Truck cylinder, ideal gas law problem

[Flip18064]

Homework Statement

One cylinder in the diesel engine of a truck has an initial volume of 500 cm^3. Air is admitted to the cylinder at 30 C and a pressure of 1.0 atm. The piston rod then does 450 J of work to rapidly compress the air. What is its final temperature? What is its final volume?

Homework Equations

PV = nRT
[tex]\Delta[/tex] E_th = (5/2)nR[tex]\Delta[/tex]T

The Attempt at a Solution

I found the temperature using the second equation to be 1100 C which is right. However i cannot find the final volume for the life of me. I tried P^{[tex]\gamma - 1[/tex]}T^{-[tex]\gamma[/tex]} and use the final pressure to find the volume but that didn't seem to work. Is there another way to go about this?

 

[Flip18064]

finally figured it out using the formula:

W = PV^{[tex]\gamma[/tex]}(V_f^{1-[tex]\gamma[/tex]} - V_i^{1-[tex]\gamma[/tex]})/(1 - _{[tex]\gamma[/tex]})

 

[nlightner]

There are a few ways to approach this problem, depending on what information you have been given. One way to find the final volume is by using the ideal gas law equation, PV = nRT. Since the initial and final temperatures are given, we can set up the following equation:

(Pinitial)(Vinitial) = (n)(R)(Tfinal)

We know the initial pressure (1.0 atm), the initial volume (500 cm^3), and the final temperature (1100 C). We also know that the number of moles (n) remains constant, so we can solve for the final volume (Vfinal):

Vfinal = (Pinitial)(Vinitial)(Tfinal)/(R)(Tinitial)

Plugging in the values, we get:

Vfinal = (1.0 atm)(500 cm^3)(1100 C)/(0.0821 L*atm/mol*K)(30 C) = 203.95 cm^3

Therefore, the final volume of the cylinder is approximately 204 cm^3.

Another way to approach this problem is by using the work-energy theorem, which states that the work done on a system is equal to the change in its internal energy. In this case, the work done by the piston rod is 450 J, and we can use this to find the change in internal energy of the air in the cylinder.

\Delta Eth = W = 450 J

We also know that \Delta Eth = (5/2)nR\DeltaT, so we can set up the following equation:

(5/2)nR\DeltaT = 450 J

We know the initial temperature (30 C) and the final temperature (1100 C), so we can solve for the change in temperature (\DeltaT):

\DeltaT = (450 J)(2)/(5nR)

Plugging in the values for n and R, we get:

\DeltaT = (450 J)(2)/(5)(1 mol)(8.314 J/mol*K) = 0.108 K

Now, we can use the ideal gas law equation again to find the final volume:

(Pinitial)(Vinitial) = (n)(R)(Tfinal)

Vfinal = (Pinitial)(Vinitial)(Tfinal)/(n)(R)(Tinitial)

Plugging in the values, we get:

Vfinal = (1