A sample of a hydrocarbon (a compound consisting of only carbon and hydrogen) contains \(2.59 \times 10^{2.3}\) atoms of hydrogen and is \(17.3 \%\) hydrogen by mass. If the molar mass of the hydrocarbon is between 55 and \(65 \mathrm{~g} / \mathrm{mol}\), what amount (moles) of compound is present, and what is the mass of the sample?

As given, we have \(2.59 \times 10^{2.3}\) atoms of hydrogen in the sample, convert this into moles by dividing by Avogadro's number (\(6.022 \times 10^{23}\ \mathrm{atoms/mol}\): Moles of hydrogen \(= \frac{2.59 \times 10^{2.3}}{6.022 \times 10^{23}}\)

Since the sample is 17.3% hydrogen by mass, that means the hydrogen mass is 17.3% of the total hydrocarbon mass. Considering the molar mass of hydrogen to be 1 g/mol, we can find the total molar mass of the hydrocarbon. Let \(M\) be the molar mass of the hydrocarbon: \(0.173 M = \) moles of hydrogen \(\times 1 \mathrm{g/mol}\)

Now we know the molar mass of the hydrocarbon, which is between 55 g/mol and 65 g/mol. We can find the moles of the hydrocarbon present in the sample by dividing the moles of hydrogen by the number of moles of hydrogen in 1 mole of hydrocarbon. Since 1 mole of hydrocarbon has x moles of hydrogen, first we need to find the formula of the hydrocarbon. Considering \(CxHy\) to be the hydrocarbon formula and since the molar mass of carbon is 12 g/mol: \(M = x(12 \ g/mol) + y(1 \ g/mol)\) In this equation, we know that the molar mass of the hydrocarbon \(M\) is between 55 and 65 g/mol. We can use this information to check different values for x and y to find suitable whole numbers for a hydrocarbon formula that lie within the given molar mass range. Also, as the molar ratio of hydrogen to carbon is \(y/x\), we can write: \(0.173 M = \frac{y}{x} \times 1\) Combining the two equations, we can find the values of x and y: \(M = 12x + y\) \(0.173(12x + y) = \frac{y}{x}\) We can find x = 5, y = 12 and M = 60 g/mol to be a solution that fulfills the given conditions. Now, since the molar mass of the hydrocarbon is 60 g/mol and there are 12 moles of hydrogen in one mole of hydrocarbon, we can calculate the moles of the sample as follows: Moles of the sample = moles of hydrogen / 12

To find the mass of the sample, we can multiply the moles of the hydrocarbon by its molar mass: Mass of the sample = moles of the sample × molar mass of hydrocarbon Putting these steps together, we can calculate the amount (moles) and mass of the hydrocarbon sample.