A container contains n 1 moles of monatomic gas in System 1 with initial tempera

Question

A container contains n1 moles of monatomic gas in System 1 with initial temperature T1i and n2 moles of diatomic gas in System 2 with initial temperature T2i.

a. Show that the equilibrium thermal energies and equilibrium temperature are as shown below

b. System 1 contains 2.0 g of helium initially at 300 K and System 2 contains 8.0 g of oxygen initially at 600 K. What is the final temperature?

c. How much heat energy is transferred between them and in which direction?:

Please show the steps to the math, thanks.

Insulation prevents heat from entering or leaving the container System System 2 N. T2 A thin barrier prevents atoms from moving from system 1 to 2 but still allows them to collide. The barrier is clamped in place and cannot move.

Explanation / Answer

As K.E of ideal gas =f/2nRT where f is degree of freedom.

Degree of freedom for mixutre of gases will be f = n1f1+ n2f2/ (n1+ n2) where n1 no. Of moles of gas 1 and f1 is degree of freedom of gas 1.

Hence in this case f = 3n1+ 5n2/ ( n1+ n2).

Hence final temperature will be calculated by

f/2(n1+ n2)RTf = f1/2n1RT1i + f2/2n2RT2i ( energy conservation)

Cancelling terms you will get desired answer.

For E1f = f1/2 n1RTf and for E2f = f2/2RTf

B). For this apply formulae for Tf as derived in A part hence answer will be Tf = 3×1×300 + 5×0.5×600/(3×1 + 5×0.5) = 436.36K

C). Heat transfered will be equal to change in internal energy for any gas which is same as change in kinetic energy for ideal gas. Hence heat = f/2nRT = 3/2×1×8.314×136.36 = 1700.59Joules

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