How Many Atoms in The Universe
The universe may perhaps be infinite, but we have no way of knowing if that is so, and so we can only sensibly discuss “the observable universe”, i.e. what we can see. Then your question becomes meaningful.
To answer “innumerable” atoms exist in an infinite universe doesn’t really tell us anything new. It is a tautology, saying the same thing twice.
The visible universe is estimated to contain between 1078 and 1080 atoms.
(One estimate at the higher end of the range is 4 x 1079.)
This is the estimated number of atoms in the observable universe, but since we do not know the absolute size of the universe, we cannot be certain. Most of the matter in the universe is still hydrogen.
The observable universe contains about 3 to 100 × 1022 stars (30 sextillion to a septillion stars), organized in more than 80 billion galaxies, which themselves form clusters and superclusters.
Two approximate calculations give the number of atoms in the observable universe to be close to 1080.
Observations of the cosmic microwave background from the Wilkinson Microwave Anisotropy Probe suggest that the spatial curvature of the universe is very close to zero, which in current cosmological models implies that the value of the density parameter must be very close to a certain critical value. A NASA page gives this density, which includes dark energy, dark matter and ordinary matter all lumped together, as 9.9×10−27 kg/m3, although the figure has not been updated since 2005 and a number of new estimates of the Hubble parameter have been made since then. The present value of the Hubble parameter is important because it is related to the value of the critical density at the present, , by the equation
where G is the gravitational constant. WMAP seven-year results from 2010 estimate the value of the at 70.4 (km/s)/Mpc or 2.28×10−18 s−1, which gives a critical density of 9.30×10−27 kg/m3.
Analysis of the WMAP results suggests that only about 4% of the critical density is in the form of normal atoms, while 22% is thought to be made of cold dark matter and 74% is thought to be dark energy, so if we make the simplifying assumption that all the atoms are hydrogen atoms (which in reality make up about 74% of all atoms in our galaxy by mass, see Abundance of the chemical elements) which each have a mass of about 1.67×10−27kg, this implies about 0.26 atoms/m3. Multiplying this by the volume of the visible universe (with a radius of 14 billion parsecs, the volume would be about 3.38×1080 m3) gives an estimate of about 8.8×1079 atoms in the visible universe, while multiplying it by the volume of the observable universe (with a radius of 14.3 billion parsecs, the volume would be about 3.60×1080 m3) gives an estimate of about 9.4×1079 atoms in the observable universe.
A typical star has a mass of about 2×1030 kg, which is about 1×1057 atoms of hydrogen per star. A typical galaxy has about 400 billion stars so that means each galaxy has 1×1057 × 4×1011 = 4×1068 hydrogen atoms. There are possibly 80 billion galaxies in the universe, so that means that there are about 4×1068 × 8×1010 = 3×1079 hydrogen atoms in the observable universe. But this is definitely a lower limit calculation, and it ignores many possible atom sources such as intergalactic gas.