An isolated system consists of N_1 molecules of type 1 of mass m_1 and N_2 molec

Question

An isolated system consists of N_1 molecules of type 1 of mass m_1 and N_2 molecules of type 2 of mass m_2 confined within a box of volume V. The molecules are supposed to interact very weakly so that they constitute an ideal gas mixture. Evidently, we have E = E_1 + E_2, where E_1 is the energy of the N_1 molecules and E_2 is the energy of the N_2 molecules while E is the total energy. Express ln Ohm(E = E_1 +E_2) as a function of E_1, E_2, N_1, N_2, and V. When the molecules of type 1 are in thermal equilibrium with the molecules of type 2, what is the derivative of Q(E = E_1 +E_2) with respect to E_2? and how is the ratio E_1/N_1 related to E_2/N_2? Use the result in (b) to find the equation of state of this system i.e. to find its mean pressure P as a function of volume V and temperature T.

Explanation / Answer

Ans:- An isolated system different type of molecule N1 mass m1 also N2 mass m2 within a box of volume V.

We are supposing intention very weakly so it is Idea gas of mixture.

E is denoted as energy E = E1 + E2  

E1 is the energy of N1 molecule & E2 is the energy N2 molecule.

(a). Suppose a molecule of mass m is freely to be move in one dimension. so we denote as a coordinate axis by x and momentum of p. Suppose the molecular energy E is confined E1 & E2 in a box so that it will be located between the box of volume V and suppose that its E = E1+ E2 energy lies between for different phase of space. the molecule indicating the volume of this space are to be accessible of the Box of volume V. the molecule is confined Express classically to the box volume  

E <p2/2m < E1 + E2

So we define 2mE < N1 + N2 < 2m(E1 + E2)

So we see E = 2mE N =(m/2E ) E1+E2

(b). the thermal equilibrium of molecule of type 2. type 2 molecule in a Box

Consider a system of volume   (E = E1 + E2) with respect to E2 we are consisting of E2 weakly interacting molecule particle of E1. so with respect to N1 molecule is much interaction as compare to N2 vice versa.

The total energy of isolated system of molecule lie between(E= E1 + E2). so our E1/N1 ratio in diffrent dimensional phase of space, Due to some light or weak coupling between the molecule of isolated system, so that momenta E1/N1 to must obey E2 with respect to (E =E1 +E2).

E < 1/2m(N21+ N22) < E1 + E2 so that the bounds ratio between isolated box in momentum space

2mE < N21+ N22< 2m(E + E). so that this equation of isolated box ratio E1/N1 3.5 times of greater than E2/N2

(c). if pressure P as a function of volume V at temperature T .

a molecule confined within a isolated box in the shape of a cube. Pressure  Px = Py = Pz. The possible Pressure energy levels of this molecule are then given by

E =(E1+E2)2/ 2m = (2 P2x / E2x +P2y / E2y +P2z / Ey )

as per above E1/N1 ration 3.5 time greater E2/N2 is in a given particular values so that three integers is decide the energy of this state. if we spreading of a molecule in isolated system

equation is H(enthalpy) = P2 (pressure=P)

if our enthalpy prepared in time t = 0 so that a state represented by the probability function

so that Normalize t=0 (X; P). hXi; hPi; X; P ;E . the random variables ^Xt; ^ Pt with ^Xt=0; ^ Pt=0. Express of variables replacement t (X; P) with t=0 (X; P) so our equation of state of isolated system is express given below.

t=0 (X; P) directly proportion exp (-a (X - X0)2 - b (p - p1)2

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.