Math Is Fun Forum

I didnt include the details of summation of v_1 because it doesnt mattter. We are taking n equal V_1's , meaning the correct expression should be :

So if this helps ,

Multiplying by Sin(theta) works too.

To understand you have to review dalton's law on partial pressures:

Take the simple case. Gas A of initial pressure p_a mixes with Gas B of initial pressure p_b.

What is the total pressure ?

Chi is a mole fraction. I think u r familiar with this from ur general / highschool chemistry literature.

Now take the case above where each gas is in its own container, and each gas has its own pressure, temp , and volume. You cannot use the same argument to find the total pressure. As I said, isolate each container.

What happens to the gas in container A after the valve is open? Its pressure, temp, and volume should change. How do they change ? according to the ideal gas equation.

PV=nRT for initial case
P'V' = nRT' for final case

but n is constant ( closed system )

so PV / T = P'V'/T'

this way you find the new partial pressure of A ( assume T is known here , V' = V_a + V_b ).

Same argument holds for container B. Next you add both new partial pressures p_a + p_b to get the total pressure of the gaseous mixture formed.

I hope this makes thing clearer.

I assume you are dealing with ideal gases. Our system is closed and no chemical reaction occurs so n = constant. The equation of state obeyed is :

Isolate each container and work on it. First we calculate the initial pressures.

'i' = initial state
'f' = final state

Hence, intial P , V , and T for both A and B are known. ( check the given )

After the two gases mix , they form a single gase of certain pressure, temperature, and volume. Each ideal gase occupies all the volume available as if they were alone. Since our gas is ideal, then the total pressure of this mixture of gases is the summation of the two gases. This is known as Dalton's law of partial pressures.

For each of the containers, the following holds true for a closed system :

Take the ratio :

I do this to form a relation bettween the partial pressures of A and B since the "ratio" is known.

We have to uknowns and a single equation, so we need another.

Again from Dalton's law,

Again, take the ratios :

We have a system of 2 equations 2 unknowns, Solve using eq(1) and eq(2) to find partial pressures of A and B. Once determined, the total pressure is determined as well by simply addition of partial pressures.

To find the final temp , apply :

Finally to find the number of moles in each container, it is simple. Each container has some unknown number of particles , but in both containers the pressure and tempreaure is the same. ( Eq is established ).

In container A , the number of particles is :

In container B, the number of particles is :

I am sorry for not doing the calculation, but I think that should no problem to you.

Incase of doubt, do return.