At Equilibrium, A Reaction Vessel Contains 4.50 Atm Of $Br_2$ And 1.10 Atm Of $NBr_3$. According To The Reaction:${2 NBr_3(g) \rightleftharpoons N_2(g) + 3 Br_2(g)}$with K P = 4.8 K_p = 4.8 K P ​ = 4.8 , Determine The Equilibrium

Introduction

In chemistry, equilibrium is a state where the rates of forward and reverse reactions are equal, resulting in no net change in the concentrations of reactants and products. The equilibrium constant, denoted by $K_p$, is a measure of the ratio of the concentrations of products to reactants at equilibrium. In this article, we will discuss how to determine the equilibrium concentrations of reactants and products in a reaction vessel using the given equilibrium constant and partial pressures of the reactants and products.

The Reaction and Equilibrium Constant

The given reaction is:

${2 NBr_3(g) \rightleftharpoons N_2(g) + 3 Br_2(g)}$

The equilibrium constant, $K_p$, is given as 4.8. The equilibrium constant is related to the partial pressures of the reactants and products by the following equation:

${K_p = \frac{(P_{N_2})(P_{Br_2}^3)}{(P_{NBr_3}^2)}}$

where $P_{N_2}$, $P_{Br_2}$, and $P_{NBr_3}$ are the partial pressures of $N_2$, $Br_2$, and $NBr_3$, respectively.

Given Partial Pressures

The reaction vessel contains 4.50 atm of $Br_2$ and 1.10 atm of $NBr_3$. We are asked to determine the equilibrium concentrations of the reactants and products.

Setting Up the Equilibrium Expression

We can set up the equilibrium expression using the given partial pressures and the equilibrium constant:

${4.8 = \frac{(P_{N_2})(P_{Br_2}^3)}{(P_{NBr_3}^2)}}$

Letting $x$ Represent the Change in Partial Pressure

Let $x$ represent the change in partial pressure of $NBr_3$. Since the reaction is:

${2 NBr_3(g) \rightleftharpoons N_2(g) + 3 Br_2(g)}$

the change in partial pressure of $NBr_3$ is $-2x$, the change in partial pressure of $N_2$ is $x$, and the change in partial pressure of $Br_2$ is $3x$.

Substituting the Changes in Partial Pressure into the Equilibrium Expression

We can substitute the changes in partial pressure into the equilibrium expression:

${4.8 = \frac{(x)(4.50 + 3x)^3}{(1.10 - 2x)^2}}$

Simplifying the Equation

We can simplify the equation by expanding the terms and combining like terms:

${4.8 = \frac{x(4.50 + 27x + 81x^2 + 135x^3 + 27x^4)}{(1.10 - 4x + 4x^2)}}$

Solving for $x$

We can solve for $x$ by equating the numerator and denominator and solving for $x$. However, this equation is difficult to solve analytically, so we will use numerical methods to find the value of $x$.

Using Numerical Methods to Find the Value of $x$

We can use numerical methods, such as the Newton-Raphson method, to find the value of $x$. The Newton-Raphson method is an iterative method that uses the following formula to find the value of $x$:

${x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}}$

where $f(x)$ is the function we are trying to solve, and $f'(x)$ is the derivative of $f(x)$.

Finding the Value of $x$

We can use the Newton-Raphson method to find the value of $x$. We start with an initial guess of $x = 0.1$ and iterate until the value of $x$ converges.

Convergence of the Newton-Raphson Method

After several iterations, the value of $x$ converges to $x = 0.05$.

Finding the Equilibrium Concentrations

Now that we have found the value of $x$, we can find the equilibrium concentrations of the reactants and products. The equilibrium concentration of $NBr_3$ is $1.10 - 2x = 1.10 - 2(0.05) = 1.00$ atm. The equilibrium concentration of $N_2$ is $x = 0.05$ atm. The equilibrium concentration of $Br_2$ is $4.50 + 3x = 4.50 + 3(0.05) = 4.65$ atm.

Conclusion

In this article, we have discussed how to determine the equilibrium concentrations of reactants and products in a reaction vessel using the given equilibrium constant and partial pressures of the reactants and products. We have used the Newton-Raphson method to find the value of $x$, which represents the change in partial pressure of $NBr_3$. We have then used the value of $x$ to find the equilibrium concentrations of the reactants and products. The equilibrium concentration of $NBr_3$ is 1.00 atm, the equilibrium concentration of $N_2$ is 0.05 atm, and the equilibrium concentration of $Br_2$ is 4.65 atm.

References

• Levine, I. N. (2012). Physical Chemistry. 6th ed. McGraw-Hill.
• Atkins, P. W., & de Paula, J. (2010). Physical Chemistry. 9th ed. Oxford University Press.
• Chang, R. (2010). Physical Chemistry for the Biosciences. Cambridge University Press.
  

Introduction

In our previous article, we discussed how to determine the equilibrium concentrations of reactants and products in a reaction vessel using the given equilibrium constant and partial pressures of the reactants and products. In this article, we will answer some common questions related to equilibrium concentrations in a reaction vessel.

Q: What is the equilibrium constant, and how is it related to the partial pressures of the reactants and products?

A: The equilibrium constant, denoted by $K_p$, is a measure of the ratio of the concentrations of products to reactants at equilibrium. It is related to the partial pressures of the reactants and products by the following equation:

${K_p = \frac{(P_{N_2})(P_{Br_2}^3)}{(P_{NBr_3}^2)}}$

Q: How do I determine the equilibrium concentrations of the reactants and products in a reaction vessel?

A: To determine the equilibrium concentrations of the reactants and products in a reaction vessel, you need to know the given equilibrium constant and partial pressures of the reactants and products. You can use the Newton-Raphson method to find the value of $x$, which represents the change in partial pressure of $NBr_3$. Once you have found the value of $x$, you can use it to find the equilibrium concentrations of the reactants and products.

Q: What is the significance of the equilibrium constant in a reaction vessel?

A: The equilibrium constant is a measure of the ratio of the concentrations of products to reactants at equilibrium. It is a useful tool for predicting the direction of a reaction and the concentrations of the reactants and products at equilibrium.

Q: How do I know if a reaction is at equilibrium?

A: A reaction is at equilibrium when the rates of forward and reverse reactions are equal, resulting in no net change in the concentrations of reactants and products. You can determine if a reaction is at equilibrium by measuring the concentrations of the reactants and products and comparing them to the equilibrium concentrations.

Q: What is the difference between the equilibrium constant and the reaction quotient?

A: The equilibrium constant is a measure of the ratio of the concentrations of products to reactants at equilibrium, while the reaction quotient is a measure of the ratio of the concentrations of products to reactants at any point in the reaction. The reaction quotient is used to predict the direction of a reaction, while the equilibrium constant is used to predict the concentrations of the reactants and products at equilibrium.

Q: Can I use the equilibrium constant to predict the direction of a reaction?

A: Yes, you can use the equilibrium constant to predict the direction of a reaction. If the equilibrium constant is greater than 1, the reaction will proceed in the forward direction, resulting in an increase in the concentrations of the products. If the equilibrium constant is less than 1, the reaction will proceed in the reverse direction, resulting in a decrease in the concentrations of the products.

Q: How do I calculate the equilibrium constant from the given partial pressures of the reactants and products?

A: To calculate the equilibrium constant from the given partial pressures of the reactants and products, you can use the following equation:

${K_p = \frac{(P_{N_2})(P_{Br_2}^3)}{(P_{NBr_3}^2)}}$

You can plug in the given partial pressures of the reactants and products into this equation to calculate the equilibrium constant.

Q: What is the relationship between the equilibrium constant and the reaction quotient?

A: The equilibrium constant is a measure of the ratio of the concentrations of products to reactants at equilibrium, while the reaction quotient is a measure of the ratio of the concentrations of products to reactants at any point in the reaction. The reaction quotient is used to predict the direction of a reaction, while the equilibrium constant is used to predict the concentrations of the reactants and products at equilibrium.

Q: Can I use the reaction quotient to predict the direction of a reaction?

A: Yes, you can use the reaction quotient to predict the direction of a reaction. If the reaction quotient is greater than 1, the reaction will proceed in the forward direction, resulting in an increase in the concentrations of the products. If the reaction quotient is less than 1, the reaction will proceed in the reverse direction, resulting in a decrease in the concentrations of the products.

Conclusion

In this article, we have answered some common questions related to equilibrium concentrations in a reaction vessel. We have discussed the significance of the equilibrium constant, how to determine the equilibrium concentrations of the reactants and products, and how to use the equilibrium constant to predict the direction of a reaction. We have also discussed the relationship between the equilibrium constant and the reaction quotient, and how to use the reaction quotient to predict the direction of a reaction.