An unknown gaseous hydrocarbon X belongs to the alkane homologous series. Under identical … — JAMB UTME

An unknown gaseous hydrocarbon $X$ belongs to the alkane homologous series. Under identical conditions of temperature and pressure, a given volume of oxygen gas ( $O_2$ ) diffuses through a micro-porous barrier in $40\text{ s}$ , whereas the exact same volume of gas $X$ requires $60\text{ s}$ to diffuse through the same barrier. What is the molecular formula of hydrocarbon $X$ ? $[H = 1,\ C = 12,\ O = 16]$

$C_5H_{12}$ Answer
B
$C_4H_{10}$
C
$C_3H_8$
D
$C_2H_6$

Answer

The molecular formula of hydrocarbon X is

C_5H_{12}

According to Graham's Law of Diffusion, the time taken for equal volumes of two gases to diffuse under identical conditions is directly proportional to the square root of their respective molar masses:

t_X / t_{O_2} = \sqrt{M_X / M_{O_2}}

. Given

t_X = 60\text{ s}

t_{O_2} = 40\text{ s}

, and

M_{O_2} = 32\text{ g/mol}

, we have

60/40 = \sqrt{M_X / 32}

. Squaring both sides yields

(3/2)^2 = 9/4 = M_X / 32

, which solves to

M_X = 72\text{ g/mol}

. Matching this to the alkane formula

C_n H_{2n+2} = 14n + 2 = 72

gives

n = 5

, identifying the gas as pentane,

C_5H_{12}

Step-by-Step Solution

Calculate the molar mass of oxygen gas (

O_2

)

M_{O_2} = 2 \times 16 = 32\text{ g/mol}

Oxygen exists as a diatomic molecule under standard conditions.

Apply Graham's Law of Diffusion relating diffusion time (

t

) and molar mass (

M

) for equal volumes

\frac{t_X}{t_{O_2}} = \sqrt{\frac{M_X}{M_{O_2}}}

Time required for diffusion of equal gas volumes is directly proportional to the square root of the molar mass.

Substitute known diffusion times (

t_X = 60\text{ s}

t_{O_2} = 40\text{ s}

) into the equation and solve for

M_X

\frac{60}{40} = \sqrt{\frac{M_X}{32}} \implies \frac{3}{2} = \sqrt{\frac{M_X}{32}} \implies \left(\frac{3}{2}\right)^2 = \frac{M_X}{32} \implies \frac{9}{4} = \frac{M_X}{32} \implies M_X = 72\text{ g/mol}

Squaring both sides removes the radical and allows direct algebraic isolation of

M_X

Determine the molecular formula using the alkane general formula

C_n H_{2n+2}

12n + (2n + 2) = 72 \implies 14n + 2 = 72 \implies 14n = 70 \implies n = 5

Each carbon atom contributes

12\text{ g/mol}

and each hydrogen atom contributes

1\text{ g/mol}

Write the molecular formula for

n = 5

C_5H_{12}

Substituting

n = 5

into

C_n H_{2n+2}

gives

C_5H_{12}

(pentane).

Key Concept

Graham's Law of Diffusion and Effusion

Estimated Time:2m 0s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic