A chamber of unknown volume is filled with an unknown amount...

A chamber of unknown volume is filled with an unknown amount of ideal gas. A cylinder with an open bottom and calibrated volume is connected to the chamber and is closed with a movable piston. A gas pressure sensor is also mounted on the chamber (Figure $P 12.72$ ). The whole experimental setup is kept at a constant temperature of $300 \mathrm{K}$. Doris moves the piston from mark 0 to mark $200 \mathrm{mL}$ in steps of $50 \mathrm{mL}$ and records the corresponding pressure in the chamber. She moves the piston very slowly and waits several minutes before recording the pressure values. Use Doris's data in the table to determine the volume of the chamber and the number of moles of gas. (Note: This is a problem that requires linearization of data.) $$ \begin{array}{cc} \hline V_{\text {Cy }}(\mathrm{mL}) & P\left(10^{5} \mathrm{N} / \mathrm{m}^{2}\right) \\ \hline 0 & 4.35 \\ 50 & 3.85 \\ 100 & 3.50 \\ 150 & 3.15 \\ 200 & 2.90 \\ \hline \end{array} $$

A chamber of unknown volume is filled with an unknown amount of ideal gas. A cylinder with an open bottom and calibrated volume is connected to the chamber and is closed with a movable piston. A gas pressure sensor is also mounted on the chamber (Figure $P 12.72$ ). The whole experimental setup is kept at a constant temperature of $300 \mathrm{K}$. Doris moves the piston from mark 0 to mark $200 \mathrm{mL}$ in steps of $50 \mathrm{mL}$ and records the corresponding pressure in the chamber. She moves the piston very slowly and waits several minutes before recording the pressure values. Use Doris's data in the table to determine the volume of the chamber and the number of moles of gas. (Note: This is a problem that requires linearization of data.)
$$
\begin{array}{cc}
\hline V_{\text {Cy }}(\mathrm{mL}) & P\left(10^{5} \mathrm{N} / \mathrm{m}^{2}\right) \\
\hline 0 & 4.35 \\
50 & 3.85 \\
100 & 3.50 \\
150 & 3.15 \\
200 & 2.90 \\
\hline
\end{array}
$$

Key Concepts

System Equilibrium

Maintaining system equilibrium, particularly thermal equilibrium, is critical in experiments involving gas measurements. By moving the piston slowly and allowing the gas to equilibrate after each alteration, the system ensures that the temperature remains constant and the pressure readings are reliable, which is a necessary condition for applying both the ideal gas law and Boyle's Law accurately.

Ideal Gas Law

This principle relates the pressure, volume, temperature, and number of moles of a gas through the equation PV = nRT. It is a cornerstone of thermodynamics and enables one to predict behavior of gases under varying conditions provided the gas behaves ideally, making it essential for determining unknown quantities when some parameters are measured.

Boyle's Law

Boyle's Law is a specific case of the ideal gas law when temperature remains constant. It states that the pressure of an ideal gas is inversely proportional to its volume. This concept is fundamental when analyzing experiments where changing the volume (such as by moving a piston) leads to predictable changes in pressure, allowing for determination of unknown volumes or gas amounts.

Data Linearization

Linearization is a technique used to transform a nonlinear relationship into a linear one, making data analysis more straightforward. In the context of gas laws, by plotting the appropriate variables (for instance, pressure versus volume or its reciprocal), one can obtain a straight-line relationship whose slope and intercept can be used in regression analysis to extract physical parameters like the chamber’s volume and the number of moles of gas.

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