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None, because grams are a measure of mass, not of weight. The mass of the earth is approx 5.97*1024 kilograms or 5.97*1027 grams, that is its mass not its weight.
It is important to note that the weight of an object depends on the [larger] object that is attracting it. For example, your weight on the moon will be around a sixth of your weight on earth even though your mass will remain the same.
It is possible to make an estimate of the weight of the earth relative to the sun.
Newton's Law of Gravitation gives us
F = G*M1*M2/R2 where
G = Universal gravitational constant = 6.673*10-11 m3kg-1s-2
M1 = mass of the earth = 5.97*1024 kg
M2 = mass of the sun = 1.99*1030 kg
R = Average radius of the earth's orbit = 1.5*1011 metres.
This gives F = 3.5*1022 Newtons approx. = Weight of the earth.
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