A nearly flat bicycle tire becomes noticeably warmer after it has been pumped up. Approximate this process as a reversible adiabatic compression. Take the initial pressure and temperature of the air before it is put in the tire to be Pi = 1.00 bar and Ti = 298 K. The final volume of the air in the tire is Vf= 1.50 L and the final pressure is Pf = 5.00 bar. Calculate the final temperature of the air in the tire.

Complete Question

A nearly flat bicycle tire becomes noticeably warmer after it has been pumped up. Approximate this process as a reversible adiabatic compression. Take the initial pressure and temperature of the air before it is put in the tire to be Pi = 1.00 bar and Ti = 298 K. The final volume of the air in the tire is Vf= 1.50 L and the final pressure is Pf = 5.00 bar. Calculate the final temperature of the air in the tire.

Assume that Cv,m = 5R/2

The value is

Explanation:

From the question we are told that

The initial pressure is

The initial temperature is

The final volume is

The final pressure is

Generally the equation for reversible adiabatic compression is

Generally from the ideal gas equation we have that

=>

So

=>

=>

=>

=>

Here is a constant mathematically represented as

Here

is the molar heat capacity at constant pressure

and is the molar heat capacity at constant volume given as

Generally for an ideal gas is mathematically represented as

So

=>  

So

=>

=>

=>

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