{2} $. The new rms speed $ u' $ can be calculated as:
$$ u' = \sqrt{\frac{3k(2T)}{\frac{M}{2}}} $$
This simplifies to:
$$ u' = \sqrt{\frac{6kT}{\frac{M}{2}}} = \sqrt{\frac{6kT \cdot 2}{M}} = \sqrt{\frac{12kT}{M}} = \sqrt{4 \cdot \frac{3kT}{M}} $$
The term $\frac{3kT}{M}$ corresponds to the square of the initial rms speed, $ u^2 $:
$$ u' = \sqrt{4} \cdot u = 2u $$
Thus, the new rms speed will be $ 2u $. Therefore, the correct answer is:
Option C: 2u
.

Answer

4u

Answer

u

Answer

2u

Answer

u$\sqrt2$

Thermodynamics & KTG

The rms speed of oxygen molecule in a gas is u, If the temperature is doubled and the molecules dissociates into two atoms, the rms speed will be

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