The root mean square speed of the molecules of a diatomic gas is v. When the temperature is doubled, the molecules dissociate into two atoms. The new root mean square speed of the atom is

Correct Option: The correct answer is 2v.

Step-by-Step Explanation:

The root mean square (rms) speed of gas molecules is given by the formula:

$v_{\mathrm{rms}}=\sqrt{\frac{3RT}{M}}$
where:
- $R$ is the universal gas constant,
- $T$ is the absolute temperature, and
- $M$ is the molar mass of the gas particles.

Let the initial absolute temperature be $T$ and the molar mass of the diatomic gas molecule be $M$. The initial rms speed is:

$v=\sqrt{\frac{3RT}{M}}$

According to the problem, two changes occur:
1. The absolute temperature is doubled:
$T^{\prime }=2T$
2. The diatomic molecules dissociate into two individual atoms. This means the mass of each new particle (atom) is half the mass of the original molecule, so the new molar mass becomes:
$M^{\prime }=\frac{M}{2}$

Now, let's write the expression for the new root mean square speed ($v^{\prime }$) of the dissociated atoms:

$v^{\prime }=\sqrt{\frac{3R{T}^{\prime }}{M^{\prime }}}$

Substitute the values of $T^{\prime }$ and $M^{\prime }$ into the new equation:

$v^{\prime }=\sqrt{\frac{3R\left(2T\right)}{M/2}}$

Simplify the expression inside the square root:

$v^{\prime }=\sqrt{4·\frac{3RT}{M}}=2\sqrt{\frac{3RT}{M}}$

Since $\sqrt{\frac{3RT}{M}}=v$, we have:

$v^{\prime }=2v$

Therefore, the new root mean square speed of the atoms is 2v.