Black Body Radiation Calculation: Unveiling My Cluelessness

[LHarriger]

I was looking over the calculation leading to the thermal average number of photons s in a mode of frequency w in a black body. The approach was pretty straightfoward: Calculate the partition function Z based on quantized energies of a harmonic oscillator, then use this to calculate:
<s> \ = \ \sum_{i=0}^{\infty}{s P(s)} \ \ \Longrightarrow \ \ \ <s> \ = \ \frac{1}{e^\frac{\hbar\omega}{\tau}-1}
I had no problem understanding the derivation. However, this result is independent of the size of the black body. For the life of me, I don't see how this could be the case. I assume that when we talk about the number of photons in a mode we are talking about the number of photons that would be emmitted for the energy of that mode to vanish. It seems to me that the larger the body, the more photon will sit in that mode. For instance, a big anvil held at a given temperature should radiate more than a penny. I am clearly missing something, could someone clue me into my cluelessness.

 

[Astronuc]

A probability density function is based on the probability of a fraction of that population being within an incremental range.

The numbers are very large of course - say for a solid, on the order of 10²² atoms / gram.

Pick a fraction like 1000 / 10²³ which is the same as 10000 / 10²⁴. The fractions of particles are the same, but obviously ²⁴ atoms radiate 10 times the energy of 10²³ atoms. The frequency distribution would the be same, the intensity, number of photons would be greater by a factor of 10 in the larger population.

 

[LHarriger]

[The] number of photons would be greater by a factor of 10 in the larger population.

That was my conclussion too. My problem is that it seems, at least to me, that this conclussion conflicts with the result:
<s>=\frac{1}{e^\frac{\hbar\omega}{\tau}-1}
which states that the average number of photons in a mode is independent of the size of the black body.