Dinitrogen tetroxide decomposes to nitrogen dioxide:
N2O4 (g) ---> 2NO2 (g) Delta H rxn: 55.3 kJ
At 298 K a reaction vessel initially containing .100 atm of N2O4. When equilibrium is reached, 58% of the N2O4 has decomposed to NO2.
-What percentage of N2O4 decomposes at 350 K? Assume that the initial pressure of N2O4 is .100 atm.
To solve this problem, we can use Le Chatelier's principle, which states that if a system at equilibrium is subjected to a change in temperature, pressure, or concentration, the system will shift its equilibrium position to counteract the effect of that change.
In this case, we are asked to find the percentage of N2O4 that decomposes at a temperature of 350 K. Let's break down the steps to solve this problem:
Step 1: Calculate the initial concentration of N2O4
Given that the initial pressure of N2O4 is 0.100 atm, we need to convert this pressure to concentration using the ideal gas law. The ideal gas law equation is PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature in Kelvin.
Since we are given the initial pressure and assume the volume and temperature remain constant, we can rewrite the ideal gas law as P = nRT/V. Given that R is a constant, we can use the equation P = [N2O4]RT/V, where [N2O4] represents the concentration of N2O4.
Step 2: Calculate the equilibrium concentration of N2O4 when 58% has decomposed to NO2
Since 58% of the original N2O4 has decomposed, this means that 42% remains unchanged at equilibrium. We can use this information to calculate the equilibrium concentration of N2O4.
Step 3: Use the equilibrium concentrations to find the percentage of N2O4 that decomposes at 350 K
Given that equilibrium is reached at 298 K and we want to find the percentage of N2O4 that decomposes at 350 K, we can assume that the equilibrium constant (K) remains constant with changing temperature. With this assumption, we can set up a ratio comparing the equilibrium concentrations of N2O4 at different temperatures.
Let's go through each step:
Step 1: Calculate the initial concentration of N2O4
Using the ideal gas law equation, P = [N2O4]RT/V, and assuming a constant volume:
[N2O4] = P / (RT/V)
Given that P = 0.100 atm, T = 298 K, R = 0.0821 L·atm/(mol·K), and V is constant, you can calculate the initial concentration of N2O4.
Step 2: Calculate the equilibrium concentration of N2O4 when 58% has decomposed to NO2
Since 58% of N2O4 has decomposed, this means that 42% remains unchanged at equilibrium. We can assume that the total pressure is still 0.100 atm (the initial pressure), and the sum of the partial pressures of N2O4 and NO2 is equal to the total pressure.
Let's assume x is the concentration of NO2 formed. Then, the change in pressure due to the decomposition of N2O4 is 0.100 - x atm. Since 2 moles of NO2 are formed for every 1 mole of N2O4 decomposed, the pressure of NO2 is 2x atm.
Based on the stoichiometry of the reaction, we can set up an expression for the equilibrium constant (K) expressed in terms of pressure (as partial pressures):
K = (NO2)^2 / (N2O4)
Substituting the partial pressures, we get:
K = (2x)^2 / (0.100 - x)
Since we know that 58% of N2O4 has decomposed, we can set up the equation:
0.42 * 0.100 = (2x)^2 / (0.100 - x)
Solving this equation will give us the equilibrium concentration of N2O4 (42% remaining) and the concentration of NO2 formed (58% decomposed).
Step 3: Use the equilibrium concentrations to find the percentage of N2O4 that decomposes at 350 K
Now that we have the equilibrium concentration of N2O4 when 58% has decomposed at 298 K, we can use the ratio of equilibrium concentrations to find the percentage of N2O4 that decomposes at 350 K.
Let's assume y is the equilibrium concentration of N2O4 at 350 K. We can set up the following ratio:
Equilibrium concentration of N2O4 at 350 K / Initial concentration of N2O4 = Equilibrium concentration of N2O4 at 298 K / Initial concentration of N2O4
Simplifying this equation will give us the percentage of N2O4 that decomposes at 350 K.
To determine the percentage of N2O4 that decomposes at 350 K, we will use the equilibrium constant (Kp) for the reaction.
The equilibrium constant expression for the given reaction is:
Kp = ([NO2]^2) / [N2O4]
We know that at 298 K, 58% of N2O4 has decomposed to NO2. This means that 42% of N2O4 remains.
Therefore, at 298 K: [N2O4] = 0.42 * 0.100 atm = 0.042 atm
And at 298 K: [NO2] = 0.58 * 0.100 atm = 0.058 atm
Next, we'll use these concentrations as initial concentrations at 350 K. We need to determine the equilibrium concentration of N2O4 at 350 K. Let's call it x.
At equilibrium, the concentration of NO2 will be twice that of N2O4, so we can write:
[NO2] = 2x
[N2O4] = 0.042 - x
Now, we can substitute these expressions into the equilibrium constant expression and solve for x:
Kp = ([NO2]^2) / [N2O4]
Kp = (2x)^2 / (0.042 - x)
Given that Kp at 298 K is 0.100:
0.100 = (2x)^2 / (0.042 - x)
Simplifying the equation:
0.100 * (0.042 - x) = 4x^2
0.0042 - 0.1x = 4x^2
Rearranging the equation:
4x^2 + 0.1x - 0.0042 = 0
Now, we can solve this quadratic equation. Using a suitable numerical method, the value of x can be approximated as x ≈ 0.0388.
Since [N2O4] at 350 K is 0.042 - x, we can calculate the percentage of N2O4 that decomposes at 350 K:
Percentage of N2O4 decomposed = ([N2O4] initial - [N2O4] equilibrium) / [N2O4] initial * 100
Percentage of N2O4 decomposed = (0.042 - 0.0388) / 0.042 * 100
Percentage of N2O4 decomposed ≈ 7.62%
Therefore, approximately 7.62% of N2O4 decomposes at 350 K.