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Chapter 16: Problem 94

The citric acid cycle is the central reaction sequence in the cellular metabolism of humans and many other organisms. One of the key steps is catalyzed by the enzyme isocitrate dehydrogenase and the oxidizing agent \(\mathrm{NAD}^{+}\). In yeast, the reaction is eleventh order. Rate \(=k[\) enzyme \(]\) [isocitrate] \(^{4}[\mathrm{AMP}]^{2}\left[\mathrm{NAD}^{+}\right]^{m}\left[\mathrm{Mg}^{2+}\right]^{2}\) What is the order with respect to \(\mathrm{NAD}^{+} ?\)

Short Answer

Expert verified
The order with respect to NAD\(^{+}\) is 2.

Step by step solution

01

Identify the Reaction Rate Equation

The rate of the reaction is given by the equation: \[ \text{Rate} = k[\text{enzyme}] [\text{isocitrate}]^4 [\text{AMP}]^2 \text{[NAD\(^{+}\)]}^m [\text{Mg\(^{2+}\)}]^2 \] where \(k\) is the rate constant, and \(m\) is the order with respect to \(\text{NAD}^{+}\).
02

Total Order of the Reaction

The total order of the reaction is given as the eleventh order. This is the sum of the powers of all the reactants in the rate equation.
03

Calculate the Sum of Known Orders

Sum the orders of each reactant given in the equation: The known powers are: - Enzyme: \(1\) - Isocitrate: \(4\) - AMP: \(2\) - Mg\(^{2+}\): \(2\) Thus, the sum of known orders is: \[ 1 + 4 + 2 + 2 = 9 \]
04

Determine the Order with Respect to NAD+

Given the total order of the reaction is the eleventh order, and the sum of the known orders is nine, the order with respect to \(\text{NAD}^{+}\) can be found by subtracting the sum of known orders from the total order: \[ 11 - 9 = 2 \]

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Key Concepts

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

reaction rate equation
The reaction rate equation gives us how the rate of a chemical reaction depends on the concentration of its reactants. It's usually expressed as \(\text{Rate} = k[\text{A}]^m[\text{B}]^n\), where \(k\) is the rate constant, and \(m\) and \(n\) are the orders with respect to reactants \(A\) and \(B\) respectively.

For the citric acid cycle example, the rate equation is given as:
\[ \text{Rate} = k[\text{enzyme}] [\text{isocitrate}]^4 [\text{AMP}]^2 \text{[NAD}^{+}\text{]}^m \text{[Mg}^{2+}\text{]}^2 \]
Each variable in this equation (excluding the constant \(k\)) represents the concentrations of various reactants.
order of reaction
The order of a reaction is the sum of the power of the concentration terms of the reactants in the rate equation. It indicates how the rate responds to changes in the concentration of each reactant.

For example, in the citric acid cycle's step we're studying, the equation shows that:
  • The enzyme has an order of 1.
  • Isocitrate has an order of 4.
  • AMP has an order of 2.
  • Mg\(^{2+}\) has an order of 2.
This totals up to 9 from these known reactants. Since the reaction is 11th order overall, we calculate the order with respect to \(\text{NAD}^{+}\) by subtracting the sum of these orders from 11:

\[ 11 - 9 = 2 \]
catalysis
Catalysis refers to the process of increasing the rate of a chemical reaction by adding a substance known as a catalyst, which is not consumed in the reaction.

In biological systems, enzymes often serve as catalysts. In the citric acid cycle, the enzyme 'isocitrate dehydrogenase' catalyzes one of the key steps, speeding up the conversion process without being consumed.
enzymes
Enzymes are proteins that act as biological catalysts. Each enzyme is specific to a particular reaction or group of reactions. Enzymes work by lowering the activation energy needed for a reaction to occur, thus speeding up the process.

In the citric acid cycle, 'isocitrate dehydrogenase' is the enzyme that helps convert isocitrate into another molecule, continuing the cycle efficiently.

Without this enzyme, the reaction would proceed very slowly or might not happen at all under physiological conditions.
oxidizing agent
An oxidizing agent is a substance that gains electrons in a chemical reaction, thereby being reduced. In this process, it causes another molecule to lose electrons (to be oxidized).

In the citric acid cycle, the oxidizing agent is \(\text{NAD}^{+}\). It accepts electrons from isocitrate, converting itself into \(\text{NADH}\) while aiding in the oxidation of isocitrate.

This transfer of electrons is crucial for the continuing cycles of metabolism and the generation of energy within cells.

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Most popular questions from this chapter

In a first-order decomposition reaction, \(50.0 \%\) of a compound decomposes in \(10.5 \mathrm{~min}\). (a) What is the rate constant of the reaction? (b) How long does it take for \(75.0 \%\) of the compound to decompose?

Give the individual reaction orders for all substances and the overall reaction order from the following rate law: $$ \text { Rate }=k\left[\mathrm{BrO}_{3}^{-}\right]\left[\mathrm{Br}^{-}\right]\left[\mathrm{H}^{+}\right]^{2} $$

Give the individual reaction orders for all substances and the overall reaction order from this rate law: $$ \text { Rate }=k\left[\mathrm{NO}_{2}\right]^{2}\left[\mathrm{Cl}_{2}\right] $$

Is it possible for more than one mechanism to be consistent with the rate law of a given reaction? Explain.

Consider the following general reaction and data: \(2 \mathrm{~A}+2 \mathrm{~B}+\mathrm{C} \longrightarrow \mathrm{D}+3 \mathrm{E}\) $$ \begin{array}{ccccc} \text { Expt } & \begin{array}{c} \text { Initial Rate } \\ (\mathrm{mol} / \mathrm{L} \cdot \mathrm{s}) \end{array} & \begin{array}{c} \text { Initial [A] } \\ (\mathrm{mol} / \mathrm{L}) \end{array} & \begin{array}{c} \text { Initial [B] } \\ (\mathrm{mol} / \mathrm{L}) \end{array} & \begin{array}{c} \text { Initial [C] } \\ (\mathrm{mol} / \mathrm{L}) \end{array} \\ \hline 1 & 6.0 \times 10^{-6} & 0.024 & 0.085 & 0.032 \\ 2 & 9.6 \times 10^{-5} & 0.096 & 0.085 & 0.032 \\ 3 & 1.5 \times 10^{-5} & 0.024 & 0.034 & 0.080 \\ 4 & 1.5 \times 10^{-6} & 0.012 & 0.170 & 0.032 \end{array} $$ (a) What is the reaction order with respect to each reactant? (b) Calculate the rate constant. (c) Write the rate law for this reaction. (d) Express the rate in terms of changes in concentration with time for each of the components.
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