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How many moles of propane gas, $\ce{C3H8}$, are contained in 11 grams of the gas at standard conditions? Also, can you please explain step-by-step (and include sigfigs?)

I think I might know how to do this, but I'm not sure if this process is correct.

I found online that $\ce{C3H8}$ has a mass of 44.11 grams per mol. So do I just use a proportion to find the number of moles this 11 grams of propane has?

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You are on the right track. Every element has a molar mass that is defined under standard conditions, which is in this case the natural abundance of an element taken into account. Pressure, volume and temperature will not change the mass of a compound. Since every element has a specific molar mass, every compound does too. The molar mass is a measurement of how much mass does one mole of a substance have.

The molar mass is usually referred to with $M$, while the mass is referred to as $m$. The amount of substance is $n$. This gives you the following relationship: $$M=\frac{m}{n}$$

Since you have given $m(\ce{C3H8})=11~\mathrm{g}$ and you already looked up $M(\ce{C3H8})=44.1~\mathrm{g\,mol^{-1}}$, you can use this formula to determine $n(\ce{C3H8})$.

In this question it is quite hard to explain the use of significant figures. Those are used to imply a certain inaccuracy. Not enough information is given by the question, as of how accurate the measurement is. It is a mere exercise of converting one property into another. Here you should not worry about it.

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