Asked and Answered 5.0

I recently became a bitters-and-club-soda drinker.  As I was to discover, club soda has the annoying habit of going flat before one finishes the bottle.  To reduce waste, one can buy small bottles of national brands but they are much more expensive than the one-liter or two-liter bottles of store brands.  What’s a club soda drinker to do?  Glad you asked, and here’s your answer.

If club soda did not go flat, the choice would be clear.  A six-pack of 10-ounce glass bottles of White Rock Club Soda costs $3.99 here — 33.25 cents per 5-ounce serving — while four one-liter bottles (135.3 ounces) of our store-brand club soda cost only $3.00 — 11.09 cents per serving.  This means club soda in the 10-ounce bottles is three times more expensive… at least before any drinks are poured.

I tried a few one-liter bottles and was surprised to see how much club soda went to waste.  (Soda without sparkle belongs in the drain, not the drink.)  So I sat down to figure out whether the one-liter variety was a bargain or a boondoggle.  The answer would involve Henry’s Law, the Ideal Gas Law, bottling industry data and a few simplifying assumptions. (Full details are provided in the Appendix.)

But let’s start with how club soda goes flat (Figure 1).  The icon at far left represents the bottle as you buy it — filled to the neck with liquid and then topped off with high-pressure carbon dioxide (CO2).  Then, you open the bottle, pour a serving, and re-close the bottle.  The headspace is now filled with air from your room.  Over time, as the bottle rests in your refrigerator, CO2 escapes from the soda and collects in the headspace, until the pressure reaches an equilibrium (as determined by Henry’s solubility constant for carbon dioxide).  This equilibrium CO2 pressure will be lower than the original pressure — as suggested by the lighter shades of blue in the diagram — due to the CO2 that escaped when you opened the bottle along with the CO2 that you poured into your drink.

FIGURE 1: How Club Soda Goes Flat

The club soda gets flatter each time you open the bottle and pour a serving, until hardly any gas is left.  At some point, you may as well water your plants with what remains.

The question is, for a given serving size and bottle size, how many servings can one pour before the contents become undrinkable?  In my case, a serving is 5 oz (148 ml), I drink one serving a day, and I consider club soda too flat to drink if it has less than half of the original fizz (your mileage may vary).  This is the case I will address first, comparing the 10-ounce bottles to the one-liter bottles.

Let’s assume we refrigerate the bottle at 4C (39F).  Once a day, we take it out of the fridge, open it, pour the 5-oz serving, then close the bottle and return it to the fridge, all done as quickly as possible so that the contents do not get a chance to warm up.  We repeat these steps every day until the bottle is empty or the soda is flat.  The following chart (Figure 2) shows the resulting carbonation level for each serving, for both sizes of bottles:

FIGURE 2: Carbonation Level vs Serving Number (One 5-Oz Serving per Day)

We see that the 10-ounce bottle delivers two acceptable servings — the carbonation level of the second serving (57%) satisfies my 50% specification.  But the real surprise is that the one-liter bottle yields just one more drinkable (61%) serving — the fourth serving has only 36% of the original fizz and the fifth 17%.

So, by my criterion, almost 56% of the one-liter bottle is undrinkable.  This increases the effective cost-per-serving of a one-liter bottle from 11 cents to 25 cents.  Even so, it is still less expensive than the 33 cents-per-serving cost of a 10-ounce bottle.

Pouring two servings at a time (1o ounces) makes the one-liter bottle an even better value, but not by much.  The first two servings would be full strength of course, and the next two would have 72% of the original fizz.  The final two servings, however, would drop to 34%. The effective cost-per-serving in this scenario would be 19 cents, and 41% of the contents would still be undrinkable.

To completely eliminate waste from a one-liter bottle, you would have to either (a) lower your fizz-level standards, or (b) belt down half the bottle (3-plus servings) each time that you open it.  Only then will it cost you 11 cents per serving.

So the bottom line is, the one-liter bottle of club soda is the better bargain, even if you have to share some of it with your houseplants.

Keep the Fizz Alive

Want to get the most fizz for your club-soda buck?  Research by Nestlé [1] suggests that soda in one-liter plastic (PET) bottles loses about 10% of its carbonation every 60 days when stored at room temperature.  This is because CO2 gas diffuses through the walls of the bottle.  (Bottlers compensate for the loss in shelf-life by adding extra carbonation to PET bottles.)  There are three takeaways from this:

(1) Buy your club soda from a busy supermarket.  This increases your chance of drinking  recently-bottled higher-carbonation product.

(2) Only buy as many bottles as you will use this week.  For the same reason.

(3) Refrigerate your club soda.  This reduces the internal pressure — and thus the rate of CO2 loss due to diffusion — by about one-third compared to room-temperature storage.  Not only that, cold liquid dissolves more CO2 so less gas is lost when you open the bottle.

Follow this advice and you will earn an A for effervescence.  And don’t forget to recycle.

Appendix

To estimate the carbonation in a bottle of club soda at each step of its product life, I made a number of reasonable simplifying assumptions:

  • The bottle is completely rigid.
  • The gases and liquids follow Henry’s Law and the Ideal Gas Law.
  • When the bottle is filled, the headspace gas is assumed to be pure CO2 .
  • The carbonic acid formed by the reaction of CO2 and water is insignificant.
  • The diffusion of CO2 through the walls of the PET bottle is ignored.
  • Evaporation of the soda is insignificant.
  • The liquid and gas are at equilibrium before the bottle is opened.
  • Bubbles that escape from the soda just before it is poured are ignored.
  • The contents of the bottle remain at 4C (39F) while the soda is poured.
  • When the soda is poured, the gas in the headspace is replaced entirely by air.
  • The amount of CO2 and water vapor in the air can be ignored.

We begin by writing down the initial conditions in each bottle (Figure 3).  I will use liters as the unit of volume, atmospheres as the unit of pressure, and moles for the mass of CO2. (One atmosphere is essentially sea-level pressure, and one mole of CO2 is about 44 grams.)

FIGURE 3: Initial Conditions

A few notes on these facts and figures.  I measured the total volume of each bottle in my kitchen by filling them to the brim with water.  I assumed that the actual amount of soda in each bottle was exactly what the label claimed (bottling equipment is very accurate).  The initial headspace is the difference between those numbers.

According to Steen and Ashurst [2], bottle-filling is usually performed at 14C (about 57F) and the carbonation pressure for club soda is about 60 psi, or just over 4 atmospheres [3].  Assuming that all the gas in the headspace is carbon dioxide, we can use the Ideal Gas Law and Henry’s constant to calculate the amount of CO2 in the gas and liquid as bottled.

Initial moles CO2 in headspace (n0) from the Ideal Gas Law [4]:

\text{(1)}\hspace{5em}n_0 = p_0V_0 / RT_0

where p0 is the CO2 pressure (4.083 atm), V0 is the initial headspace volume (0.010 liters or 0.85 liters depending on bottle size), R is the gas constant (0.08206 liter-atm/mol-deg) and T0 is the filling temperature in degrees Kelvin (14C + 273.15 ≈ 287K).

Initial concentration of CO2 in liquid (c0) from Henry’s Law [5]:

\text{(2)}\hspace{5em}c_0 = p_0H_{14\text{C}}

where H14c is Henry’s constant at 14C (refer to above table). 

We now know the total moles (m0) of CO2 in the bottle:

\text{(3)}\hspace{5em} m_0 = n_0 + c_0L_0

where L0 is the initial liquid volume (0.296 or 1.000 liters depending on bottle size).

Chilling the bottle to 4C does not change the moles of CO2 in the bottle but it does affect the CO2 liquid-to-gas ratio.  We find the new liquid concentration (c1) at 4C by combining the Ideal Gas Law and Henry’s Law, then substituting for pressure p1 and rearranging:

\text{(4)}\hspace{5em} n_1 = p_1V_1/RT_1 = m_1 - c_1L_1\;,

\text{(5)}\hspace{5em} p_1 = c_1/H_{4\text{C}}\;,\hspace{.75em}\text{and so...}

\text{(6)}\hspace{5em} c_1 = m_1H_{4\text{C}}/(L_1H_{4\text{C}}+V_1/RT_1)

where m1m0 L1 = L0 and V1 = V0 (since nothing has been removed from the bottle).  Knowing ci lets us back-calculate n1 and p1 from Equations (4) and (5).

We now pour the first serving of soda, whose volume is LS .  The pour reduces the liquid in the bottle to L2 = L1Land the total moles of CO2 to m2 = m1n1c1 LS , as we assume all CO2 in the headspace is lost.  The headspace volume is now V2 = V1 + LS .  We quickly close the bottle and return it to the refrigerator, so that T2T1 .  While the bottle rests, the CO2 concentration of the liquid gradually decreases from c1 to cand the CO2 gas pressure in the headspace rises to p2 .  The values of c2 and p2 are unknown, but we can solve for them using Equations (5) and (6) with new subscripts:

\text{(5')}\hspace{5em} p_2 = c_2/H_{4\text{C}}\;,\hspace{.75em}\text{and...}

\text{(6')}\hspace{5em} c_2 = m_2H_{4\text{C}}/(L_2H_{4\text{C}}+V_2/RT_2)

This is the procedure I used to generate the CO2 concentration results in this article.

References

[1] Profaizer, Mauro. “Shelf life of PET bottles estimated via a finite elements method simulation of carbon dioxide and oxygen permeability.” Italian Food and Beverage Technology, vol. 48, 2007.

[2] Steen, David P., and Ashurst, Philip R. (editors).  Carbonated soft drinks: formulation and manufacture.  Oxford Ames, Iowa: Blackwell Publishers, 2006.

[3] Spangenberg, Craig. “Exploding Bottles.”  Ohio State Law Journal, vol. 24, 1963, p. 513.

[4] Khan Academy (khanacademy.org) is one of thousands of sites with info on the Ideal Gas Law.  The Ideal Gas Law (pV=nRT) is to chemistry what Newton’s Second Law (F=ma) is to physics.

[5] Choose your own reference: Henry’s Law on Wikipedia or Henry’s Sparkling Water on ice.

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Insomniac with headphonesMy spouse is one of those enviable persons who falls asleep minutes after her head hits the pillow.  And I would be her prime envier, as it is much harder for me to disengage at night.  Things to do, trips to take, ideas to pursue, problems to fix… so much food for thought but so little time to clean the mental plate.

Many people play soft music to help them sleep, but I tend to listen to music analytically, so that usually doesn’t help.  One technique I do use to induce sleep is listening to long, dull podcasts — voices droning on about topics I don’t care about or barely comprehend are effective soporifics for me.

I have bookmarked several such podcasts, which I decided to share here for the benefit of my fellow nerd-insomniacs.  To make this list, a podcast must be at least 30 minutes long and have no ads or jarring theme music.  Speech must be clear and understandable so that one’s brain does not need to work to decipher it.  And extra credit for low-vocal-intonation British or Australian presenters — Americans try too hard not to sound dull.

I give each podcast a Z score, where more Zs correspond to greater monotony.  But I must warn you: repeated listening may lower your ability to remain disinterested.  Naturally, this means the podcast will lose effectiveness as a sleep aid, and you will run the risk of being entertained or learning something.

The Science of Everything   Z Z Z 

A podcast that befits its name, S.O.E. is a series of 98 (to date) 45-to-60-minute monologs by Australian researcher/secularist/philosopher James Fodor.  His podcasts cover topics as wide-ranging as Political Ideologies, Minerals and Rocks, Cell Division, Magnetism and Disturbing Social Psychology Experiments.  Fodor’s comprehensive overviews of his topics resemble long-form Cliff’s Notes.  I do admire the breadth of his interests and the quirky, if not quixotic, nature of his endeavor.  But for insomniacs, many Zs are available here.

Philosophy Now Radio Show   Z Z Z Z Z 

This is a series of 41 radio shows produced between 2011 and 2014 by the London-based magazine Philosophy Now.  Most of them were hosted by the magazine’s editor, the genial and pleasant-voiced Grant Bartley.  He and his guests discussed philosophy basics such as  right and wrong, tragedy in life, free will, and all the usual philosopher-suspects: Socrates, Kant, Nietzsche, Schopenhauer, Wittgenstein, Hume and Hegel.  Thankfully, for a show about philosophy, there are no annoying arguments about the meaning of trivial words.  However, Grant would often include a musical interlude halfway into the show — best to fall asleep before then.

In Our Time   Z Z Z Z 

In Our Time is a long-running show on BBC Radio 4 in the UK, touching on philosophy, culture, religion, history and science.  Each 40-minute-long show features Melvyn Bragg and two or three guests engaging in a rather orchestrated discussion of topics as diverse as Papal Infallibility, The Muses, Circadian Rhythms, Cicero, and Pauli’s Exclusion Principle.  With over 800 podcasts at your fingertips, it is easy to find something uninteresting.

MCMP Philosophy of Physics   Z Z Z Z 

The Munich Center for Mathematical Philosophy, part of the five-centuries-old Ludwig-Maximilians-Universität in Germany, recorded various workshop lectures on cosmology, gravity, time and relativity, from 2013 to 2015.  Although these podcasts are not posted at the university’s website, 65 of them are still available at player.fm.  All are in English but they vary in understandability.  I listened to several lectures all the way through, including Leibniz, Mach and Barbour and On Causal Explanations of Quantum Correlations, which shows that some talks are more accessible than others.  But most listeners can easily Z out.

Perimeter Institute Video Library    Z Z Z Z Z 

The Perimeter Institute for Theoretical Physics in Waterloo, Ontario, Canada, records all of its public lectures, classroom lectures, and student dissertations.  The public lectures, as the name suggests, are intended for anyone with a sense of curiosity and so are unsuitable for insomniacs.  But the other lectures and dissertations are inscrutable enough to make most people fall asleep — as I am sure many of the attendees do.  The background clack of chalk on chalkboard only enhances the monotony.  Hundreds of lectures are posted here — head for the ones with “field theory” or “condensed matter” in the title.

Your Money & You   Z Z Z 

Your Money & You is a Sunday morning talk-radio show presented on KDKA-AM by the Pittsburgh financial advisory firm Hefren-Tillotson.  The show is hosted by 70-year-old Executive VP James Meredith, who tends to offer reasonable but often rambling advice to listeners and callers.  Meredith can get a little cranky at times and is quite evidently an Anti-Regulation Republican.  The show is fairly boring during the first half-hour when Meredith reviews the week’s financial news — but it reaches and maintains peak boredom for 90 minutes if there is a substitute host.  The firm’s website has the latest four shows.

A Way with Words   Z Z 

This show would seem to have a lot going for it.  Billed as “a public radio program about language examined through history, culture, and family,” A Way with Words sounds like it should be an easy snooze any time of day.  Co-hosts Martha Barnette and Grant Barrett field calls (Hi, this is Sophia, I’m calling from Naperville, Illinois, and I’m thirteen!) about the origins of various American idioms and funny-sounding names.  The hosts keep things light and bubbly and inconsequential.  The problem for insomniacs is not that the show is engaging but that it is so gosh-darn annoyingly bland, one can barely stand listening to it.

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• On a recent edition of CBS Sunday Morning, host Jane Pauley interviewed cartoonist Garry Trudeau.  My spouse remarked how odd it must be to interview one’s own husband.  “Trudeau is Jane Pauley’s husband?” I said.  “Yes, how could you not know that?” she said. In my defense, I do know who was the first to verify Einstein’s Theory of General Relativity (namely, astronomer Arthur Eddington, in 1919).  But how could anyone not know that?

• Speaking of Einstein: Alan Lightman, author of the 1993 novel Einstein’s Dreams, made a seemingly innocuous observation in the third paragraph of his prologue as he described “… the young man’s desk … cluttered with half-opened books …”  I had nearly finished the paragraph when I stopped to ask myself, what exactly is a half-opened book?

• Life is like calculus: a lot of unnoticeable differences make a difference.

• Rank these Nabisco snack products from highest to lowest fat content: Oreos cookies, Cheese Nips crackers, Chicken in a Biskit crackers, Ritz crackers, Nutter Butter cookies.  Answers below (and no fair looking at their so-called “Nutrition Facts” labels).

• The actor Meryl Streep has appeared in over 55 films and has won three Oscars and eight Golden Globes.  Netflix streams only three of her films, Julie and Julia, Mamma Mia! and The River Wild.  These may have been very good films, but they clearly do not measure up to Happy Gilmore or The Ridiculous 6 — just two of the thirteen films on Netflix starring zero-time Oscar or Golden Globe winner Adam Sandler.

Ritz wins with 28% fat, followed by Chicken in a Biskit at 26%, Nutter Butter at 21.4%, Oreos at 20.5% and Cheese Nips at 20.0%.  See, Oreos aren’t as bad as you thought.

• Saying “oops” when you accidentally honk your horn and no one is around.  Closing the bathroom door for privacy when you are alone in the house.  Reaching for the light switch when you enter a room during a power outage.  These are a few of my behavioral things.

• A guy with an identity crisis answers the phone and says, “Hello, is this me?”

• In early 2017, I wrote here about the so-called Trump Rally in the U.S. stock market and pointed out how the rise in stock prices after his election was nothing unusual, despite all the chest-thumping by Trump and his shills.  Now, as Trump enters his third year, there is not even a mirage of a rally for him to take undeserved credit for — the S&P 500 index is exactly where one would expect it to be based on its long-term trend (7.1% per year).

• I like to think of myself as an evidence-based person — I am skeptical of product (and political) claims, I never buy extended warranties (companies would not sell them if they didn’t make money) and I don’t pray.  Yet I harbor my own superstitions.  I take lutein every day on the off-chance it will preserve my eye health.  I don’t root for athletes to fail (except for Tom Brady) because of the what-goes-around-comes-around principle.  And I always tip well, in the belief that those extra couple of dollars will matter to someone.

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