A gas cylinder is mounted vertically and sealed by a freely...

A gas cylinder is mounted vertically and sealed by a freely movable light piston. The external air pressure is negligible. A mass $m_1$ is placed on the piston, and initially the system is in equilibrium, such that the weight of this mass is supported by pressure forces provided by the gas. The piston is then temporarily prevented from moving, while the mass is increased to $m_2$, and then the piston is released. Treating the gas as ideal with $\gamma=5 / 3$, find, in terms of $m_1$ and $m_2$, the ratios by which the volume and temperature of the gas change: (i) in the case $\mathrm{A}$, where the piston moves freely and the total system energy $U_{\text {tot }}$ stays constant (where by $U_{\text {tot }}$ we mean the sum of gravitational potential energy of the mass and internal energy of the gas) (ii) in the case B, where the piston is released gradually, such that the change in volume is quasistatic and without heat exchange. [Ans. $V_2 / V_1=\left(3 m_1+2 m_2\right) / 5 m_2$ and $\left(m_1 / m_2\right)^{3 / 5} ; T_2 / T_1=\left(3 m_1+\right.$ $\left.2 m_2\right) / 5 m_1$ and $\left.\left(m_2 / m_1\right)^{2 / 5}\right]$ This experiment is explored further in Exercise 7.2 of chapter 17 .

A gas cylinder is mounted vertically and sealed by a freely movable light piston. The external air pressure is negligible. A mass $m_1$ is placed on the piston, and initially the system is in equilibrium, such that the weight of this mass is supported by pressure forces provided by the gas. The piston is then temporarily prevented from moving, while the mass is increased to $m_2$, and then the piston is released. Treating the gas as ideal with $\gamma=5 / 3$, find, in terms of $m_1$ and $m_2$, the ratios by which the volume and temperature of the gas change:
(i) in the case $\mathrm{A}$, where the piston moves freely and the total system energy $U_{\text {tot }}$ stays constant (where by $U_{\text {tot }}$ we mean the sum of gravitational potential energy of the mass and internal energy of the gas)
(ii) in the case B, where the piston is released gradually, such that the change in volume is quasistatic and without heat exchange.
[Ans. $V_2 / V_1=\left(3 m_1+2 m_2\right) / 5 m_2$ and $\left(m_1 / m_2\right)^{3 / 5} ; T_2 / T_1=\left(3 m_1+\right.$ $\left.2 m_2\right) / 5 m_1$ and $\left.\left(m_2 / m_1\right)^{2 / 5}\right]$ This experiment is explored further in Exercise 7.2 of chapter 17 .

Key Concepts

Specific Heat Ratio (?)

The specific heat ratio, ?, is the ratio of the specific heat capacities at constant pressure and constant volume (Cp/Cv). This parameter is critical in determining the behavior of an ideal gas during adiabatic processes, as it influences the relationship between pressure, volume, and temperature changes, and plays a central role in the derived relationships for the gas dynamics in phenomena such as piston movements.

Energy Conservation in Combined Systems

In systems where different forms of energy coexist, such as the internal energy of a gas and the gravitational potential energy of an overlying mass, the principle of energy conservation must account for all these contributions. Understanding how energy is transferred among different forms—like the conversion between potential energy and internal energy—is essential for analyzing the complete thermodynamic behavior of the system.

Quasistatic Process

A quasistatic process occurs so slowly that the system remains nearly in equilibrium at each stage of the transformation. This allows the use of equilibrium thermodynamic relationships throughout the process and is particularly important for accurately applying adiabatic and other thermodynamic relations, as reversible conditions are approximated.

Adiabatic Process

An adiabatic process is one in which no heat is exchanged between the system and its surroundings. For ideal gases undergoing adiabatic transformations, the relationship PV^? = constant holds. This concept is crucial for understanding how changes in pressure and volume affect the temperature of the gas when the process occurs without any heat input or output.

Ideal Gas Law

The ideal gas law relates pressure, volume, and temperature of a gas through the equation PV = nRT. It is fundamental in understanding how gases behave when subjected to changes in external conditions, and forms the basis for analyzing processes where these state variables change in response to work being done or energy being transferred within the system.

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