If V^' is the volume of a vessel, which is to be evacuated...

If $V^{\prime}$ is the volume of a vessel, which is to be evacuated by means of a piston air pump, then how many strokes are required to reduce the pressure in the vessel $\eta$ times? One piston stroke captures the volume $\Delta V$. Assume the process to be isothermal and the gas ideal: (a) $n=\frac{\log \eta}{\log \left(1-\frac{\Delta V}{V}\right)}$ (b) $n \cdot=\frac{\log \eta}{\log \left[1+\frac{\Delta V}{V}\right)}$ (c) $n=\frac{\log \eta}{\log \left(\frac{\Delta V}{V}\right)}$ (d) None of these

If $V^{\prime}$ is the volume of a vessel, which is to be evacuated by means of a piston air pump, then how many strokes are required to reduce the pressure in the vessel $\eta$ times? One piston stroke captures the volume $\Delta V$. Assume the process to be isothermal and the gas ideal:
(a) $n=\frac{\log \eta}{\log \left(1-\frac{\Delta V}{V}\right)}$
(b) $n \cdot=\frac{\log \eta}{\log \left[1+\frac{\Delta V}{V}\right)}$
(c) $n=\frac{\log \eta}{\log \left(\frac{\Delta V}{V}\right)}$
(d) None of these

Key Concepts

Isothermal Process

An isothermal process is one in which the temperature remains constant throughout the process. In the context of gas dynamics, this means that the product of pressure and volume remains constant (as per the ideal gas law), allowing the pressure to be directly related to changes in volume. This concept is crucial for understanding how the pressure in a vessel changes when some volume is removed repeatedly, especially when the system is allowed to equilibrate at constant temperature.

Ideal Gas Law

The ideal gas law (PV = nRT) relates the pressure (P), volume (V), amount (n) of gas, and temperature (T) in an idealized gas. In problems involving gas evacuation, the ideal gas law is used to predict changes in pressure when the gas volume changes, assuming the behavior of the gas follows this law. It forms the foundation for analyzing how successive removal of a given volume affects the overall pressure in the vessel.

Exponential Decay of Pressure

During an evacuation process where a constant fraction of gas is removed each stroke, the pressure in the vessel decreases in an exponential (or geometric) manner with respect to the number of strokes. This exponential decay arises because the remaining gas after each stroke is a fixed proportion of the previous amount, leading to a multiplicative decrease in pressure over successive strokes.

Logarithmic Analysis

Logarithmic analysis is applied to solving problems where the outcome is exponentially related to the number of steps or strokes, as in the case of pressure reduction via a piston pump. By taking the logarithm of the exponential decay formula, one can solve for the number of strokes needed to achieve a desired reduction in pressure. This technique is essential when rearranging formulas to isolate the number of strokes in evacuation problems.

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