Let’s add hydrogen gas

Hydrogen: a page from Theodore Gray's book, The Elements
Hydrogen: a page from Theodore Gray’s book, The Elements

I’m going to add 1.00 mol of hydrogen gas, H2(g), to our 10-litre vessel. We’ll assume that the entire experiment is carried out at normal room temperature – let’s say it’s 20°C.

How much does a mole of hydrogen cost?

Hydrogen gas is a relatively cheap element, and my one mole of H2(g) would cost less than one cent at wholesale prices. That said, the shipping, handling and service fee would be a couple of orders of magnitude greater than the cost of the gas itself, and I’d probably need to give the store about a dollar for the privilege of taking one cent’s worth of hydrogen gas.

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What does a mole of hydrogen look like?

We’ll assume the temperature is 20°C and the atmospheric pressure is 102.3 kPa, which is what the Weather app on my phone is reading right now. After the hydrogen gas has been released from its high-pressure storage cylinder, my one mole of H2(g) would have a volume of 23.8 litres at these conditions. That’s about enough hydrogen gas to fill up a party balloon.

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Hydrogen is a colourless, odourless gas that’s lighter than air. It’d float upwards very quickly if I opened the valve in the store. I’m now going to squeeze all that gas into my 10-litre vessel.

What’s the resulting pressure of the vessel?

If I squeeze that 23.8 litres of hydrogen gas into my 10-litre vessel, the resulting pressure in the vessel must be greater than atmospheric pressure (1 atm) because I’ve compressed the gas. We can calculate the final pressure precisely by using the ideal gas law: PV=nRT.

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That’s significantly higher pressure than atmospheric pressure, which varies from 100 kPa to 102 kPa under normal weather conditions.

Interestingly, the pressure in the vessel, 243 kPa, is equal to 35.2 pounds per square inch (psi), which is the same as the recommended pressure for a car tyre.

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Other than making random movements inside the vessel, the hydrogen molecules won’t really do anything else.

How fast are the molecules moving about?

We can calculate the average speed of the molecules by using the following equation:

*Note that R is the gas constant, 8.31, and M is the molar mass in kg/mol

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The molecules are travelling at about 1760 metres per second (on average).

How much distance will the gas molecules travel before they collide with one another?

For this question, we need to calculate something called mean free path. The mean free path is the average distance we can expect each molecule to travel before it collides with another molecule. Mean free path is quite long in a vacuum, and very short at high pressure conditions. One of the formulae used to calculate mean free path, λ, is shown below.

**Note that in this formula, pressure (P) must be measured in pascals (Pa)- not kilopascals (kPa). We therefore need to multiply our kilopascal pressure by 1000 to convert it from kPa to Pa.

*** Note also that d is the diameter of the molecules being studied in metres. Wikipedia tells us that hydrogen molecules have a diameter of 120 picometres. I’ve used this value in the equation below.

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The molecules in our vessel collide with each other roughly every 260 nanometres. That’s tiny: it’s just a few percent of the width of a cell nucleus!

How often do the molecules collide?

Let’s go right back to Year 10 Physics for this one. The time between collisions will be equal to the average distance travelled between collisions divided by the average speed of the molecules:

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The molecules collide with each other roughly every 0.1478 nanoseconds.

How many times do the molecules bump into each other each second?

By taking the reciprocal of the average collision time, we can find out how many times the molecules collide with each other every second, on average:

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Each molecule in our ten-litre vessel makes 6.765 billion collisions per second with neighbouring molecules.

Apart from lots of uneventful particle collisions – a total of 4.07 decillion uneventful collisions per second to be precise – not much else is happening in our ten-litre vessel at this stage.

Conclusion

  • Hydrogen gas, H2(g): 1.00 mol

Next week, we’ll add some helium to the vessel and see what happens.

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