Question

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the cylinder in the drawing contains 4.53 mol of an ideal gas. by moving the piston, the volume of the gas is reduced to one - fourth its initial value, while the temperature is held constant. how many moles δn of the gas must be allowed to escape through the valve, so that the pressure of the gas does not change?
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Explanation:

Step1: Recall Ideal Gas Law (PV = nRT)

For constant pressure (\(P\)) and temperature (\(T\)), \(V \propto n\) (since \(R\) is constant). So \(\frac{V_1}{n_1} = \frac{V_2}{n_2}\).

Step2: Define Initial and Final Volumes/Moles

Let initial volume be \(V_1\), final volume \(V_2 = \frac{V_1}{4}\), initial moles \(n_1 = 4.53\ \text{mol}\), final moles \(n_2\).

Step3: Substitute into Proportion

From \(\frac{V_1}{n_1} = \frac{V_2}{n_2}\), substitute \(V_2 = \frac{V_1}{4}\):
\(\frac{V_1}{4.53} = \frac{\frac{V_1}{4}}{n_2}\). Cancel \(V_1\) (non - zero), solve for \(n_2\):
\(n_2 = \frac{4.53}{4} = 1.1325\ \text{mol}\).

Step4: Calculate Moles Escaped (\(\Delta n\))

\(\Delta n = n_1 - n_2 = 4.53 - 1.1325 = 3.3975\ \text{mol}\) (≈ 3.40 mol).

Answer:

\(\Delta n = 3.40\ \text{mol}\) (or 3.3975 mol, depending on precision)