The root mean square (R.M.S) speed V of the molecules of an ideal gas is given by the expression, v = sqrt(3RT/M) and v = sqrt(3KT/m) where R is universal gas constant, T is the absolute Kelvin temperature, m is the molar mass, K is Boltzmann’s constant and M is the molecular mass. The R.M.S speed of oxygen molecules O2 at temperature T1 is V1. When the temperature is doubled, if the oxygen molecules are dissociated into atomic oxygen, what will be R.M.S speed of oxygen atoms? (Treat the gas as ideal)

To determine the root mean square (R.M.S) speed of oxygen atoms after the temperature is doubled and the oxygen molecules are dissociated, we can follow a systematic approach using the formulas provided. Let's break this down step by step.

Understanding the Initial Conditions

Initially, we have diatomic oxygen (O₂) at a temperature T₁. The R.M.S speed of the oxygen molecules is given by the formula:

V₁ = sqrt(3RT₁/M)

Here, R is the universal gas constant, T₁ is the initial temperature, and M is the molar mass of O₂.

Effect of Temperature Change

When the temperature is doubled, the new temperature becomes T₂ = 2T₁. The R.M.S speed of the oxygen molecules at this new temperature can be calculated using the same formula:

V₂ = sqrt(3R(2T₁)/M)

By simplifying this expression, we find:

V₂ = sqrt(6RT₁/M) = sqrt(2) * V₁

Dissociation of Oxygen Molecules

When the temperature is increased, the diatomic oxygen molecules (O₂) dissociate into atomic oxygen (O). The molar mass of atomic oxygen (O) is half that of molecular oxygen (O₂), so:

m = M/2

Now, we need to find the R.M.S speed of the atomic oxygen at the new temperature T₂. The formula for the R.M.S speed of the atomic oxygen becomes:

V_O = sqrt(3RT₂/m)

Substituting m with M/2, we have:

V_O = sqrt(3RT₂/(M/2)) = sqrt(6RT₂/M)

Calculating the Final R.M.S Speed

Now, substituting T₂ = 2T₁ into the equation:

V_O = sqrt(6R(2T₁)/M) = sqrt(12RT₁/M)

We can express this in terms of V₁:

V_O = sqrt(4) * V₁ = 2V₁

Final Result

Thus, the R.M.S speed of the oxygen atoms after the temperature is doubled and the molecules are dissociated will be:

V_O = 2V₁

This means that the R.M.S speed of the atomic oxygen is twice that of the original R.M.S speed of the diatomic oxygen molecules at the initial temperature.