Question

Question #b23fd

Answer 1

`K_p = "0.0407 atm"^(-1)`

Explanation:

You can derive an expression for `K_p` using the ideal gas law and `K_c`.

Your equilibrium reaction looks like this

`"PbCl"_text(3(g]) + "Cl"_text(2(g]) rightleftharpoons "PbCl"_text(5(g])`

By definition, what is `K_c` equal to?

`K_c = ( ["PbCl"_5])/( ["PbCl"_3] * ["Cl"_2])`

The equilibrium concentrations featured in the expression of `K_c` can be written as a ratio between the number of moles of each species that takes part in the rection and the volume of the container in which said reaction takes place.

If you take `V` to be the volume of the container, you can say that

`[PbCl_3] = n_text(PbCl_3)/V" "` and `" "[Cl_(2)] = n_(Cl_2)/V`

`[PbCl_5] = n_(PbCl_5)/V`

Now take a look at the expression for `K_p`

`K_p = ( (PbCl_5))/((PbCl_3) * (Cl_2))`

The partial pressure of each component of the mixture can be expressed using the ideal gas law

`PV = nRT implies P = (nRT)/V = n/V * RT`

This means that you have

`(PbCl_5) = underbrace(n_(PbCl_5)/V)_(color(blue)(=[PbCl_5])) * RT`

`(Cl_2) = underbrace(n_(Cl_2)/V)_(color(blue)(=[Cl_2])) * RT`

`(PbCl_3) = underbrace(n_(PbCl_3)/V)_(color(blue)(=[PbCl_3])) * RT`

This means that you can write

`K_p = ( [PbCl_5] * color(red)(cancel(color(black)(RT))))/( [Cl_2] * color(red)(cancel(color(black)(RT))) * [PbCl_3] * RT)`

`K_p = overbrace(( [PbCl_5])/( [PbCl_3] * [Cl_2]))^(color(green)(=K_c)) * 1/(RT)`

`K_p = K_c * (RT)^(-1)`

If you take `R = 0.082("atm" * "L")/("mol" * "K")`, and knowing that your reaction takes plac at `"500 K"`, you get

`K_p = K_c * [0.082("atm" * "L")/("mol" * color(red)(cancel(color(black)(K)))) * 500color(red)(cancel(color(black)(K)))]^(-1)`

`K_p = K_c * 0.0244"mol"/("atm" * "L")`

What about the units?

Well, if you look at the expression for `K_c`, you will see that it is not unitless, because

`K_c = ( [PbCl_5]color(red)(cancel(color(black)("M"))))/( [PbCl_3]color(red)(cancel(color(black)("M"))) * [Cl_2]" M") = [1/"M"]`

SInce `"M"` is `"mol"/"L"`, you get that

`K_p = 1.67 color(red)(cancel(color(black)("L")))/color(red)(cancel(color(black)("mol"))) * 0.0244 color(red)(cancel(color(black)("mol")))/("atm" * color(red)(cancel(color(black)("L")))) = color(green)("0.0407 atm"^(-1))`