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Chapter 10: Problem 2

One hundred fifty kmol of an aqueous phosphoric acid solution contains 5.00 mole\% \(\mathrm{H}_{3} \mathrm{PO}_{4}\). The solution is concentrated by adding pure phosphoric acid at a rate of \(20.0 \mathrm{L} / \mathrm{min}\). (a) Write a differential mole balance on phosphoric acid and provide an initial condition. [Start by defining \(n_{\mathrm{p}}(\mathrm{kmol})\) to be the total quantity of phosphoric acid in the tank at any time.] Without solving the equation, sketch a plot of \(n_{\mathrm{p}}\) versus \(t\) and explain your reasoning. (b) Solve the balance to obtain an expression for \(n_{\mathrm{p}}(t) .\) Use the result to derive an expression for \(x_{\mathrm{p}}(t)\) the mole fraction of phosphoric acid in the solution. Without doing any numerical calculations, sketch a plot of \(x_{\mathrm{p}}\) versus \(t\) from \(t=0\) to \(t \rightarrow \infty,\) labeling the initial and asymptotic values of \(x_{\mathrm{p}}\) on the plot. Explain your reasoning. (c) How long will it take to concentrate the solution to \(15 \% \mathrm{H}_{3} \mathrm{PO}_{4} ?\)

Short Answer

Expert verified
The mole balance and mole fraction equations turn out to be \(\frac{dn_{p}}{dt} = r_{in}\) and \(x_{p}(t) = \frac{n_{p}(t)}{n_{p}(t) + n_{water}(t)}\) respectively. To reach 15% \(H_{3}PO_{4}\), a certain amount of time, \(t\), will be calculated using these. Total moles of acid will continuously increase due constant acid addition, while mole fraction will increase steadily from 0.05 until it eventually becomes 1. The exact time required to reach 15% concentration depends on the numerical solution of these equations.

Step by step solution

01

Formulate Differential Mole Balance

The rate of change of acid is given by the rate in (rate of pure phosphoric acid added) minus the rate out (none in this case). This can be written as a differential equation: \(\frac{dn_{p}}{dt} = r_{in}\). The rate at which phosphoric acid is added, \(r_{in}\), is constant at 20 L/min. Initially, \(n_{p0}\) is known to be \(n_{p}(0) = 0.05 * 150 = 7.5 \, kmol\).
02

Plot of \(n_{p}\) versus t and Explanation

Since the rate of addition of phosphoric acid is constant, \(n_{p}\) will increase linearly with time, thus the plot of \(n_{p}\) versus \(t\) will be a straight line with a positive slope.
03

Solve Balance to Obtain Expression for \(n_{p}(t)\)

Solving the differential equation, one gets: \(n_{p}(t) = r_{in} * t + n_{p0}\).
04

Derive Expression for \(x_{p}(t)\) and plot

The mole fraction \(x_{p}(t)\) would be given by the total moles of phosphoric acid divided by the total moles in the mixture: \(x_{p}(t) = \frac{n_{p}(t)}{n_{p}(t) + n_{water}(t)}\). Initially, \(x_{p}(0)\) will be 0.05 and as \(t -> \infty\), \(x_{p}->1\). The plot of \(x_{p}\) versus \(t\) will be a curve starting from 0.05 and tending to 1, with an initially steep slope that gradually flattens.
05

Concentration Time Calculation

For the solution to be 15% \(H_{3}PO_{4}\), \(x_p = 0.15\). This can be substituted into the expression derived above for \(x_{p}(t)\) to solve for \(t\).
06

Solve for \(t\)

The time calculation will result in a positive value, denoting the time it will take to achieve a concentration of 15% \(H_{3}PO_{4}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Phosphoric Acid Concentration
Phosphoric acid concentration is a key factor in various chemical industries, especially in the production of fertilizers, detergents, and even in food processing. The concentration denotes how much phosphoric acid \(\mathrm{H}_{3}\mathrm{PO}_{4}\) is present in a solution as compared to the rest of the components, usually water.

The concentration of phosphoric acid can be expressed in different ways, such as by weight percentage, molarity, normality, or mole fraction. Mole fraction, in particular, is a dimensionless number that represents the ratio of the number of moles of phosphoric acid to the total number of moles of all species in the solution. Increasing the phosphoric acid concentration typically means increasing the purity or strength of the solution. This process can involve adding pure acid into a solution, which changes the overall composition and amount of the acid in the system, as illustrated in the example problem.

It鈥檚 important to understand that the concentration can either be represented in terms of a static value at a given moment or dynamically as it changes over time due to a chemical process such as evaporation or dilution. In industrial processes, maintaining the correct concentration of phosphoric acid is crucial for the efficiency and quality of the product.
Mole Fraction Calculation
Mole fraction calculation is a fundamental concept in chemistry that helps quantify the composition of mixtures. It is expressed as the number of moles of a component divided by the total number of moles of all components in the mixture. To put it simply, it is a way of expressing how much of one substance is present in comparison to the whole.

In our example, the mole fraction \(x_{p}(t)\) represents the fraction of phosphoric acid in the solution. The initial mole fraction was given as 5 mole%, which means for every 100 moles of solution, there are 5 moles of \(\mathrm{H}_{3}\mathrm{PO}_{4}\). Mathematically, this is expressed as \(x_{p}(0) = 0.05\). When additional pure \(\mathrm{H}_{3}\mathrm{PO}_{4}\) is added to the solution, the mole fraction changes over time, and this alteration can be tracked using the equation derived from the mole balance.

Calculating mole fraction is not only key in getting precise mixing ratios but also essential for understanding various properties of the solution such as boiling point elevation, freezing point depression, and vapor pressure which all depend on the solution's composition.
Chemical Process Dynamics
Chemical process dynamics refer to how the concentration, temperature, volume, and other properties of a chemical system change over time. It involves understanding and applying dynamic principles to predict the behavior of chemical reactions and processes as they evolve. This concept is especially critical in the context of continuous flow processes and reactions occurring in closed systems such as reactors.

In the context of the phosphoric acid addition in our exercise, the dynamics are simple since there is a constant addition rate and no removal. The phosphoric acid concentration increases at a steady rate, and if we continue this indefinitely, the concentration of phosphoric acid in the tank would approach that of pure \(\mathrm{H}_{3}\mathrm{PO}_{4}\), disregarding any volume limitations or practical considerations. In practice, such a system would be subject to constraints like reaction kinetics, tank volume, and the influence of temperature and pressure on reaction rates.

Concepts of chemical process dynamics are crucial for designing chemical reactors and are vital for scaling up processes from the laboratory to industrial manufacturing. They require a deep understanding of thermodynamics, kinetic theories, and mass transfer alongside mole balance equations to analyze and optimize chemical reactions and processes.

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Most popular questions from this chapter

A radioactive isotope decays at a rate proportional to its concentration. If the concentration of an isotope is \(C(\mathrm{mg} / \mathrm{L}),\) then its rate of decay may be expressed as $$r_{\mathrm{d}}[\mathrm{mg} /(\mathrm{L} \cdot \mathrm{s})]=k C$$ where \(k\) is a constant. (a) A volume \(V(\mathrm{L})\) of a solution of a radioisotope whose concentration is \(C_{0}(\mathrm{mg} / \mathrm{L})\) is placed in a closed vessel. Write a balance on the isotope in the vessel and integrate it to prove that the half-life \(t_{1 / 2}\) of the isotope \(-\) by definition, the time required for the isotope concentration to decrease to half of its initial value- equals ( \(\ln 2\) )/ \(k\). (b) The half-life of \(^{56} \mathrm{Mn}\) is \(2.6 \mathrm{h}\). A batch of this isotope that was used in a radiotracing experiment has been collected in a holding tank. The radiation safety officer declares that the activity (which is proportional to the isotope concentration) must decay to \(1 \%\) of its present value before the solution can be discarded. How long will this take?

A liquid-phase chemical reaction with stoichiometry \(\mathrm{A} \rightarrow \mathrm{B}\) takes place in a semibatch reactor. The rate of consumption of A per unit volume of the reactor contents is given by the first-order rate expression (see Problem 10.19) $$r_{\mathrm{A}}[\operatorname{mol} /(\mathrm{L} \cdot \mathrm{s})]=k C_{\mathrm{A}}$$ where \(C_{\Lambda}(\text { mol } A / L)\) is the reactant concentration. The tank is initially empty. Beginning at a time \(t=0\) a solution containing \(\mathrm{A}\) at a concentration \(\mathrm{C}_{\mathrm{A} 0}(\mathrm{mol} \mathrm{A} / \mathrm{L})\) is fed to the tank at a constant rate \(\dot{V}(\mathrm{L} / \mathrm{s})\) (a) Write a differential balance on the total mass of the reactor contents. Assuming that the density of the contents always equals that of the feed stream, convert the balance into an equation for \(d V / d t\) where \(V\) is the total volume of the contents, and provide an initial condition. Then write a differential mole balance on the reactant, A, letting \(N_{\mathrm{A}}(t)\) equal the total moles of A in the vessel, and provide an initial condition. Your equations should contain only the variables \(N_{\mathrm{A}}, V,\) and \(t\) and the constants \(\dot{V}\) and \(C_{\mathrm{A} 0}\). (You should be able to eliminate \(C_{\mathrm{A}}\) as a variable.) (b) Without attempting to integrate the equations, derive a formula for the steady-state value of \(N_{\mathrm{A}}\). (c) Integrate the two equations to derive expressions for \(V(t)\) and \(N_{\mathrm{A}}(t),\) and then derive an expression for \(C_{\mathrm{A}}(t)\). Determine the asymptotic value of \(N_{\mathrm{A}}\) as \(t \rightarrow \infty\) and verify that the steady-state value obtained in \(\operatorname{Part}(\mathbf{b})\) is correct. Briefly explain how it is possible for \(N_{\mathrm{A}}\) to reach a steady value when you keep adding A to the reactor and then give two reasons why this value would never be reached in a real reactor. (d) Determine the limiting value of \(C_{\mathrm{A}}\) as \(t \rightarrow \infty\) from your expressions for \(N_{\mathrm{A}}(t)\) and \(V(t) .\) Then explain why your result makes sense in light of the results of Part (c).

A ventilation system has been designed for a large laboratory with a volume of \(1100 \mathrm{m}^{3}\). The volumetric flow rate of ventilation air is \(700 \mathrm{m}^{3} / \mathrm{min}\) at \(22^{\circ} \mathrm{C}\) and 1 atm. (The latter two values may also be taken as the temperature and pressure of the room air.) A reactor in the laboratory is capable of emitting as much as 1.50 mol of sulfur dioxide into the room if a seal ruptures. An \(\mathrm{SO}_{2}\) mole fraction in the room air greater than \(1.0 \times 10^{-6}(1 \mathrm{ppm})\) constitutes a health hazard. (a) Suppose the reactor seal ruptures at a time \(t=0,\) and the maximum amount of \(\mathrm{SO}_{2}\) is emitted and spreads uniformly throughout the room almost instantaneously. Assuming that the air flow is sufficient to make the room air composition spatially uniform, write a differential SO_ balance, letting \(N\) be the total moles of gas in the room (assume constant) and \(x(t)\) the mole fraction of \(\mathrm{SO}_{2}\) in the laboratory air. Convert the balance into an equation for \(d x / d t\) and provide an initial condition. (Assume that all of the \(\left.\mathrm{SO}_{2} \text { emitted is in the room at } t=0 .\right)\) (b) Predict the shape of a plot of \(x\) versus \(t\). Explain your reasoning, using the equation of Part (a) in your explanation. (c) Separate variables and integrate the balance to obtain an expression for \(x(t)\). Check your solution. (d) Convert the expression for \(x(t)\) into an expression for the concentration of \(\mathrm{SO}_{2}\) in the room, \(C_{\mathrm{SO}_{2}}\) (mol \(\mathrm{SO}_{2} / \mathrm{L}\) ). Calculate (i) the concentration of \(\mathrm{SO}_{2}\) in the room two minutes after the rupture occurs, and (ii) the time required for the \(S O_{2}\) concentration to reach the "safe" level. (e) Why would it probably not yet be safe to enter the room after the time calculated in Part (d)? (Hint:One of the assumptions made in the problem is probably not a good one.)

A stirred tank contains \(1500 \mathrm{lb}_{\mathrm{m}}\) of pure water at \(70^{\circ} \mathrm{F}\). At time \(t=0,\) two streams begin to flow into the tank and one is withdrawn. One input stream is a \(20.0 \mathrm{wt} \%\) aqueous solution of \(\mathrm{NaCl}\) at \(85^{\circ} \mathrm{F}\) flowing at a rate of \(15 \mathrm{lb}_{\mathrm{m}} / \mathrm{min},\) and the other is pure water at \(70^{\circ} \mathrm{F}\) flowing at \(10 \mathrm{lb}_{\mathrm{m}} / \mathrm{min} .\) The mass of liquid in the tank is held constant at \(1500 \mathrm{lb}_{\mathrm{m}}\). Perfect mixing in the tank may be assumed, so that the outlet stream has the same \(\mathrm{NaCl}\) mass fraction \((x)\) and temperature \((T)\) as the tank contents. Also assume that the heat of mixing is zero and the heat capacity of all fluids is \(C_{p}=1 \mathrm{Btu} /\left(\mathrm{lb}_{\mathrm{m}} \cdot^{\circ} \mathrm{F}\right)\) (a) Write differential material and energy balances and use them to derive expressions for \(d x / d t\) and \(d T / d t\) (b) Without solving the equations derived in Part (a), sketch plots of \(T\) and \(x\) as a function of time \((t)\) Clearly identify values at time zero and as \(t \rightarrow \infty\)

The following chemical reactions take place in a liquid-phase batch reactor of constant volume \(V\). $$\begin{aligned} &\mathrm{A} \rightarrow 2 \mathrm{B} \quad r_{1}[\mathrm{mol} \mathrm{A} \text { consumed } /(\mathrm{L} \cdot \mathrm{s})]=0.100 C_{\mathrm{A}}\\\ &\mathbf{B} \rightarrow \mathbf{C} \quad r_{2}[\mathrm{mol} \mathbf{C} \text { generated } /(\mathbf{L} \cdot \mathbf{s})]=0.200 C_{\mathrm{B}}^{2} \end{aligned}$$ where the concentrations \(C_{\mathrm{A}}\) and \(C_{\mathrm{B}}\) are in mol/L. The reactor is initially charged with pure \(\mathrm{A}\) at a concentration of 1.00 mol/L. (a) Write expressions for ( \(i\) ) the rate of generation of \(\mathrm{B}\) in the first reaction and (ii) the rate of consumption of \(\mathrm{B}\) in the second reaction. (If this takes you more than about 10 seconds, you're missing the point.) (b) Write mole balances on A, B, and C, convert them into expressions for \(d C_{\mathrm{A}} / d t, d C_{\mathrm{B}} / d t\), and \(d C_{\mathrm{C}} / d t,\) and provide boundary conditions. (c) Without doing any calculations, sketch on a single graph the plots you would expect to obtain of \(C_{\mathrm{A}}\) versus \(t, C_{\mathrm{B}}\) versus \(t,\) and \(C_{\mathrm{C}}\) versus \(t .\) Clearly show the function values at \(t=0\) and \(t \rightarrow \infty\) and the curvature (concave up, concave down, or linear) in the vicinity of \(t=0 .\) Briefly explain your reasoning. (d) Solve the equations derived in Part (b) using a differential equation- solving program. On a single graph, show plots of \(C_{\mathrm{A} \text { versust }}, C_{\mathrm{B}}\) versus \(t,\) and \(C_{\mathrm{C}}\) versus \(t\) from \(t=0\) to \(t=50\) s. Verify that your predictions in Part (c) were correct. If they were not, change them and revise your explanation.
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