How to use physics to make yourself look good

idealgasBack in August, the UK government administration (collectively known as Whitehall) was criticized by Members of Parliament for failing to meet their own carbon emission targets. On September 15, UK Cabinet Office minister Angela Smith claimed in a BBC article that Whitehall had “saved enough carbon dioxide to fill almost 2,500 Olympic-sized swimming pools.”

2,500 Olympic-sized swimming pools sounds like a lot. Swimming pools are big, after all, and 2,500 of them would hold lots of water. But when I dug a little further, I found that Whitehall’s carbon dioxide (CO2) emissions were actually reduced by only 12,000 tonnes, a nearly negligible amount.

To put this into perspective, in 2006, the Energy Information Administration estimated that the total emissions for the United Kingdom was 585.71 million metric tons (aka tonnes). 12,000 tonnes is only 2 thousandths of one percent of the UK’s total emissions three years ago.

So how did we get from 12,000 tonnes to “2,500 Olympic-sized swimming pools?” It’s called the ideal gas equation.

Look at the image above – the equation shown is the ideal gas equation, where P is pressure, V is volume, n is the number of moles of gas (a measurement of the number of atoms of gas, which id directly proportional to the mass of the gas), R is the universal gas constant, and T is the temperature of the gas on the Kelvin scale (Kelvin is equal to degrees Celsius + 273.15). While the ideal gas equation doesn’t perfectly represent the real behavior of gases, it’s close enough that it’s pretty commonly used by scientists and engineers alike.

If you look deeper into the equation, you find that it shows essentially three relationships:

  • Gas pressure increases when you add more gas to a constant volume at a constant temperature. Gas pressure also increases when you heat the gas up at a constant volume and constant mass of gas. And gas pressure increases if you reduce the volume the gas is contained in for a constant amount of gas and a constant gas temperature. (Pressure varies proportionally to mass and temperature and inversely proportionally with volume.)
  • Gas volume increases when you add more gas at a constant temperature and pressure. Gas volume goes up if you heat it up at a constant pressure and mass of gas. And gas volume goes up if you reduce the pressure for a constant amount of gas and gas temperature. (Volume varies proportionally to mass and temperature and inversely proportionally to pressure.)
  • Gas temperature increases if you increase the pressure but while holding the volume and mass of gas constant. Temperature also goes up if you increase the gas’ volume while holding the pressure and mass of gas constant. And temperature increases if you decrease the amount of gas in a constant volume and at a constant pressure. (Temperature varies proportionally to pressure and volume and inversely proportionally to mass.)

Some of these seem counter-intuitive upon first examination, but these properties of ideal gases are used all the time to produce liquid gases, pressurize oxygen for use in medical O2 canisters, even to cool your body by evaporation.

Using the density of CO2 gas, a value that is calculated using the ideal gas law, Smith or her staff calculated the volume that 12,000 tonnes of CO2 would take up at a given atmospheric pressure and temperature. At one atmosphere (atm) of pressure and a temperature of 273.15 Kelvin (0 °C), 12,000 tonnes of CO2 takes up 2,427.9 Olympic-sized swimming pools. And that’s close enough to the reported value of “2,500 Olympic-sized swimming pools” that it’s reasonable to say that this calculation is almost certainly where Smith got her swimming pool number from. (When I looked up the volume of an Olympic-sized swimming pool, I found that the volume is at least 2,500 cubic meters (or 2.5 million liters), but could be greater if the pool is deeper than the minimum 2.0 meters required by Olympic standards.)

But that’s not the end of it. The density of CO2 gas is defined at a particular pressure and temperature, specifically 1 atm and 0 °C. If you cut the mass of gas to 6,000 tonnes (a reduction of n by 1/2) and held the temperature and the volume the same (0 °C and 2,500 Olympic-sized swimming pools respectively), all that would happen is that the pressure would fall to 0.5 atm. Similarly, cutting the mass of CO2 by a factor of 100 (from 12,000 to 120 tonnes) would still fill up 2,500 Olympic-sized swimming pools – at an atmospheric pressure of only 0.01 atmospheres.

So Smith could just as accurately, from the standpoint of physics, claimed that nearly any amount of reductions produced a volume of “2,500 Olympic-sized swimming pools.” And given the fact that the BBC article neglected to mention that the mass of the reductions was 12,000 tonnes, no-one who just read the BBC would have caught the deception.

It seems reasonably likely that the reason that Ms. Smith or her staff converted mass into volume specifically because Whitehall would look better, and thus deflect some criticism, using the larger volume number. But the problem is that the exact same physics games can be used to make exceptional progress in cutting emissions look insignificant. Here’s three quick examples.

Just as 2,500 Olympic-sized swimming pools can be the volume of a gas at 0.01 atm, it can be the volume of CO2 gas at 10 atm too – crank the pressure up to 10 atm, hold the temperature and volume the same, and shazam! those swimming pools now hold 120,000 tonnes of CO2 gas.

Better yet – compress CO2 until it becomes a liquid (at 56 atm and 20 °C) and you increase the density of the CO2 from 0.001977 kg/L to 0.77 kg/L, an increase of 389x. Suddenly someone who is trying to downplay a reduction of 4.81 million tonnes of CO2 emissions can claim, perfectly accurately according to the physics, that it only fits into 2,500 Olympic-sized swimming pools (the math is 2.5 million L/ossp * 0.77 kg/L * 2,500 ossp, where “ossp” is “Olympic-sized swimming pool”). Even better – turn CO2 into dry ice at 1 atm and -78.5 °C and you more than double the density again. Now those same 2,500 Olympic-sized swimming pools represent 9.76 million tonnes of CO2.

For comparison, 9.76 million tonnes of CO2 represents about 1.67% of the UK’s entire CO2 emissions in 2006. That’s a far cry from the 0.002% that Smith attempts to trumpet by way of her “2,500 Olympic-sized swimming pools” quote in the BBC and Kable articles

Clearly, using volume as a proxy for the value you really care about is almost entirely meaningless. Sticking with mass, even when it makes you look bad, is by far the more accurate and directly comparable measurement.

If Cabinet Office minister Angela Smith doesn’t want her opponents to start using her own physical manipulations against her, then she should probably not attempt to disguise Whitehall’s poor track record of cutting CO2 emissions.