The task is to order the relative diffusion rates of the given gases from the slowest to the fastest diffusion rate. We follow Graham's law of effusion, which states that the rate of effusion of a gas (and consequently its rate of diffusion) is inversely proportional to the square root of its molar mass.
Given gases and their labels:
1. \(Cl_2\)
2. \(F_2\)
3. \(He\)
4. \(Kr\)
5. \(N_2\)
6. \(O_2\)
To determine the order from the slowest to the fastest diffusion rate, we should consider the molar mass of each gas. Generally, gases with higher molar mass will diffuse slower than those with lower molar mass. Here's the given detailed order based on their respective molar masses:
1. \(Cl_2\) - labeled as 6
2. \(Kr\) - labeled as 3
3. \(O_2\) - labeled as 4
4. \(F_2\) - labeled as 1
5. \(N_2\) - labeled as 2
6. \(He\) - labeled as 5
Thus, arranging these in the order of diffusion rate from slowest to fastest, we get:
[tex]\[ He < N_2 < O_2 < F_2 < Cl_2 < Kr \][/tex]
When converted to their labels, the order is:
[tex]\[ 5-2-4-1-6-3 \][/tex]
So, the final ordered list from slowest to fastest diffusion rate is:
[tex]\[ 5^{\text{th }} - 2^{\text{nd }} - 4^{\text{th }} - 1^{\text{st }} - 6^{\text{th }} - 3^{\text{rd }} \][/tex]
Therefore, the numbered order from slowest to fastest diffusion rate is:
[tex]\[ \boxed{5, 2, 4, 1, 6, 3} \][/tex]
