Two insulated tanks A and B are connected by a valve, shown...

Two insulated tanks $A$ and $B$ are connected by a valve, shown in Fig. P12.53. Tank $A$ has a volume of $1 \mathrm{m}^{3}$ and initially contains argon at 300 $\mathrm{kPa}$ and $10^{\circ} \mathrm{C} .$ Tank $B$ has a volume of $2 \mathrm{m}^{3}$ and initially contains ethane at $200 \mathrm{kPa}$ and $50^{\circ} \mathrm{C}$ The valve is opened and remains open until the resulting gas mixture comes to a uniform state. Determine the final pressure and temperature.

Key Concepts

Internal Energy of Ideal Gases and Specific Heats

For ideal gases, the internal energy is a function of temperature and can be computed using the specific heat at constant volume. By knowing or calculating these specific heats, the internal energy of each gas prior to mixing can be determined and then used in the energy conservation equation to find the final temperature of the mixture.

Mixing of Ideal Gases in a Closed System

Mixing two ideal gases that are initially at different conditions leads to a new uniform state characterized by a common pressure and temperature. The final state is determined by simultaneously applying conservation of mass (or moles) and conservation of energy, making it necessary to consider the properties of each gas in the mixture and their contributions to the overall final equilibrium.

Conservation of Energy in Adiabatic Processes

Since the tanks are insulated, no heat is transferred to or from the surroundings, meaning that the process is adiabatic. In an adiabatic process with no external work done, the total internal energy of the system remains constant. This concept is crucial when determining the final temperature of the gas mixture by applying energy balance based on the initial internal energies of the individual gases.

Conservation of Mass

In any mixing problem involving fluids, the total mass of each gas (or the total mass of the mixture) is conserved. This principle is important because the total number of moles of gas present in the combined system remains the same before and after the mixing process, even though the gases started in different tanks with separate conditions.

Ideal Gas Law

The ideal gas law (PV = nRT) is used to relate pressure, volume, temperature, and number of moles for each component gas. This law allows us to calculate additional properties, such as the total number of moles in each tank, which is essential for calculating the final state of the mixed gases.

Transcript

00:01 Hi, everybody.

00:02 So we are asked, determine the final pressure and temperature.

00:06 So for here, we can find the ma, the mass of pressure 1a, volume of a, divided by the ra, and the t1a, temperature 1a.

00:19 So we have 300 times 1, divided by 0 .2081, times 10 plus 273 .15.

00:35 And the reason we're doing that, because it's in celsius, and we need it in kelvin's, okay? and this is going to be 5 .0913 kilograms.

00:48 And then we can find the mass of the ethane.

00:54 And that is going to be 200 times 2 divided by 0 .27.

01:04 Times 50 plus 273 .15.

01:10 And we get 4 .476 kilograms.

01:15 Okay.

01:17 And so then the mass total here is going to be 5 .0913 plus 4 .44767 and we get 9 .68 kilograms.

01:32 And now what we can do is do the energy equation.

01:36 Okay.

01:37 So we have u2 minus u1 equals zero.

01:41 Okay.

01:42 And from here what we can do is m -a -c -b -c -b, which is u -a -t -2 minus t -1a, plus m -b times c -b times 2 -t - minus t to be equals zero and from here we can do 5 .0913 times 0 .312 times t2 minus 283 .15 plus 4 .4767 times 1 .49 times t2 minus 3223.

02:38 Oops, if i can write three for everybody correctly here...

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