Individual order of reactants is a variable used to determine the rate of a reaction. Each reactant in the rate-limiting step of a reaction is assigned an order (typically zeroth, 1st, or 2nd). The reaction order is the sum of all individual orders. The rate of a reaction is defined as follows.

Where  is the rate of the reaction,  and  are reactants,  is rate constant, and  and  are individual orders. The reaction order of of this reaction would be . Note that the rate of a reaction depends on the concentration of reactants, rate constant, and the individual orders. It doesn’t depend on the reaction order.

The half-life is defined as the amount of time it takes to reduce the concentration of a substance (in this case a drug). After one half-life, 50% of the drug will be eliminated and 50% will remain. After the second half-life, half of the remaining drug (25%) will be eliminated; therefore, at the end of second half-life 75% of drug will be eliminated. To solve for this time, we can simply calculate the half-life and multiply it by 2.

To solve this question, we need to know the equation for half-life. It is stated that the reaction follows first-order; therefore, the half-life for a first order reaction is defined as

Where  is half-life and  is the rate constant. Since we are not given the rate constant we cannot calculate the half-life.

The half-life of a first order reaction is as follows:

Where  is half-life and  is the rate constant. The half-life only depends on the rate constant. Since rate constant is in the denominator, half-life and rate constant are inversely proportional. This means that half-life increases as rate constant decreases and vice versa. Concentration of reactants does not affect the half-life for a first order reaction.

Rate constant is a unique constant for each reaction that determines the rate of a reaction. It is one of several variables that determines the rate of reaction. As rate constant increases the rate of reaction increases. Rate constant depends on two main factors: temperature and activation energy. The rate constant increases as the temperature is increased. Recall that increasing temperature increases the kinetic energy of the molecules; therefore, an increase in temperature will increase the amount of molecules that reach activation energy. This will increase the rate constant and, subsequently, the rate of the reaction.

Rate constant increases as the activation energy is decreased. Activation energy is the energy hill that reactants must overcome to produce products. If the activation energy is decreased then it will be easier for reactants to overcome the energy hill and convert into products; therefore, decreasing activation energy will increase the rate constant and, subsequently, the rate of the reaction. Recall that catalysts speed up a reaction (increase rate of reaction) by lowering activation energy; therefore, catalysts increase the rate constant.

Condition I is false. The reactant concentration does change over time (can be seen the integrated rate law, which has a  term), it just does not depend on the reactant concentration.

Condition II is true. The rate is independent of reactant concentration for 0th order reactions.

Condition III is false.  does have an affect on the overall reactant concentration over time (can be seen by in the integrated rate law, which has a  term).

The only statements that are true are that a plot of ln[A] vs t will be linear for a first order reaction (this can be seen from the integrated rate law given, since the form of the equation is  

With  and  and 

Additionally, for a first order reaction, the rate is proportional to the concentration of [A]. 

All other statements are not true. A plot of [A] vs. t will be exponential.

Michaelis constant, or , is defined as the concentration of substrate at which the reaction rate is half the maximum (). It is a useful measure of how much substrate is needed for reaction to proceed rapidly. A reaction with a high Michaelis constant will need lots of substrate to reach high reaction rates whereas a reaction with low Michaelis constant will need small amounts of substrate to reach high reaction rates. 

Reaction rate, according to Michaelis-Menten model is as follows.

where  is reaction rate,  is maximum reaction rate,  is substrate concentration, and  is the Michaelis constant. If we analyze the given options, we will observe that the greatest increase in  occurs when  is doubled (increased by a factor of 2). Increasing substrate concentration by a factor of 2 will have nearly the same effect; however, since  is also found in the denominator it will only slightly contribute to an increase in .

Note that the units for  is molarity,  is molarity, and  is . Solving for  will give us units of .

To solve this problem we need to use the Michaelis-Menten equation.

where  is reaction rate,  is maximum reaction rate,  is substrate concentration, and  is the Michaelis constant. If we plug in the given values we get a reaction rate of

Note that the Michaelis-Menten equation implies that the  will never exceed . Regardless of how high the substrate concentration is, the reaction rate will approach  but will never equal or exceed it. You can try this by substituting very high values for substrate concentration. The  will get very close to 0.2 () but will never equal or exceed it.

The Michaelis-Menten equation is as follows.

Where  is reaction rate,  is maximum reaction rate,  is substrate concentration, and  is the Michaelis constant. Since the Michaelis constant, , is in the denominator, the reaction rate is inversely proportional to the Michaelis constant; therefore, increasing the Michaelis constant will decrease the reaction rate.