You work with the following system:
2X(g) <--> Y(g)
At, equilibrium, [Y]=3[X]. If K=10, calculate the [Y] at equilibrium.
K = 10 = (Y)/(X)^2 = 3X/(X)^2 = 3/X
10 = 3/X; X = 3/10 = 0.30
To solve this question, we can use the formula for equilibrium constant (K):
K = [Y]^y / [X]^x
Given that K = 10 and [Y] = 3[X], we can substitute these values into the formula:
10 = (3[X])^y / [X]^x
To simplify this equation, we need to determine the values of x and y. Looking at the balanced equation, we have:
2X(g) <--> Y(g)
This means that the coefficients in front of X and Y are 2 and 1, respectively. Therefore, x = 2 and y = 1.
Now, we can substitute these values into the equation:
10 = (3[X])^1 / [X]^2
Simplifying further, we have:
10 = 3[X] / [X]^2
To eliminate the fraction, we multiply both sides of the equation by [X]^2:
10[X]^2 = 3[X]
Now, we can rearrange the equation:
10[X]^2 - 3[X] = 0
Factoring out an [X], we get:
[X] * (10[X] - 3) = 0
From this equation, we have two possible solutions:
1) [X] = 0
2) 10[X] - 3 = 0
Solving the second equation for [X], we get:
10[X] = 3
[X] = 3/10
Now that we have the value of [X], we can calculate the value of [Y] using the given relationship:
[Y] = 3[X] = 3 * (3/10) = 9/10
Therefore, the [Y] at equilibrium is 9/10.
To calculate the concentration of Y at equilibrium, we can use the equilibrium constant (K) expression and the given information.
The equilibrium constant expression for the reaction is:
K = ([Y]/[X]^2)
Given that [Y] = 3[X] at equilibrium, we can substitute this into the expression:
K = ([Y]/[X]^2) = (3[X])/[X]^2 = 3/[X]
Since the equilibrium constant (K) value is given as 10, we can set up the following equation:
10 = 3/[X]
To solve for [X], we can rearrange the equation as follows:
[X] = 3/10
Now that we have the concentration of X at equilibrium, we can use the given relationship between [X] and [Y] to find the concentration of Y. Since [Y] = 3[X], we can substitute the value of [X] we just calculated:
[Y] = 3 * (3/10) = 9/10
Therefore, the concentration of Y at equilibrium is 9/10.