Given $K_c = 0.36$ at $2000^@ "C"$ for $"N"_2"O"_4(g) rightleftharpoons "NO"_2(g)$, if the initial concentration of $"NO"_2$ is $"1 M"$, what are the equilibrium concentrations of $"N"_2"O"_4(g)$ and $"NO"_2(g)$?

Question

Given $K_c = 0.36$ at $2000^@ "C"$ for $"N"_2"O"_4(g) rightleftharpoons "NO"_2(g)$, if the initial concentration of $"NO"_2$ is $"1 M"$, what are the equilibrium concentrations of $"N"_2"O"_4(g)$ and $"NO"_2(g)$?

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Md Ariful Islam · Answer 1 · Apr 02, 2024 09:48:30

We're asked to find the equilibrium concentrations of $"N"_2"O"_4$ and $"NO"_2$, given the initial $"NO"_2$ concentration.

The equilibrium constant expression is given by

$K_c = (["NO"_2]^2)/(["N"_2"O"_4]) = ul(0.36)" "$ $(2000color(white)(l)""^"o""C")$

We'll do our I.C.E. chart in the form of bullet points, for fun. Then, our initial concentrations are

INITIAL

$"N"_2"O"_4:$ $0$
$"NO"_2:$ $1$ $M$

According to the coefficients of the reaction, the amount by which $"NO"_2$ decreases is two times as much as the amount by which $"N"_2"O"_4$ increases:

CHANGE

$"N"_2"O"_4:$ $+x$
$"NO"_2:$ $-2x$

And so the final concentrations are

FINAL

$"N"_2"O"_4:$ $x$
$"NO"_2:$ $1$ $M$ $- 2x$

Plugging these into the equilibrium constant expression gives us

$K_c = ((1-2x)^2)/(x) = ul(0.36)" "$ $(2000color(white)(l)""^"o""C"$, excluding units$)$

Now we solve for $x$:

$(4x^2 - 4x + 1)/x = 0.36$

$4x^2 - 4x + 1 = 0.36x$

$4x^2 - 4.36x + 1 = 0$

Use the quadratic equation:

$x = (4.36+-sqrt((4.36)^2 - 4(4)(1)))/(8) = 0.328color(white)(l)"or"color(white)(l)0.762$

If we plug the larger solution in for $x$ in the final $"NO"_2$ concentration $(1-2x)$, we would obtain a worthless negative value. Therefore, we use the smaller solution to calculate the final concentrations:

$color(red)("final N"_2"O"_4) = x = color(red)(ulbar(|stackrel(" ")(" "0.33color(white)(l)M" ")|)$

$color(blue)("final NO"_2) = 1-2x = 1-2(0.328) = color(blue)(ulbar(|stackrel(" ")(" "0.34color(white)(l)M" ")|)$

each rounded to $2$ .

Given $K_c = 0.36$ at $2000^@ "C"$ for $"N"_2"O"_4(g) rightleftharpoons "NO"_2(g)$, if the initial concentration of $"NO"_2$ is $"1 M"$, what are the equilibrium concentrations of $"N"_2"O"_4(g)$ and $"NO"_2(g)$?

1 Answers