A catalyst is a substance that either accelerates a chemical...

A catalyst is a substance that either accelerates a chemical reaction or is necessary for the reaction to occur. Suppose an enzyme $E$ (a catalyst) combines with a substrate $S$ (a reacting chemical) to form an intermediate product $X$ that then produces a product $P$ and releases the enzyme. If initially there are $x_{0}$ moles/liter of $S$ and there is no $P$, then based on the theory of Michaelis and Menten, the concentration of $P, p(t),$ after $t$ hr is given by the equation $$V t=p-k \ln \left(1-\frac{p}{x_{0}}\right)$$ where the constant $V$ is the maximum possible speed of the reaction and the constant $k$ is called the Michaelis constant for the reaction. Find the rate of change of the formation of the product $P$ in this reaction.

Key Concepts

Catalysis

Catalysis is the process by which a substance called a catalyst accelerates a chemical reaction without being consumed in the process. It works by lowering the activation energy of the reaction, thereby increasing the reaction rate without altering the overall equilibrium.

Enzyme Kinetics

Enzyme kinetics is the study of the rates of biochemical reactions that are catalyzed by enzymes. It involves understanding how various factors, such as substrate concentration and enzyme activity, influence the speed of reactions in biological systems.

Michaelis-Menten Equation

The Michaelis-Menten equation is a fundamental model in enzyme kinetics that describes the relationship between the reaction rate and the substrate concentration. It introduces key parameters such as the maximum reaction velocity and the Michaelis constant, which reflects the substrate concentration at which the reaction rate is half of its maximum.

Differentiation in Kinetics

Differentiation is a mathematical tool used to determine the instantaneous rate of change in a system. In the context of kinetics, implicit differentiation—often using the chain rule—is applied to integrated rate equations to find how fast a product is being formed at any given moment.

Transcript

00:01 So we're told that a catalyst is a substance that either accelerates a chemical reaction or is necessary a chemical for the reaction to occur.

00:08 So say suppose we have an enzyme e, which is a catalyst, combines with a substrate s, a reacting chemical to form an intermediate product x and then produces a product p and releases the enzyme.

00:23 If initially there are x not moles per liter of s and there is no p, then based on the theory of michaelis and meton, the concentration of p, which is a function of time, after t hours is given by this.

00:42 And what are all these variables in here? well, v is the maximum possible speed of the reaction.

00:51 K is a constant.

00:53 And then we told the x not is the initial number of moles of s.

00:57 So we're basically going, so we have an enzyme, the same, the same, the same, the same, the we're trying to get the, there's a product p and releases in.

01:06 So we're trying to get, what am we trying to do here? e is the catalyst.

01:12 S is the substrate to a reacting chemical to form an intermediate product x and then produce s or a p and releases the enzyme.

01:27 So basically, i guess the substrate is reacting with something.

01:31 I don't know what it is, but we're forming something.

01:35 The substrate reacts with the catalyst, and then that thing reacts for something else, and releases the enzyme, so then it can, you know, react more, you know, can basically do itself some more, do some more work as a catalyst.

01:52 Anyway, this is the formula.

01:58 Now, they want us to know, where, let's see, what do they want? find the rate of change of the formation of the product p in this reaction.

02:07 Okay.

02:08 So we can explicitly differentiate this thing with respect to time.

02:14 So over here we just get v, so that's an easy one.

02:17 Here, you know, this is just p dot, and now we need to take the time derivative of this.

02:22 So we have our constant k here, we have a minus sign there, and then the derivative of this is one over the argument, so we get that, and then we need to take the derivative of the argument, and that's minus p.

02:33 Dot over x -not...

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