
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
A cubic tank with rigid walls, with an edge length of $a=1.2$ meters, is divided by a thin, lightweight, non-deformable piston that moves vertically without friction. Initially, the tank is completely filled with methyl alcohol. How much of an ideal gas needs to be pumped into the space under the piston for it to move a distance of $\Delta h=0.8$ mm? The temperatures of the gas and alcohol are the same, constant, and equal to $t=27^\circ \mathrm{C}$. Given values: the universal gas constant $R=8.31$ J/(K·mol), acceleration due to gravity $g=10$ m/s², density of alcohol $\rho=810$ kg/m³, and the compressibility of alcohol (relative volume change under isothermal pressure change) $\varepsilon=8\cdot 10^{-10}$ Pa⁻¹.

Possible solution. When the piston moves, the relative volume change is

$$
\delta=\frac{a^2 \Delta h}{a^3}=\frac{\Delta h}{a}
$$

The initial pressure of the water at the bottom is

$$
p'=\rho g a
$$

After filling the space under the piston with neon, the pressure will be

$$
p=p'+\frac{\delta}{\varepsilon}=\rho g a + \frac{\Delta h}{a \varepsilon}
$$

Using the Clapeyron-Mendeleev equation, the amount of substance of the ideal gas is obtained:

$$
v=\frac{p V'}{R T}=\frac{a^3 \Delta h}{R T}\left(\rho g + \frac{\Delta h}{\varepsilon a^2}\right) \approx 0.39 \text{ moles}
$$

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