How to compute the energy needed to compress the water isothermally?

In summary, the discussion involves calculating the energy required to compress water by 46.4 parts per million at a pressure of 100 atmospheres. The original answer given is different from the one provided by the Chemistry math expert/Professor, who takes into account the changing pressure and finds an upper estimate of 506 Joules. However, the actual answer is likely to be around half of that, approximately 229 Joules. The question of whether the temperature of the water at 20 degrees Celsius needs to be considered is also raised.
  • #1
WMDhamnekar
MHB
376
28
Hi,
Answer given is $E_n=29.4 Joules$ Here is the question.

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Answer provided by the Chemistry math expert/Professor is as follows but it is different from the answer given. How is that?

Compressibility is the fractional change in volume per unit increase in pressure. For each atmosphere increase in pressure, the volume of water would decrease 46.4 parts per million.
I'll pick a shape for the device, calculate distance traveled and force required, and use $work = force \times distance.$
with 100 atm, volume would decrease by 4640 PPM or by a factor of 0.00464

10 kg of water is about 10 liters or $0.01 m^3$

$100 atm = 1.013e7 Pa$ or $1.013e7 N/m^2$
assume a cube shape, height is $\sqrt[3]{0.01 m^3} = 0.2154435 m$ and base area is$ 0.0464159 m^2$ (coincidence that "464" appears as two different values)
force on piston is $1.013e7 N/m^2 \times 0.0464 m^2 = 470000 N$
that change in volume causes what change in height

new volume $= 0.01 m^3 – 0.01 m^3(0.00464) = 0.00995 m^3$

which has a height of $\frac{0.00995 m^3}{0.0464159 m^2} = 0.2143663 ,$

change is 0.2154435 – 0.2143663 = 0.00108 m
energy = Fd = (470000 N)(0.00108 m) = 506 JIs temperature of water $20^\circ C$ to be considered?
 
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  • #2
Dhamnekar Winod said:
Answer given is $E_n=29.4 Joules$ Here is the question.

Answer provided by the Chemistry math expert/Professor is as follows but it is different from the answer given. How is that?
(snip)
force on piston is $1.013e7 N/m^2 \times 0.0464 m^2 = 470000 N$
that change in volume causes what change in height
new volume $= 0.01 m^3 – 0.01 m^3(0.00464) = 0.00995 m^3$
which has a height of $\frac{0.00995 m^3}{0.0464159 m^2} = 0.2143663 ,$
change is 0.2154435 – 0.2143663 = 0.00108 m

energy = Fd = (470000 N)(0.00108 m) = 506 J
Your expert's answer assumes that the force/pressure is constant at 100 atmosphere, but that is not the case.
Instead it will build up from 1 atmosphere up to 100 atmosphere.
So we can expect the actual answer to be about half of that $506\,J$, which is really an upper estimate.

I found $229\,J$ myself while taking the changing pressure into account with Calculus, which is indeed in the neighborhood of half of that $506\,J$.
Either way, it looks as if the answer of $29.4\,J$ is not correct.

Is temperature of water $20^\circ C$ to be considered?

We take the coefficient of isothermal compressibility of water at $20^\circ C$.
Wikipedia mentions that it is $4.4$ to $5.1×10^{-10}\, Pa^{-1}$ in ordinary conditions.
Close enough to that 46.4 parts per million that you mentioned.
 

Related to How to compute the energy needed to compress the water isothermally?

1. How do you calculate the energy needed to compress water isothermally?

The energy needed to compress water isothermally can be calculated by using the equation W = PΔV, where W is the work done, P is the pressure, and ΔV is the change in volume.

2. What is the difference between isothermal and adiabatic compression?

Isothermal compression is a process where the temperature of the system remains constant, while adiabatic compression is a process where there is no heat exchange between the system and its surroundings. In isothermal compression, the energy needed to compress the water is used to maintain the constant temperature, while in adiabatic compression, the energy is used to increase the temperature of the system.

3. How does the compressibility of water affect the energy needed to compress it isothermally?

The compressibility of water refers to how much the volume of water changes in response to a change in pressure. The higher the compressibility, the more energy is needed to compress the water isothermally. This is because a more compressible substance requires more work to be done to change its volume.

4. Can the energy needed to compress water isothermally be calculated using the ideal gas law?

No, the ideal gas law (PV = nRT) cannot be used to calculate the energy needed to compress water isothermally. This is because water is not an ideal gas and does not follow the assumptions of the ideal gas law.

5. How does the temperature of the water affect the energy needed to compress it isothermally?

The temperature of the water does not affect the energy needed to compress it isothermally. This is because isothermal compression is a process where the temperature remains constant, so the energy needed is solely determined by the change in volume and pressure.

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