### Mathematical Issues for Chemists

A brief outline of the mathematical issues faced by chemistry students.

### Reaction Rates

Explore the possibilities for reaction rates versus concentrations with this non-linear differential equation

### Catalyse That!

Can you work out how to produce the right amount of chemical in a temperature-dependent reaction?

# Smoke and Daggers

##### Stage: 5 Challenge Level:

Meet Ellie Holt, a rather loud and vivacious A-level student. She is also a smoker. Ellie is a typical female and has a total lung capacity of 4.7 litres exactly when she last checked. Upon taking a deep breath, she completely fills this volume up with regular air.

Given that the air pressure in her lungs is 760mmHg, and that the percentage of oxygen in the air by volume is 20.95%, how many moles of oxygen are contained in her fully inflated lungs?

(1 mmHg = 133.322 Pa) NB. This will also be important in later questions

Dr Mathew Greyham is a scientist at the anti-smoking research unit at St. Bartholomew's Hospital. He is an expert in his field and has decided to carry out an analysis of the components of cigarette smoke. His results prove to be quite interesting:

C$\equiv$O 16 mg/cigarette
HCN 300 $\mu$ g/cigarette
Nicotine 10 mg/cigarette
Cyclohexane 300 ng/cigarette

As you may have noticed, the units used by Matt to record his data aren't that accessible.

Can you calculate the relative molecular masses for all of these compounds? What units does this value have?

Now calculate the number of moles of each in the volume of smoke produced on burning one cigarette.

Luckily you are inventive enough to calculate the volume of mainstream smoke produced by a single cigarette. You look on eShore and buy a "cigarette smoke generator" for a ridiculously small amount of money! You place a cigarette in the generator and let it work its magic. As you're also a good scientist, you decide that some repeats would be useful.

Number of puffs recorded per cigarette
17 21 23 19 23 20 18 19

These data were obtained by setting puff durations of 5 seconds at a rate of 1 puff per minute. It is known that the machine takes puff volumes of 45 ml if the puff duration was set at 2 seconds.

Calculate the volume of cigarette smoke produced by a single cigarette.
What is the concentration of each of the four products in cigarette smoke? Use standard units.

Defying NHS advice, Ellie decides to stop trying to quit smoking and lights up a cigarette. She inhales 390 ml of smoke which corresponds to a normal breath. Her lungs are normal human lungs which mean they have a residual volume of 0.93 litres and an expiratory reserve volume of 0.93 litres.

What factor of dilution does the cigarette smoke undergo?

Calculate the number of moles of each compound present in her lungs after she inhales one breath of smoke.

Each compound contributes to the total pressure in her lungs, known as the partial pressure. Can you work out the partial pressure that each of the compounds contributes?

What is the density of the cigarette smoke containing air in her lungs? Assume that these are the only components of the smoke and use the following percentages as the approximate composition of alveolar air.
CO$_2$ 4.05%
O$_2$ 16.10%
N$_2$ 79.85%

Ellie is rather addicted to smoking and smokes continuously and deeply, taking large drags which mean that the largest possible lung volume she is capable of eventually becomes uniformly filled with cigarette smoke. The inspiratory reserve volume which is the additional air that can be inhaled after a normal tidal breath is 2.45 litres.

How many molecules of each compound are present in her lungs at this point?

The nicotine in her lungs diffuses into her blood at a rate of 400$\mu$gs$^{-1}$.

How long does it take until all of the nicotine is absorbed into the blood stream (assuming that the rapid flow of blood is able to maintain this diffusion gradient)?

What is the average concentration of nicotine in Ellie's blood given that systolic blood pressure is 120 mmHg, the diastolic blood pressure is 80 mmHg? [Hint: You may assume that systolic and diastolic phases are of equivalent length in the human body] The volume of blood in Ellie's body is 4.7 dm$^3$.

What modelling have you used in the above calculation? Which of these modelling assumptions are unlikely to be true for the scenario given?

Nicotine has a half life of 2 hours in the human body. Ellie has smoked a cigarette every 30 minutes for 3 hours today. She has also just realised she has an appointment with her physician which involves a blood test to check if Ellie has kept to her promise of not smoking.

How long does it take for levels of nicotine to drop below the 200 ng/ml detectable level so that Ellie can cheat her test? List any assumptions inherent in your calculation.

According to Dr Jeremy Sharman, a research fellow at The Hospital of St. Radegund and St. Rubensti$\tilde{\text{n}}$o, the percentage of dangerous compounds in the smoke coming off the burning end of a cigarette is 86%! The percentage of dangerous compounds in the mainstream smoke that is inhaled by a smoker is only 11.2%.

Ellie is not the most responsible person and never one to take shocking advertisements to heart. She sometimes smokes indoors in a room close to her baby brother Pete who is often unaware of the situation being only 4 years old.

Here are Pete's vital statistics:

Total Lung Volume: 1.45 litres
Vital Capacity: 1.15 litres
Tidal Volume: 125 ml
Residual Volume: 300 ml
Expiratory Reserve Volume: 300 ml
Inspiratory Reserve Volume: 730 ml

Ellie takes 15 drags on her cigarette, all of which are 390 ml. Each time the smoke is diluted with the functional residual capacity of her lungs and 390 ml of air in the lungs is expired. Her brother, Pete, is in the room and takes 15 tidal breaths in a similar fashion. Assuming that the diffusion of passive smoke is such that a constant diluted amount of cigarette smoke is present in the air breathed in by Pete:

Give a reasonable estimation of this dilution and thus the percentage of dangerous compounds in the air breathed in by Pete.

Assuming that none of these dangerous compounds is absorbed into the blood stream over this period, what is the percentage of dangerous compounds in the alveolar air for Ellie and her brother, Pete, after the last outward breath?

Extension
Repeat the calculation with different dilutions for the passive smoke. Is there a value for the dilution which means that the percentage of dangerous compounds in Pete and Ellie is the same?

Ellie is still in the process of trying to quit smoking. Her physician has told her that all she needs is will power and determination.