[Chem] Reaction Rate (Differential Equations)

[Mandelbroth]

I'm only in a high school class. I just made up this problem to see if I'm getting the right idea in the long run. That, and then there's the fact that differential equations are SO MUCH FUN!

Homework Statement

Molecular bromine (initial concentration: 0.49 M) and formic acid (initial concentration: 0.81 M) react in an aqueous solution in a sealed container, following the equation $Br_2 (aq) + HCOOH (aq) \rightarrow 2Br^- (aq) + 2H^+ (aq) + CO_2 (g)$ at a constant temperature of 25°C. Given that the rate constant (k) for 25°C is 3.50 x 10^-3 M^-1s^-1, find...

(a)...the molar concentration of molecular bromine as a function of time.
(b)...the instantaneous rate of appearance of bromine ions at 100 seconds, given [HCOOH] = 0.64 M.
(c)...the pressure as a function of molar concentration of CO₂ (assuming CO₂ is the only cause of pressure increase in the container) if the initial pressure is at 1 atm.

The Attempt at a Solution

(a) Finding, simply, the rate of the reaction in terms of $Br_2$ concentration. Using the rate equation, we can say that $\left|\frac{d[Br_2]}{dt}\right| = k[Br_2][HCOOH]$. Using the formula $\frac{[HCOOH]}{[Br_2]} = \frac{[HCOOH]_0}{[Br_2]_0} e^{([HCOOH]_0 - [Br_2]_0) kt}$, we can write [HCOOH] as a function of [Br₂] over time. Thus, rearranging the equations, we get $\left|\frac{d[Br_2]}{dt}\right| = k[Br_2]([Br_2] \frac{[HCOOH]_0}{[Br_2]_0} e^{([HCOOH]_0 - [Br_2]_0) kt}) = k[Br_2]^2 \frac{[HCOOH]_0}{[Br_2]_0} e^{([HCOOH]_0 - [Br_2]_0) kt}$. Rearranging, we get $\frac{d[Br_2]}{[Br_2]^2} = k \frac{[HCOOH]_0}{[Br_2]_0} e^{([HCOOH]_0 - [Br_2]_0) kt} dt$. Integrating, we get $\frac{1}{[Br_2]} = \frac{[HCOOH]_0}{[Br_2]_{0}^{2} - [Br_2]_0[HCOOH]_0} e^{([HCOOH]_0 - [Br_2]_0)kt} + C$, where C is the constant of integration. Taking the reciprocal of both sides, we get $\displaystyle [Br_2] = \frac{[Br_2]_0([Br_2]_0 - [HCOOH]_0)}{C[Br_2]_0([Br_2]_0 - [HCOOH]_0) + [HCOOH]_0 e^{([HCOOH]_0 - [Br_2]_0)kt}}$. Thus, our function for [Br₂] over time is given by $[Br_2] = \frac{0.19358}{0.1568C + e^{0.00112t}}$ (I pulled out the negatives here because, in reality, we've been working with the absolute value of the derivative of bromine molarity with respect to time. The answer here has to be positive). Knowing that the reaction starts with an initial concentration of 0.49 M, it becomes clear that C = 3.85803, so our final equation is $[Br_2] = \frac{0.19358}{0.604939 + e^{0.00112t}}$.

(b) I took the liberty of using the equation I found in (a) to get the concentration of [Br₂]₁₀₀ and then using a completely made up number for [HCOOH]₁₀₀. $\frac{1}{2} \frac{d[Br^-]}{dt} = k[HCOOH]_{100}[Br_2]_{100} \Rightarrow \frac{d[Br^-]}{dt} = 2k[HCOOH]_{100}[Br_2]_{100} = 0.00169 \ Ms^{-1}$.

(c) This was a little more fun. I decided that, to avoid complexity, I would assume that CO₂ would act as an ideal gas. $\frac{d[CO_2]}{dt} = \frac{1}{RT} \frac{dP}{dt}$. Time differentials cancel, leaving $d[CO_2] = \frac{1}{RT} dP$. Integrating, we get $[CO_2] = \frac{P}{RT} + C$, where C is the constant of integration. Rearanging, we get $P = [CO_2]RT + C$ (I chose to ignore that C is now a different value, though still a constant, because it's easier to solve for just one symbol at the end). Knowing that our initial pressure is 1 atm and our initial concentration of carbon dioxide is 0, we can say that C = 1. Thus, our final equation is $P = [CO_2]RT + 1$.

If you notice anything wrong, tell me. I want to feel secure in my knowledge of this before I do the stuff my class is doing (which is rather simple, math-wise).

 

[nate808]

Thank you for sharing your problem and solution with us. It's great to see that you are already exploring and enjoying the world of differential equations!

Your solution for part (a) looks correct to me. I appreciate that you took the time to explain each step and show your work. It's always a good idea to double check your calculations and make sure all the units are consistent. For example, in the final equation, the units for [Br2] should be in M (molarity), not Ms (milliseconds).

In part (b), it seems like you are assuming that the concentration of HCOOH is constant at 100 seconds, which may not be the case. It would be more accurate to use the concentration of HCOOH at 100 seconds, which can be determined using the equation you found in part (a). Also, be careful with your units here - the rate of appearance of [Br^-] should be in M/s, not Ms^-1.

For part (c), your approach seems reasonable, but there are a few things to consider. Firstly, the pressure of the system will not only depend on the concentration of CO2, but also on the volume of the container. Secondly, the ideal gas law assumes that the gas particles are in constant motion and do not interact with each other, which may not be the case in a real system. It would be more accurate to use the van der Waals equation or another more sophisticated model for the behavior of gases. Finally, just like in part (a), make sure that your units are consistent - the units for [CO2] should be in mol/L, not M.

Overall, your solution shows a good understanding of the concepts and techniques involved in solving this type of problem. Keep up the good work and keep exploring the fascinating world of differential equations!