10. Gas [tex]\( P \)[/tex] diffuses through a porous material at a rate of [tex]\( 12 \, \text{cm}^3 \, \text{s}^{-1} \)[/tex], whereas gas [tex]\( Q \)[/tex] diffuses through the same material at a rate of [tex]\( 7.2 \, \text{cm}^3 \, \text{s}^{-1} \)[/tex]. Given that the molar mass of [tex]\( P \)[/tex] is 16, calculate the molar mass of [tex]\( Q \)[/tex].

[tex]\( \text{(3 marks)} \)[/tex]



Answer :

Certainly! Let's solve this problem step by step.

Step 1: Understanding Graham's Law of Effusion

Graham's Law of Effusion states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass. Mathematically, it can be written as:

[tex]\[ \frac{\text{Rate}_P}{\text{Rate}_Q} = \sqrt{\frac{\text{Molar mass}_Q}{\text{Molar mass}_P}} \][/tex]

Step 2: Plugging in the Given Data

We are given:
- The rate of diffusion of gas [tex]\(P\)[/tex] ([tex]\(\text{Rate}_P\)[/tex]) = [tex]\(12 \, \text{cm}^3/\text{s}\)[/tex]
- The rate of diffusion of gas [tex]\(Q\)[/tex] ([tex]\(\text{Rate}_Q\)[/tex]) = [tex]\(7.2 \, \text{cm}^3/\text{s}\)[/tex]
- The molar mass of gas [tex]\(P\)[/tex] ([tex]\(\text{Molar mass}_P\)[/tex]) = [tex]\(16 \, \text{g/mol}\)[/tex]

Step 3: Applying Graham's Law

Using Graham's Law, we can write:

[tex]\[ \frac{12}{7.2} = \sqrt{\frac{\text{Molar mass}_Q}{16}} \][/tex]

Step 4: Simplifying the Equation

First, calculate the ratio on the left side:

[tex]\[ \frac{12}{7.2} = \frac{5}{3} \][/tex]

This gives us:

[tex]\[ \frac{5}{3} = \sqrt{\frac{\text{Molar mass}_Q}{16}} \][/tex]

Step 5: Solving for [tex]\(\text{Molar mass}_Q\)[/tex]

Next, square both sides to eliminate the square root:

[tex]\[ \left(\frac{5}{3}\right)^2 = \frac{\text{Molar mass}_Q}{16} \][/tex]

Calculate [tex]\(\left(\frac{5}{3}\right)^2\)[/tex]:

[tex]\[ \left(\frac{5}{3}\right)^2 = \frac{25}{9} \][/tex]

So we have:

[tex]\[ \frac{25}{9} = \frac{\text{Molar mass}_Q}{16} \][/tex]

Now, solve for [tex]\(\text{Molar mass}_Q\)[/tex] by multiplying both sides by 16:

[tex]\[ \text{Molar mass}_Q = \frac{25}{9} \times 16 \][/tex]

Calculate the value:

[tex]\[ \text{Molar mass}_Q = \frac{25 \times 16}{9} = \frac{400}{9} \approx 44.44 \, \text{g/mol} \][/tex]

Conclusion

The molar mass of gas [tex]\(Q\)[/tex] is approximately [tex]\(44.44 \, \text{g/mol}\)[/tex].