- Identify the expression for the equilibrium constant based on the balanced equation: \(K_c = \frac{[NH_3]^2}{[N_2][H_2]^3}\).
- Substitute the given molar concentrations: \([NH_3] = 0.105 \, M\), \([N_2] = 1.1 \, M\), and \([H_2] = 1.50 \, M\).
- Calculate the numerical value of the expression: \(K_c = \frac{0.105^2}{1.1 \times 1.5^3} \approx 0.00297\).
- Conclude that the correct value from the given options is \(\boxed{0.0030}\).
Explanation
- Defining the Equilibrium Constant Expression
The reaction for the synthesis of ammonia is given by:
\[N_2(g) + 3H_2(g) \rightleftharpoons 2NH_3(g)\]
The equilibrium constant \(K_c\) for a chemical reaction \(aA + bB \rightleftharpoons cC + dD\) is defined as the product of the concentrations of the products raised to their stoichiometric coefficients divided by the product of the concentrations of the reactants raised to their stoichiometric coefficients:
\[K_c = \frac{[C]^c [D]^d}{[A]^a [B]^b}\]
Applying this to our specific reaction, the expression for the equilibrium constant is:
\[K_c = \frac{[NH_3]^2}{[N_2][H_2]^3}\]
The equilibrium concentrations are given as \([NH_3] = 0.105 \, M\), \([N_2] = 1.1 \, M\), and \([H_2] = 1.50 \, M\).
- Calculating the Equilibrium Constant
Now, substitute the provided equilibrium concentrations into the formula for \(K_c\):
\[K_c = \frac{(0.105)^2}{(1.1) \times (1.50)^3}\]
First, calculate the square of the ammonia concentration:
\[(0.105)^2 = 0.011025\]
Next, calculate the cube of the hydrogen concentration:
\[(1.50)^3 = 3.375\]
Then, multiply this by the nitrogen concentration:
\[1.1 \times 3.375 = 3.7125\]
Finally, divide the concentration of the products by the concentration of the reactants:
\[K_c = \frac{0.011025}{3.7125} \approx 0.00297\]
Comparing this result to the given options (340, 0.0030, 0.030, 34), we see that \(0.00297\) is equivalent to \(0.0030\) when rounded to two significant figures.
- Conclusion and Final Answer
The calculation shows that the equilibrium constant \(K_c\) is approximately \(0.0030\). Therefore, the correct answer among the given choices is \(0.0030\).
\[\boxed{0.0030}\]
Examples
The concept of an equilibrium constant is fundamental in chemical engineering and industrial chemistry. For example, in the Haber-Bosch process for the mass production of ammonia, chemists and engineers must constantly calculate \(K_c\) at various temperatures and pressures. These calculations determine the yield of ammonia under specific conditions, helping to optimize the process for better efficiency and higher production rates of fertilizers and other essential nitrogen-based chemicals. By understanding the equilibrium state, industries can design better reactors to shift the reaction to favor the desired product.