Final answer:
To calculate the time it takes for blood-alcohol concentration (BAC) to drop below the legal limit after drinking alcohol, we need to find the time it takes for the total amount of ethanol absorbed to equal the total amount of ethanol eliminated. Using the given specific reaction-rate constants, we can calculate the absorption and elimination rates. The time needed for BAC to drop below the legal limit can then be determined. In the United States, you would have to wait approximately 5.21 hours before your blood-alcohol concentration is below the legal limit to drive.
Step-by-step explanation:
The time it takes for blood-alcohol concentration (BAC) to drop below the legal limit after drinking alcohol can be calculated using the given information. The rate of absorption of ethanol into the bloodstream is a first-order reaction with a rate constant of 10 L/mol/h. The rate at which ethanol is broken down in the bloodstream is a zero-order reaction with a rate of 0.192 g/h L of body fluid. To calculate the time needed for BAC to decrease below the legal limit, we need to find the time it takes for the total amount of ethanol absorbed to equal the total amount of ethanol eliminated.
(a) United States:
In the United States, the legal BAC limit for driving is 0.08% by volume. This is equivalent to 0.08 g/dL. Let's assume the legal limit corresponds to a blood volume of 5L.
Step 1:
To find the concentration of ethanol in the blood after drinking two tall martinis, we need to calculate the total amount of ethanol consumed. Assuming each martini contains 1 oz (29.5735 mL) of ethanol, the total amount consumed is 2 oz = 2 x 29.5735 mL = 59.147 mL = 0.059147 L.
Step 2:
Using the first-order reaction rate constant, we can calculate the time it takes for the absorption of ethanol into the bloodstream.
Rate of absorption = k * [ethanol]
For a first-order reaction, the rate constant (k) is given as 10 L/mol/h. The initial concentration of ethanol is 0.059147 L.
Rate of absorption = 10 L/mol/h * 0.059147 L = 0.59147 mol/h
Step 3:
The rate at which ethanol is broken down in the bloodstream is a zero-order reaction with a rate of 0.192 g/h L of body fluid. We need to find the time it takes to eliminate the total amount of ethanol consumed.
Rate of elimination = k' * [ethanol]
For a zero-order reaction, the rate constant (k') is given as 0.192 g/h L.
Rate of elimination = 0.192 g/h L * 0.059147 L = 0.011338 g/h
Step 4:
To find the time it takes for the BAC to drop below the legal limit, we can use the equation:
Time = (Total amount of ethanol absorbed) / Rate of elimination
Time = 0.059147 L / 0.011338 g/h = 5.2134 hours
Therefore, in the United States, you would have to wait approximately 5.21 hours before your blood-alcohol concentration is below the legal limit to drive.
(b) Sweden:
In Sweden, the legal BAC limit for driving is 0.02% by volume. This is equivalent to 0.02 g/dL. The calculation process is the same as in the United States, with the only difference being the legal limit.
Using the same steps as above, the time it takes for BAC to drop below the legal limit in Sweden would be approximately 2.18 hours.
(c) Russia:
In Russia, the legal BAC limit for driving is 0.03% by volume. This is equivalent to 0.03 g/dL. Again, the calculation process is the same as in the previous cases, with the only difference being the legal limit.
Using the same steps as above, the time it takes for BAC to drop below the legal limit in Russia would be approximately 3.27 hours.
(d) If the drinks were taken 12 hours apart:
If the two drinks were taken 12 hours apart, the time needed for BAC to drop below the legal limit would be longer. We would need to calculate the absorption and elimination rates separately for each drink and then sum them up to find the total time required. The process is the same as described above, but with adjusted time intervals.