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Chapter 5: Problem 14

Air is allowed to escape from an inflated tire. Is the temperature of the escaping air higher than, lower than, or equal to the temperature of the air inside the tire? Why?

Short Answer

Expert verified
Answer: The temperature of the escaping air is lower than the temperature of the air inside the tire.

Step by step solution

01

Understand the initial situation

The air inside the inflated tire is under high pressure as it is compressed into a small volume. This higher pressure is due to the air molecules colliding more frequently with the walls of the tire, resulting in an increase in temperature. The temperature of the air inside the tire is assumed to be in equilibrium, meaning it is neither increasing nor decreasing.
02

Analyze the change in pressure and volume

As air is allowed to escape from the tire, the pressure inside the tire decreases. Simultaneously, the volume of the air will increase as it escapes the confined space of the tire. According to the ideal gas law, the relationship between pressure, volume, and temperature for an ideal gas is given by: \[PV = nRT\] where \(P\) is the pressure, \(V\) is the volume, \(n\) is the amount of substance of the gas (in moles), \(R\) is the ideal gas constant, and \(T\) is the temperature (in Kelvin). Since this is an adiabatic process (meaning no heat is exchanged with the environment), the numbers of moles (\(n\)) and the gas constant (\(R\)) will remain constant.
03

Analyze the change in temperature

As air escapes the tire, the pressure decreases, and the volume increases. Thus, the product of the pressure and volume, \(PV\), will also change. By looking at the ideal gas law equation, it can be inferred that the temperature of the gas should change as well to maintain a constant product: \[\Delta(PV) = nR\Delta T\] As \(\Delta P < 0\) and \(\Delta V > 0\), for the product \(PV\) to remain constant, it implies that \(\Delta T < 0\).
04

Conclude the temperature relationship

Since we found that the change in temperature \(\Delta T < 0\), it means the temperature of the escaping air is lower than the temperature of the air inside the tire. This conclusion follows from the reduction in pressure and the corresponding increase in volume as air escapes the confined space of the tire in an adiabatic process.

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