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Chapter 18: Problem 19

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

Expert verified
Answer: During adiabatic compression, the pressure and temperature of the gas increase, while other properties, such as the number of moles and gas constant, remain constant.

Step by step solution

01

Recall the Ideal Gas Law

The Ideal Gas Law is given by the equation: PV = nRT where P is the pressure, V is the volume, n is the number of moles of gas, R is the ideal gas constant, and T is the temperature (in Kelvin). This equation helps us understand the relationship between the gas properties during the compression process.
02

Analyze the impact of compression on pressure

As the piston is pushed in, the volume of the gas (V) decreases. According to the Ideal Gas Law (PV = nRT), if V decreases while n, R, and T remain constant, then the pressure (P) must increase for the equation to be balanced. Therefore, the pressure of the gas increases.
03

Consider the adiabatic process

Since the thermos is well insulated, the compression occurs without any heat exchange, which is called an adiabatic process. In an adiabatic process, the following relationship holds: PV^γ = constant where γ is the adiabatic index (also known as the heat capacity ratio, equal to the ratio of specific heat capacities, Cp/Cv).
04

Analyze the impact of compression on temperature

In the adiabatic process, as the piston is pushed in and the gas volume decreases, the pressure increases. From the equation PV^γ = constant, we observe that if V decreases and P increases, T must also increase to maintain the equation's balance. Therefore, during adiabatic compression, the temperature of the gas increases.
05

Determine any other property changes

From the Ideal Gas Law and the adiabatic process equation, we can conclude that the main properties of the gas that change are pressure and temperature. Other variables, such as the number of moles (n) and the gas constant (R) remain constant during the process. So, no other properties of the gas change. In summary: a) The pressure of the gas increases. b) The temperature of the gas increases. c) No other properties of the gas change.

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