Okay, struggled on this for an hour. We are clueless as to where to go with any

Question

Okay, struggled on this for an hour. We are clueless as to where to go with any of it. Please help us guys! We need to figure this out.

A particle gas consists of N monatomic particles each of mass m all contained in a volume V at temperature T. Your answers should be written in terms of the Boltzmann constant kB and Avagadro's number NA rather than R=NAkB.

Part A

Find

?v2?, the average speed squared for each particle.

Express the average speed squared in terms of the gas temperature T and any other given quantities.

Part B

Find U, the internal energy of the gas. Express the internal energy in terms of the gas temperature T and any other given quantities.

Part E

What will be the ratio of the new molar mass M?to the old molar mass M?

Part F

What will be the ratio of the new rms speed v?rms to the old rms speed vrms?

Part G

What will be the ratio of the new molar heat capacity C?V to the old molar heat capacity CV?

Explanation / Answer

This is how the kinetic theory was developed. Suppose the N partcles do not interact with each other.

Consider the force one particle exerts on the wall of the container of volume V= L^3 (volume is a cube with one side L).

Fx = delta(Px)/delta(t)

P is the linear momentum, v(x) is the speed in the x direction

delta(P)= mvi-mvf = 2*m*v(x) for a totally elestic colission on the x direction

delta(t) = (2*L)/v(x)

is the time require for a particle to impact the wall (Volume = L^3 and the particle need to travel 2 back and forth from the oposing wall)



Thus F = 2m*v(x)^2/2L = mv(x)^2/L

is the force one partcle exerts on a wall. For N particles we have

F(tot) = N*F =N*m*v(x)^2/L

The pressure on the wall is

P =F/S =F/L^2 = Nmv(x)^2/L^3

Now, here comes the trick. Since all 3 spatial direction are equivalent we can write

v(x)^2 = v^2/3

which means the AVERAGE square speed in x direction is 1/3 of the total average sqaure speed.

Thus the pressure becomes

P = N*m*v^2/(3L^3) = Nmv^2/(3V)

Again, uppercase V is volume, lowecase v is average square speed.

By rearranging we get

P*V =N*m*v^2/3

but we know that for an ideal gas we have

P*V =N*k*T    (K is the Boltzmann constant, T is the temperature)

Thus

N*m*v^2/3 = N*k*T    (1)

(which means that the kinetic energy of a particle of a gas is proportional to the Absolute temperature T. The pproportionality constant is (3*k/2) )

mv^2/2 =3/2*k*T    (2)

Now the average speed squared is simply

v^2 = 3*k*T/m

B)

The total internal energy is the sum of all kinetic energies of particles.

U = N*mv^2/2 = (3/2)*N*k*T   

which results from (2).

For parts E,F, G text data is missing.

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