Energy of gas as the system

[huskydc]

A cylinder (cross section is 0.2m2) with a free moving piston is filled with gas. The piston is attached to a heavy weight W = 10000N. Outside the cylinder, the air is at 300K and 1atm. Initially the gas is at 300K, then it is heated to 400K. The heat capacity of the gas under the constant pressure is 500J/K.
If the length of the gas in the cylinder l increases by 20cm during the heating, find the change in the internal energy of the gas in Joule J.



i figured...heat is being transferred to the gas, and work is done by the gas, so both Q and W should be postive...

delta U = Q - W

i'm not really sure how to start this...but i tried the following:

C = Q/ delta T

500 = Q / (400-300)

Q = 50000 J

W = P delta V (i'm not sure if the P is of the system or the surrounding)

but with the problem, I've been given a clue that says this:

P system = (P surrounding - W force)/cross area...

and I came up with 456500 J...

Q- W

50000 - 456500 = delta U...apparently the answer didn't work, any clues?

 

[huskydc]

can i even use the equation: delta U = 1.5 n R delta T
but that would include finding out n...which i don't have enough info to calculate...any one help??
i'm lost... =(

 

[quicknote]

I would approach this problem by first understanding the concept of energy in a gas system. The internal energy of a gas is the sum of its kinetic and potential energies of the particles within the system. In this case, the gas in the cylinder is the system and the piston and weight act as external forces.

To find the change in internal energy of the gas, we can use the equation:

ΔU = Q - W

Where ΔU is the change in internal energy, Q is the heat transferred to the system, and W is the work done by the system.

We are given the heat capacity of the gas (500 J/K), which tells us that it takes 500 Joules of energy to raise the temperature of 1 mole of gas by 1 Kelvin. Since we know the initial and final temperatures, we can calculate the heat transferred to the gas:

Q = CΔT = (500 J/K)(400K-300K) = 50000 J

Next, we need to calculate the work done by the system. Since the piston is attached to a heavy weight, it is acting against gravity and doing work to lift the weight. The work done by the system can be calculated as:

W = mgh = (10000N)(0.2m)(0.2m) = 4000 J

Now we can plug these values into the equation to find the change in internal energy:

ΔU = Q - W = 50000 J - 4000 J = 46000 J

But wait, we also need to take into account the change in volume of the gas. The length of the gas in the cylinder has increased by 20 cm (0.2m) during heating. This means that the gas has done work in expanding against the piston. We can calculate the work done by the gas as:

W = PΔV = (1 atm)(0.2m2)(0.2m) = 0.04 J

Since the gas is expanding, it is doing work on the surroundings and thus the work done by the gas should be negative. Therefore, we need to subtract this value from the previous calculation of ΔU:

ΔU = Q - W = 46000 J - 0.04 J = 45999.96 J

This is the final answer for the change in internal energy of the gas in Joules. It is