Question

100 g of acetylene (`"C"_2"H"_2`) gas is placed in a 20.0 L container at 298 K. (i) Use the ideal gas equation to calculate the pressure? (ii) Use van der Waals equation to calculate the pressure? (`a="4.390 L"^2cdot"atm/mol"^2`, `b="0.0513 L/mol"`)

Answer 1

(i) `p_text(ideal) = "4.70 atm"`: (ii) `p_text(vdW) = "4.58 atm"`

This tells you that the van der Waals equation of state picks up on the attractive interactions in acetylene that the ideal gas law assumes is not present.

Explanation:

`"Moles of C"_2"H"_2 = 100 color(red)(cancel(color(black)("g C"_2"H"_2))) × ("1 mol C"_2"H"_2)/(26.04 color(red)(cancel(color(black)("C"_2"H"_2)))) = "3.840 mol C"_2"H"_2`

(a) Ideal gas

The equation for the Ideal Gas Law is

`color(blue)(bar(ul(|color(white)(a/a)pV = nRTcolor(white)(a/a)|)))" "`

Then,

`p = (nRT)/V`

`n ="3.840 mol"`
`R = "0.082 06 L·atm·K"^"-1""mol"^"-1"`
`T = "298 K"`
`V = "20.0 L"`

`p = (3.840 color(red)(cancel(color(black)("mol"))) × "0.082 06" color(red)(cancel(color(black)("L")))·"atm"·color(red)(cancel(color(black)("K"^"-1""mol"^"-1"))) × 298 color(red)(cancel(color(black)("K"))))/(20.0 color(red)(cancel(color(black)("L")))) = "4.70 atm"`

The pressure predicted by the Ideal Gas Law is 4.70 atm.

(b) van der Waals Equation

The van der Waals equation is

`color(blue)(bar(ul(|color(white)(a/a) (P + (n^2a)/V^2)(V - nb) = nRTcolor(white)(a/a)|)))" "`

`P + (n^2a)/V^2 = (nRT)/(V - nb)`

`P = (nRT)/(V - nb)- (n^2a)/V^2`

For this problem,

`n = "3.840 mol"`
`R = "0.082 06"color(white)(l)"L·atm·K"^"-1""mol"^"-1"`
`T = "298 K"`
`V = "20.0 L"`
`a = "4.390 atm·L"^2"mol"^"-2"`
`b = "0.0513 L·mol"^"-1"`

`P = (nRT)/(V-nb) – (n^2a)/V^2`

`= (3.840 color(red)(cancel(color(black)("mol"))) × "0.082 06 atm"color(red)(cancel(color(black)("L·""K"^"-1""mol"^"-1")))× 298 color(red)(cancel(color(black)("K"))))/(20.0 color(red)(cancel(color(black)("L"))) – 3.840 color(red)(cancel(color(black)("mol")))× 0.0513 color(red)(cancel(color(black)("L·mol"^(-1))))) - ((3.840 color(red)(cancel(color(black)("mol"))))^2 × "4.390 atm" color(red)(cancel(color(black)("L"^2"mol"^"-2"))))/(20.0 color(red)(cancel(color(black)("L"))))^2`
`= "93.90 atm"/19.80 - "0.1618 atm "= "4.742 atm" - "0.1618 atm"= "4.58 atm"`

The pressure predicted by the van der Waals equation is 4.58 atm, lower than for the ideal gas law.