Question

A thermos bottle fitted with a piston is filled with a gas. Since the thermos bottle is well insulated, no heat can enter or leave it. The piston is pushed in, compressing the gas. a) What happens to the pressure of the gas? Does it increase, decrease, or stay the same? b) What happens to the temperature of the gas? Does it increase, decrease, or stay the same? c) Do any other properties of the gas change?

Short answer

Answer: When a gas inside an insulated thermos bottle is compressed adiabatically, the pressure and temperature of the gas both increase.

Step-by-step solution

Understand the given information

We have an insulated thermos bottle with a gas inside it. Since the bottle is well insulated, no heat can enter or leave it, and therefore, the process happening inside is an adiabatic process. A piston is used to compress the gas.

Apply Adiabatic Process Concepts

For adiabatic processes, the heat transfer (Q) is zero. The ideal gas law can be written as PV^n = constant, where P is the pressure, V is the volume, and n is the adiabatic index (ratio of specific heats) for the gas.

Determine Change in Pressure of the Gas

Since we are compressing the gas in an adiabatic process, the volume decreases. From the equation PV^n = constant, as the volume decreases, the pressure must increase to keep the product of the two constant. Thus, the pressure of the gas increases.

Determine Change in Temperature of the Gas

We can use the ideal gas law (PV = nRT) to determine the change in temperature. When compressing the gas, the pressure increases. Since the gas amount (n) and the gas constant (R) are constant, then, as the pressure increases, the temperature must increase as well to maintain the proportionality. Thus, the temperature of the gas increases.

Determine Change in Other Properties of the Gas

In this adiabatic compression process, we have discussed the change in volume, pressure, and temperature of the gas. Since no other properties of the gas have been mentioned in the problem statement, we can assume that no other properties change significantly during the process.

Key concepts

Thermodynamics

Thermodynamics is the branch of physics that deals with the relationships between heat and other forms of energy. In the scenario described, the gas inside the thermos bottle undergoes a transformation due to the work being done on it by the piston. This is a core concept in thermodynamics, where energy transfer takes place in various processes.

An adiabatic process is a particular type of thermodynamic process in which no heat is exchanged with the surroundings. Since the thermos bottle is well-insulated, the process of compressing the gas is adiabatic; the system is isolated from its environment. In such a process, the First Law of Thermodynamics states that the work done on the system will result in an increase in its internal energy, which, for this gas, manifests as an increase in temperature and pressure.

It is important to recognize that in thermodynamics, energy is conserved. Therefore, whatever energy is added to the system by compressing the gas doesn't disappear but transforms, affecting the state variables of the gas—specifically, its pressure and temperature—according to the principles of thermodynamics.

Ideal Gas Law

The ideal gas law is a fundamental equation that describes the state of an ideal gas. It is typically expressed as \(PV = nRT\), where:\begin{itemize}\item\(P\) stands for pressure\item\(V\) stands for volume\item\(n\) is the number of moles of the gas\item\(R\) is the universal gas constant\item\(T\) is the absolute temperature in Kelvin.\end{itemize}In the context of the textbook problem, when the piston is pushed into the thermos bottle compressing the gas, we can use the ideal gas law to predict the behavior of the gas under these new conditions. Since the bottle is sealed and no heat escapes, we consider the number of moles (\(n\)) and the gas constant (\(R\)) to remain constant.

As the volume (\(V\)) decreases due to compression, the pressure (\(P\)) must increase to maintain the equality in the \(PV=nRT\) equation. A direct consequence of this increase in pressure is the increase in temperature (\(T\)), as seen in the provided solution. Understanding the ideal gas law allows us to determine the relationship between the different properties of a gas and predict changes in its state when subjected to different conditions, such as compression.

Heat Transfer

Heat transfer is the movement of thermal energy from one body or substance to another. In typical scenarios, this can occur via three main mechanisms: conduction, convection, and radiation. However, in the given exercise, heat transfer is notably absent due to the thermos bottle's insulation.

In thermodynamics, a system permits heat transfer when it's not perfectly insulated, leading to changes in energy within the system. But in an adiabatic process like the one described, heat transfer to and from the surroundings is zero (\(Q=0\)). Without heat transfer, any change in the gas's energy is solely due to work done on or by the gas.

This has a significant educational implication. When you're solving problems related to thermodynamics, recognizing whether a process is adiabatic is crucial because it simplifies the analysis. You can exclude any calculations involving heat exchange, focusing solely on how work input alters the gas's temperature and pressure. Overall, understanding the principles of heat transfer is essential because it dictates how energy is distributed within a system, influencing the system's behavior and state changes.