Given:
- A hydrocarbon contains 10.5 g of carbon per gram of hydrogen.
- The vapor of the hydrocarbon at 127°C and 1 atm pressure weighs 2.8 g.

To find:
The molecular formula of the hydrocarbon.

Solution:
We are given the weight of the hydrocarbon vapor and the ratio of carbon to hydrogen in the hydrocarbon. From this information, we can determine the molecular formula of the hydrocarbon.

Step 1: Calculate the number of moles of the hydrocarbon
To calculate the number of moles of the hydrocarbon, we need to use the ideal gas law equation:

PV = nRT

Where:
P = pressure of the gas (1 atm)
V = volume of the gas (1 L)
n = number of moles of the gas
R = ideal gas constant (0.0821 L•atm/mol•K)
T = temperature of the gas in Kelvin (127°C + 273.15 = 400.15 K)

Plugging in the values, we have:

(1 atm)(1 L) = n(0.0821 L•atm/mol•K)(400.15 K)

Simplifying the equation, we get:

n = (1 atm)(1 L) / (0.0821 L•atm/mol•K)(400.15 K)
n = 0.0293 mol

Step 2: Calculate the moles of carbon and hydrogen
From the given ratio of carbon to hydrogen (10.5 g C / 1 g H), we can calculate the moles of carbon and hydrogen in the hydrocarbon.

Moles of carbon = 10.5 g C / 12.01 g/mol = 0.874 moles
Moles of hydrogen = 0.874 moles * (1 g H / 10.5 g C) = 0.083 moles

Step 3: Determine the empirical formula
To find the empirical formula, we need to divide the moles of each element by the smallest number of moles.

Moles of carbon = 0.874 moles / 0.083 moles = 10.5
Moles of hydrogen = 0.083 moles / 0.083 moles = 1

Therefore, the empirical formula is CH.

Step 4: Determine the molecular formula
To find the molecular formula, we need to know the molar mass of the hydrocarbon. We can calculate it using the given weight of the vapor and the number of moles.

Molar mass = mass / moles
Molar mass = 2.8 g / 0.0293 mol = 95.6 g/mol

The molar mass of the hydrocarbon is 95.6 g/mol.

Now we need to find a hydrocarbon with a molar mass close to 95.6 g/mol. Looking at the answer choices, option 'A' (C7H8) has a molar mass of approximately 92 g/mol.

Conclusion:
Therefore, the molecular formula of the hydrocarbon is C7H8.