A 4-ft insulated tank contains nitrogen gas at 30 Ibf/in. ^2...

A 4-ft insulated tank contains nitrogen gas at 30 Ibf/in. $^{2}$ and ambient temperature 77 F. The tank is connected by a valve to a supply line flowing carbon dioxide at $180 \mathrm{lbf} / \mathrm{in.}^{2}, 190 \mathrm{F}$. A mixture of 50 mole percent nitrogen and 50 mole percent carbon dioxide is to be obtained by opening the valve and allowing flow into the tank until an appropriate pressure is reached, when the valve is closed. What is the pressure? The tank eventually cools to ambient temperature. Calculate the net entropy change for the overall process.

Key Concepts

Ideal Gas Law

The ideal gas law (PV = nRT) provides the fundamental relationship between pressure, volume, temperature, and the number of moles for gases. It is essential for analyzing processes involving gases, since it allows one to relate the initial and final states of the system as the gases mix, and to determine properties like pressure when gas quantities and temperature are known.

Mass and Energy Balances in Closed Systems

In any process involving the mixing of gases in a closed container, mass conservation is crucial as the total number of moles in the container must equal the sum of the individual moles introduced. Energy balance is equally important, especially when the process is adiabatic (insulated initially) and later brought to a final temperature by interaction with the surroundings. This balance helps in determining state parameters of the final mixture.

Gas Mixing in Closed Systems

Gas mixing involves combining two or more different gases so that the resulting mixture attains a uniform composition. When gases mix in a closed system, their partial pressures and mole fractions combine according to Dalton’s law, and equilibrium properties such as total pressure can be determined by considering the contributions of each gas along with the ideal gas law.

Entropy Change and the Second Law of Thermodynamics

The calculation of net entropy change is a key aspect of analyzing irreversible processes, such as gas mixing. The second law of thermodynamics indicates that for irreversible processes, the total entropy of the system and its surroundings increases. In mixing processes at constant temperature, the entropy change can be evaluated using the relationship that involves the change in the number of molecules and their states, reflecting the increased disorder as the gases combine.

Determination of Final Pressure in Gas Mixtures

Determining the final pressure in a gas mixture involves combining the contributions of each gas in the container. By using the ideal gas law together with mass conservation and mole fraction analysis, one can compute the mixture's overall pressure after mixing and equilibration. This concept is critical when designing processes where specific mixtures or operating pressures are targeted.

Transcript

00:01 Hi everybody, so we need to know find the pressure here.

00:05 So we need to know the number of moles in the tank.

00:08 So we have n2 -c -o -2 equals n -1 -2 equals n2 and 2 equals p1, v -1, r -t -1, equals 130 times 144 times 4 equals 1545 times 535 times 5336 .7 equals 0 .02084.

00:55 Okay.

00:57 Okay.

00:58 And so the number of total moles and state 2 we have n 2 equals n 2 n 2 plus n 2 c o 2 equals 0 .0284 plus 0 .02084 which equals 0 .0168 now what we have to do is apply the first law into the system.

01:42 So here we have n1 equals n -i.

01:46 So we have c -p -i -t -i -c -o -2 plus c -v -v -t -1 of n2 equals c -v of t2 equals n -2, into plus cv times t 2 of c o 2 me pull down so we have 44 .01 times 0 .201 times 649 .67 plus 28 .033 times 2 .13 times 0 .178 times 536 .67 equals 28 .0133 013 times 0 .178 t2 plus 44 .01 times 0 .156 times t2.

03:05 And your t2 is in 710, 710, not 718, 710 .801r.

03:18 And so your final pressure in the tank is going to be p2 in to r, t2 divided by b2.

03:28 So we get 0 .04168 times 1 ,554, 5, 5, 5, 710.

03:45 801 4 equals 11444 43 .114 i bf feet squared and we need this of inches so 79.

04:04 0 .47 ibf inches squared okay and just going to scroll down so the tank is from t2 to t for ambient temperature.

04:25 And so we need to find the cv mix here, which is just going to be your y -i -c -e -i...

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