Question

nitrogen forms a surprising number of compounds with oxygen. a number of these, often given the collective symbol nox (for nitrogen + x oxygens) are serious contributors to air pollution. they can often be interconverted, sometimes by reaction with oxygen or ozone (o₃) in the air.
an atmospheric scientist decides to study the reaction between nitrogen trioxide and nitrogen monoxide that produces nitrogen dioxide. he fills a stainless steel reaction chamber with 7.3 atm of nitrogen trioxide gas and 5.2 atm of nitrogen monoxide gas and raises the temperature considerably. at equilibrium he measures the mole fraction of nitrogen dioxide to be 0.46.
calculate the pressure equilibrium constant ( k_{p} ) for the equilibrium between nitrogen trioxide, nitrogen monoxide, and nitrogen dioxide at the final temperature of the mixture.
round your answer to 2 significant digits.
( k_{p}= )

Explanation:

Step1: Write the balanced chemical equation

$$\ce{NO3 + NO <=> 2NO2}$$

Step2: Set up the initial and equilibrium pressures

Let \(x\) be the change in pressure.
Initial pressures: \(P_{\ce{NO3}} = 7.3\space atm\), \(P_{\ce{NO}} = 5.2\space atm\), \(P_{\ce{NO2}}=0\space atm\)
Equilibrium pressures: \(P_{\ce{NO3}}=(7.3 - x)\space atm\), \(P_{\ce{NO}}=(5.2 - x)\space atm\), \(P_{\ce{NO2}} = 2x\space atm\)
The total pressure at equilibrium \(P_{total}=(7.3 - x)+(5.2 - x)+2x=12.5\space atm\)
Given mole fraction of \(\ce{NO2},y_{\ce{NO2}} = 0.46\)
Since \(y_{\ce{NO2}}=\frac{P_{\ce{NO2}}}{P_{total}}\), and \(P_{\ce{NO2}} = 2x\), \(P_{total}=12.5\space atm\)
We have \(2x=0.46\times12.5\)

Step3: Solve for \(x\)

\(2x = 5.75\), so \(x = 2.875\)
Equilibrium pressures:
\(P_{\ce{NO3}}=7.3 - 2.875=4.425\space atm\)
\(P_{\ce{NO}}=5.2 - 2.875 = 2.325\space atm\)
\(P_{\ce{NO2}}=5.75\space atm\)

Step4: Calculate \(K_p\)

The formula for \(K_p\) is \(K_p=\frac{P_{\ce{NO2}}^2}{P_{\ce{NO3}}\times P_{\ce{NO}}}\)
Substitute the values: \(K_p=\frac{(5.75)^2}{4.425\times2.325}\)
\(K_p=\frac{33.0625}{10.398125}\approx3.2\)

Answer:

\(3.2\)