Need help on this gas flowing into a vessel problem

[gharrington44]

1. Homework Statement

An evacuated rigid vessel is filled adiabatically through a valve from a large constant pressure source of a certain gas, until the pressures inside and outside are equal. If the P-U-V relation for the gas is given by the equation:
U = 831 + 2PV where U is joules, P is pascals, and V is m^3
Find the ratio of the initial and final volumes of the gas which enters.
(Hint: This is an application of the non-steady flow energy equation)


2. Homework Equations
So we need to use the Non-steady flow energy equation. Does that mean I use the equation
dE/dt = Q - W + (dm/dt)*(h2-h1) + .5*(dm/dt)*(V1^2 - V2^2)
In this case, kinetic and potential energy contributions would be negligible and I'm guessing the equation reduces to:
ΔU = Q - W + (dm/dt)*(h2 - h1)

3. The Attempt at a Solution
Also since the process is adiabatic, then Q = 0 and W = 0 for the process so the above equation simplifies to:
dU/dt = (dm/dt)*(h2-h1)
I also think that the internal energy of the system is 831 since the initial pressure and volume are both 0. In other words, U(0,0) = 831, but I don't know how to figure out the rest of the problem. Please let me know if I'm doing this right or not.

 

[anathema]

Your approach to using the non-steady flow energy equation is correct. Since the process is adiabatic, there is no heat transfer (Q = 0) and no work done (W = 0). Therefore, the change in internal energy (ΔU) is equal to the change in enthalpy (Δh), which is given by (h2 - h1) in the equation.

To find the ratio of initial and final volumes, we can use the ideal gas law, which states that PV = nRT, where n is the number of moles of gas and R is the gas constant. We can rearrange this equation to get V = nRT/P. Since the process is adiabatic, the number of moles of gas remains constant (dm/dt = 0), so the ratio of initial and final volumes is simply the ratio of initial and final pressures (P1/P2).

Using the P-U-V relation given in the problem, we can find the initial and final pressures by substituting the initial and final volumes into the equation. This gives us:

U1 = 831 + 2P1V1
U2 = 831 + 2P2V2

Since the internal energy of the system remains constant throughout the process (ΔU = 0), we can set U1 equal to U2 and solve for the ratio of initial and final pressures:

831 + 2P1V1 = 831 + 2P2V2
P1V1 = P2V2
P1/P2 = V2/V1

Therefore, the ratio of initial and final volumes is 1/P1. I hope this helps in solving the problem. Let me know if you have any further questions or need clarification. Happy problem-solving!