Question

In an equation element `"A"(s)rightleftharpoons 2"B"(g)+"C"(g)+3"D"(g)`. If the partial pressure of `"D"` at equilibrium is `P_1`, calculate the partial pressures of `"B"` and `"C"` . Also, calculate the value of `K_p` in terms of `P_1`?

Answer 1

Here's what I got.

Explanation:

You know that you have

`"A"_ ((s)) rightleftharpoons color(red)(2)"B"_ ((g)) + "C"_ ((g)) + color(blue)(3)"D"_ ((g))`

This tells you that for every `color(blue)(3)` moles of `"D"` that are produced by the reaction, you also get

• `color(red)(2)` `"moles of B"`
• `"1 mole of C"`

Now, when volume and temperature are kept constant, the pressure of a gas is directly proportional to the number of moles present in the sample.

Consequently, you can say that the partial pressure of a gas that's part of a gaseous mixture depends on the mole fraction of said gas and on the total pressure of the mixture `->` this is known as Dalton's Law of Partial Pressures.

If you take `P_"total"` to be the total pressure of the mixture, i.e. of `"B"`, `"C"`, and `"D"`, you can say that the partial pressure of `"D"`, let's say `P_"D"`, is equal to

`P_"D" = overbrace( (color(blue)(3)color(red)(cancel(color(black)("moles"))))/((color(red)(2) + 1 + color(blue)(3))color(red)(cancel(color(black)("moles")))))^(color(purple)("the mole fraction of D")) * P_"total"`

`P_"D"= 3/6 * P_"total"`

`P_"D" = 1/2 * P_"total"`

Rearrange to get the value of `P_"total"` in terms of `P_1` and use the fact tha `P_"D" = P_1`

`P_"total" = 2 * P_1`

Next, use this value to find an expression for the partial pressure of `"B"`, let's say `P_"B"`, and the partial pressure of `"C"`, let's say `P_"C"`.

You will have

`P_"B" = overbrace( (color(red)(2)color(red)(cancel(color(black)("moles"))))/((color(red)(2) + 1 + color(blue)(3))color(red)(cancel(color(black)("moles")))))^(color(purple)("the mole fraction of B")) * P_"total"`

`P_"B" = 2/6 * P_"total"`

`P_"B" = 1/3 * P_"total"`

And so

`P_"B" = 1/3 * (2 * P_1) = 2/3 * P_1`

Similarly, you will have

`P_"C" = overbrace( (1color(red)(cancel(color(black)("mole"))))/((color(red)(2) + 1 + color(blue)(3))color(red)(cancel(color(black)("moles")))))^(color(purple)("the mole fraction of C")) * P_"total"`

`P_"C" = 1/6 * P_"total"`

And so

`P_"C" = 1/6 * (2 * P_1) = 1/3 * P_1`

You can thus say that you have

`{(P_"B" = 2/3 * P_1), (P_"C" = 1/3 * P_1), (P_"D" = P_1) :}`

Finally, the equilibrium constant for this equilibrium can be written using the equilibrium partial pressures of the three gases

`K_p = (P_"B")^color(red)(2) * P_"C" * (P_"D")^color(blue)(3)`

Plug in your values to find

`K_p = (2/3 * P_1)^color(red)(2) * 1/3 * P_1 * (P_1)^color(blue)(3)`

`color(darkgreen)(ul(color(black)(K_p = 4/27 * (P_1)^6)))`