This is out-of-date. For the up-to-date estimates see:
https://sealevel.info/PgC_vs_Gt_CO2.html  and
https://sealevel.info/conversion_factors.html
 

The Earth's atmosphere is variously estimated to have a mass of 5.1 to 5.3 × 10¹⁸ kg = 5.3 × 10⁶ Gt = 5.3 million Gt, so (using the “5.3” estimate) one ppmm (part-per-million by mass) weighs about 5.3 Gt.

However, atmospheric gas concentrations are customarily expressed in ppmv (parts-per-million by volume, a/k/a molar fraction, µ mol/mol), so to calculate the mass of one ppmv requires scaling according to the molecular weight of the gas in question. (Note: if water vapor is ignored this is properly called the dry molar fraction.)
 
The average molecular weight of the Earth's atmosphere is 28.966 g/mole (≈29). So, for example: ↑

Carbon Dioxide:
1 ppmv CO₂ (molecular wt 44.01) has mass ≈(44/29) × 5.3 Gt = 8.053 Gt, of which 12/44-ths or 2.196 Gt^† is carbon.
1 Pg = 1 Gt, so 1 PgC (“petagrams carbon”) is contained in (44/12) = 3.667 Gt CO₂, and is equivalent to 3.667/8.053 = 0.4553 ppmv CO₂ in the atmosphere.
416 ppmv CO₂^‡ has mass 416 × 8.053 Gt/ppmv = 3350 Gt.
That much CO₂ contains (12/44)×3350 = 914 PgC. ↑

Methane:
1 ppmv CH4 (molecular wt 16.044) has mass ≈(16/29) × 5.3 Gt = 2.9356 Gt.
1.9 ppmv CH₄^‡ has mass 1.9 × 2.9356 Gt/ppmv = 5.578 Gt. ↑

 

^†IPCC AR5 WGI uses a slightly different figure: 2.12 PgC per ppmv (Prather et al, 2012).

^‡which are the approximate current (2022) average atmospheric concentrations of the two gasses: 416 ppmv CO₂ and 1.90 ppmv CH₄.

 

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