Grand partition function (Volume divided into N spaces)

[anaisabel]

Homework Statement

Homework: I have a model with volume V divided into N spaces, each space only has one atom. It is considered an ideal gas. I have to write the partition function and the average number of particles wich I did. But then I cant solve the next exercise(I will post picture of the equation I have to proof). the first picture is the calculations i did to obtain the partition function and the number of particles and the second picture is the equation i have to obtain, considering the grand partion function of the average number of particles.

Relevant Equations

Grand partition function and average number of particles

equation i need to proof. the N in here, is the avarege number of particles, N0 is the total number of particles,V is total volume, v0 I am not quite sure what it is because it isn't mentioned in the homework, but I am assuming it is the volume of which space.

 

[NateD]

\frac{N}{N_0}=\frac{V}{V_0}Proof:Let N be the average number of particles, N0 be the total number of particles, V be the total volume and V0 be the volume of a given space.By definition, we know that N is equal to the total number of particles divided by the total volume. Thus, N = N0/V. Likewise, V0 is equal to the total volume divided by the volume of the given space. Thus, V0 = V/V0.Substituting these equations into our initial equation gives us: \frac{N}{N_0}=\frac{V}{V_0} \frac{N_0/V}{N_0}=\frac{V/V_0}{V_0}Simplifying both sides of the equation yields:\frac{1}{N_0} = \frac{1}{V_0}which proves the original equation.