# Water Pressure in a Sealed Vessel

#### Dyslexic Poet

Hello,

I am a PhD student in chemistry, and need to determine the pressure of a reaction to be carried out

I have a steel pressure vessel with an inner Teflon container (total volume = 270 mL) which is placed in an oven at 250°C. I wish to put 240 mL of water into this container (along with 1 g of material). At this temperature, I know that the head-space pressure to be approximately 40 bar. However, the relative specific volume of water at 250°C (to that at RT) is 1.25. I believe this means that my water wants to take up a volume of 300 mL. As this is 30 mL larger than the container itself, I am not sure if it will therefore be exerting an extra pressure much larger than the head-space pressure.

I've tried a few different ways to calculate a value for this second pressure, but to no avail. One was to use water's compressibility and bulk modulus, but keep ending up with a value upwards of 2500 bar, which is just ridiculous.

Any help would be appreciated in helping me solve my problem. Please ask if I have left out any information required of the system.

Thanks,

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#### Nugatory

Mentor
I've tried a few different ways to calculate a value for this second pressure, but to no avail. One was to use water's compressibility and bulk modulus, but keep ending up with a value upwards of 2500 bar, which is just ridiculous.
If the liquid wants to occupy a volume greater than the 270 cc that you have available for it, that number is not at all unreasonable. Liquids are pretty much incompressible.

To really appreciate just how incompressible liquids can be, you have to tear down and rebuild an automobile engine that has been destroyed by hydrolock: If the volume of the combustion chamber at maximum compression is 40 cc and somehow you've allowed more than 40 cc of liquid into the cylinder before the piston comes up... The piston doesn't come up and compress the liquid down to 40 cc. Instead, the piston stops cold and the connecting rod (which is a no-kidding serious piece of steel) bends or breaks.

#### AlephZero

Homework Helper
The bulk modulus of steel is about 160 GPa compared with 2.2 GPa for water, but to get the "bulk stiffness" of the pressure vessel you have to reduce the 160 by a factor of the order of (thickness/length) of the vessel. So the effective bulk modulus of the vessel could be a similar order of magnitude as the bulk modulus of the liquid inside.

The fact that you would probably need to expand the volume of the pressure vessel by a few percent without bursting it should be enough to see this isn't a realistic situation, without doing a more "accurate" calculation.

#### Chestermiller

Mentor
Does the container also expand with temperature? By how much?

Chet

#### Dyslexic Poet

Thank you for the replies.

I can tell you that the width of the vessel walls are 10 mm. The Teflon container will expand, but I don't believe by any significant amount.

The same experiment has been carried out successfully (many dozens of times), in a similar container with a total volume of 90 mL (it's walls are 15 mm thick). For this, only 80 mL of water was used, so the ratios are kept constant. This vessel is sitting on my desk looking absolutely fine, which suggests to me that the pressure couldn't be that high.

I have this calculation, but I am dubious of it:

Volume of bomb: 270 mL
Volume of water at 20-25 C placed in bomb: 240 mL
Volume of water at 250 C: 1.25x240 = 300 mL (www.engineeringtoolbox.com/water-thermal-properties-d_162.html)

Isothermal compressibility of water, k: 4.8 x 10-10 Pa-1 (at 100°C - this value I believe to be incorrect, and I will explain in a moment why I am using its value at only 100°C)

k = - (1/V)(dV/dp)T
Δp = - (1/k) (ΔV/V) = (1/4.8x10-10)(30/270)
Δp = 2300 bar

The following link is to the site where I took this k value from (which unfortunately only goes up to 100°C): http://physchem.kfunigraz.ac.at/sm/Service/Water/H2Obetat.htm [Broken]

However, I have in front of me three literature articles that state that at 100°C, k = 49.1 x 106 bar-1 (4.91 x 1012 Pa-1). Which is 20 orders of magnitude apart. My thermal physics knowledge runs out completely here, and have no idea which is right and why.

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#### Chestermiller

Mentor
Thank you for the replies.

I can tell you that the width of the vessel walls are 10 mm. The Teflon container will expand, but I don't believe by any significant amount.

The same experiment has been carried out successfully (many dozens of times), in a similar container with a total volume of 90 mL (it's walls are 15 mm thick). For this, only 80 mL of water was used, so the ratios are kept constant. This vessel is sitting on my desk looking absolutely fine, which suggests to me that the pressure couldn't be that high.

I have this calculation, but I am dubious of it:

Volume of bomb: 270 mL
Volume of water at 20-25 C placed in bomb: 240 mL
Volume of water at 250 C: 1.25x240 = 300 mL (www.engineeringtoolbox.com/water-thermal-properties-d_162.html)

Isothermal compressibility of water, k: 4.8 x 10-10 Pa-1 (at 100°C - this value I believe to be incorrect, and I will explain in a moment why I am using its value at only 100°C)

k = - (1/V)(dV/dp)T
Δp = - (1/k) (ΔV/V) = (1/4.8x10-10)(30/270)
Δp = 2300 bar

The following link is to the site where I took this k value from (which unfortunately only goes up to 100°C): http://physchem.kfunigraz.ac.at/sm/Service/Water/H2Obetat.htm [Broken]

However, I have in front of me three literature articles that state that at 100°C, k = 49.1 x 106 bar-1 (4.91 x 1012 Pa-1). Which is 20 orders of magnitude apart. My thermal physics knowledge runs out completely here, and have no idea which is right and why.
The 4.8x10-10 is correct.

Incidentally, the coefficient of volume thermal expansion for teflon is 3x10-4 per degree C. So the volume of the teflon container will increase significantly also. See what happens if you take this into account.

Chet

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