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.

### The Picture Frame Safety Factor Calculator

At last, the calculator.  This calculator lets you estimate the tension in the wire and the inward deflection of the sides of the frame, based on your dimensions and wiring setup.  This necessarily involves a number of assumptions, which I will discuss after presenting the calculator.

To evaluate the safety factors in your frame, you will need to enter the dimensions shown in the figure below:

First, indicate whether you have one wall hook or two.  (Before doing this, you might want to consult The “Hang It with Two Hooks” Calculator for my two-hook recommendation.)  Next, select whether you will enter the weight of your frame or let the calculator estimate the weight from its construction.

Now enter the frame dimensions, starting with the overall width and height (W and H) and the total length of wire (L).  If you are using D-rings, enter the length (D) from the hole to the tip.  But if your wire is attached directly to the frame, enter zero for that value.

Next, enter the distance (B) between the D-ring fastening screws (or wire fastening points if there are no D-rings).   If you indicated you are using two wall hooks, you will be asked to enter the distance (X) between the hooks.

Finally, enter the dimensions of the frame molding and the breaking strength of the wire.  It is possible you may not know these values, so here is some guidance:

For the cross-section of the molding, enter the face width of the molding (F) and the average thickness of the molding (T).  Frame moldings can have complicated profiles, so do your best to estimate average thickness.  The more curves in the molding profile, the greater uncertainty there will be in the estimated deflection.

For the breaking strength of the wire, enter the value if you know it; otherwise enter 2.5 times the rated weight.  If you don’t know that, make a conservative guess such as 50 lbs. or less.  Corroded or kinked wire is likely to have a lower breaking strength than new wire — any wire is only as strong as the weakest point along its length.

When you are finished, click CALCULATE to validate your entries and show the results. The calculator will estimate the tension in the wire and tell you what percentage of the breaking strength this represents.  (With wire and cable, it is common to use a 5x safety factor, which implies the tension should be no more than 20% of its breaking strength.)  The calculator will also estimate the inward deflection of the sides of your frame.  I suggest that if the deflection is more than one-third the typical clearance (1/16th-inch all around) between the frame and its contents, then you are in danger of damaging the artwork and/or glass.  Do not count on the glass to reinforce a frame: it is the job of the frame to support the art and the glass.

As promised, here is a list of my assumptions:

1. The estimated weight (if selected) assumes 2.5 mm soda-lime glass (if selected) with 2.5 specific gravity, wood frame molding with 0.4 specific gravity, and other materials at 0.002 lb / in².
2. Elongation of the wire due to tension in the wire is ignored.
3. The force pulling inward on the side of the frame is assumed to be concentrated at a point one-third of the way down from the top of the frame.  The corners of the frame are assumed to be stationary.
4. The calculator does not evaluate the integrity of the miter joints or the fasteners.
5. The side of the frame is assumed to bend as if it had a rectangular cross-section.
6. The amount of bend in the frame is inversely proportional to the elastic modulus of the wood.  The elastic modulus is assumed to be 1,500,000 lbf / in², which is a mid-range value for typical framing woods (see reference).

If the calculator warns you about tension or excessive bending of your frame, I suggest you buy some #5 vinyl-coated wire and consult The “Hang It with Two Hooks” Calculator to find a more frame-friendly wiring method.  Also, be aware that the taller the frame or the narrower the molding, the more that its sides will bend inward for a given tension.

And now I must add my usual disclaimer.  This calculator makes it easy for you to estimate the safety factors in your framing situation — but because of the assumptions involved, the results should only be treated as estimates.   The calculator may indicate a problem where there is none, or it may fail to warn you that a problem exists.  I offer this calculator as a convenience but I assume no liability for damage of any kind, even if my suggestions are followed exactly.  You bear full responsibility for choosing to use this information.

That concludes my three-part series on framing with wall hooks and wires.  I believe this is one of the most exhaustive (hopefully not exhausting) treatments of this topic that you will find on the internet.  I have tried to make it as accessible as possible.  Please let me know if you find the calculator useful, or if you have problems or discover bugs while using it.