Five hundred(500)cm³ of hydrogen gas (H₂) diffused through a small hole in 50 seconds. 853 cm³ of a gas P diffused through the same hole under the same conditions of temperature and pressure in 400 seconds.Calculate the relative molecular mass (R.M.M) of gas P. (3mks)

Kiwango cha usambazaji wa hidrojeni ($Rate_{H_2}$):

$$Rate_{H_2} = \frac{500\, \text{cm}^3}{50\, \text{s}} = 10\, \text{cm}^3/\text{s}$$

Kiwango cha usambazaji wa gesi P ($Rate_P$):

$$Rate_P = \frac{853\, \text{cm}^3}{400\, \text{s}} = 2.1325\, \text{cm}^3/\text{s}$$

Uzito wa molekuli wa hidrojeni ($M_{H_2}$) ni $2 \times 1 = 2\, \text{g/mol}$.

$$\frac{Rate_{H_2}}{Rate_P} = \sqrt{\frac{M_P}{M_{H_2}}}$$ $$\frac{10\, \text{cm}^3/\text{s}}{2.1325\, \text{cm}^3/\text{s}} = \sqrt{\frac{M_P}{2}}$$ $$4.6894 = \sqrt{\frac{M_P}{2}}$$

$$(4.6894)^2 = \frac{M_P}{2}$$ $$21.9805 = \frac{M_P}{2}$$ $$M_P = 21.9805 \times 2$$ $$M_P = 43.961$$

Uzito wa molekuli wa gesi P ni takriban $\boxed{\text{44}}$.

$$\frac{Rate_{H_2}}{Rate_{O_2}} = \sqrt{\frac{M_{O_2}}{M_{H_2}}}$$ $$\frac{10}{Rate_{O_2}} = \sqrt{\frac{32}{2}}$$ $$\frac{10}{Rate_{O_2}} = \sqrt{16}$$ $$\frac{10}{Rate_{O_2}} = 4$$

$$Rate_{O_2} = \frac{10}{4} = 2.5\, \text{cm}^3/\text{s}$$

Kiasi cha $O_2$ = $Rate_{O_2} \times Time$
Kiasi cha $O_2$ = $2.5\, \text{cm}^3/\text{s} \times 20\, \text{s}$
Kiasi cha $O_2$ = $\boxed{\text{50\, cm}^3}$

$$\frac{Rate_{H_2}}{Rate_{C_5H_{12}}} = \sqrt{\frac{M_{C_5H_{12}}}{M_{H_2}}}$$ $$\frac{10}{Rate_{C_5H_{12}}} = \sqrt{\frac{72}{2}}$$ $$\frac{10}{Rate_{C_5H_{12}}} = \sqrt{36}$$ $$\frac{10}{Rate_{C_5H_{12}}} = 6$$

$$Rate_{C_5H_{12}} = \frac{10}{6} = 1.6667\, \text{cm}^3/\text{s}$$

Kiasi cha $C_5H_{12}$ = $Rate_{C_5H_{12}} \times Time$
Kiasi cha $C_5H_{12}$ = $1.6667\, \text{cm}^3/\text{s} \times 30\, \text{s}$
Kiasi cha $C_5H_{12}$ = $\boxed{\text{50\, cm}^3}$

$$\frac{Rate_{H_2}}{Rate_{C_7H_{16}}} = \sqrt{\frac{M_{C_7H_{16}}}{M_{H_2}}}$$ $$\frac{10}{Rate_{C_7H_{16}}} = \sqrt{\frac{100}{2}}$$ $$\frac{10}{Rate_{C_7H_{16}}} = \sqrt{50}$$ $$\frac{10}{Rate_{C_7H_{16}}} \approx 7.071$$

$$Rate_{C_7H_{16}} = \frac{10}{7.071} \approx 1.4142\, \text{cm}^3/\text{s}$$

Kiasi cha $C_7H_{16}$ = $Rate_{C_7H_{16}} \times Time$
Kiasi cha $C_7H_{16}$ = $1.4142\, \text{cm}^3/\text{s} \times 28\, \text{s}$
Kiasi cha $C_7H_{16}$ = $\boxed{\text{39.6\, cm}^3}$

Kiwango cha usambazaji wa gesi X ($Rate_X$):

$$Rate_X = \frac{200\, \text{cm}^3}{20\, \text{s}} = 10\, \text{cm}^3/\text{s}$$

Kiwango cha usambazaji wa hidrojeni ($Rate_{H_2}$):

$$Rate_{H_2} = \frac{200\, \text{cm}^3}{5\, \text{s}} = 40\, \text{cm}^3/\text{s}$$

$$\frac{Rate_X}{Rate_{H_2}} = \sqrt{\frac{M_{H_2}}{M_X}}$$ $$\frac{10\, \text{cm}^3/\text{s}}{40\, \text{cm}^3/\text{s}} = \sqrt{\frac{2}{M_X}}$$ $$\frac{1}{4} = \sqrt{\frac{2}{M_X}}$$

$$\left(\frac{1}{4}\right)^2 = \frac{2}{M_X}$$ $$\frac{1}{16} = \frac{2}{M_X}$$ $$M_X = 2 \times 16$$ $$M_X = 32$$

Uzito wa molekuli wa gesi X ni $\boxed{\text{32}}$.