Question

If the density of ethylene is \(1.25 \mathrm{~g}\) /liter at S.T.P. and the ratio of carbon to hydrogen atoms is \(1: 2\), what is molecular weight and formula of ethylene?

Short answer

The molecular weight of ethylene is 28.02 g/mol, and its molecular formula is \(C_2H_4\).

Step-by-step solution

Use the Ideal Gas Law to find the molar mass of ethylene

The Ideal Gas Law is given as PV = nRT, where P is pressure, V is volume, n is the number of moles of gas, R is the gas constant, and T is the temperature. Since ethylene is at Standard Temperature and Pressure (S.T.P.), we know that the pressure is 1 atm, temperature is 273.15 K, and R is the gas constant (0.0821 L atm/mol K). The Ideal Gas Law can be rearranged to solve the molar mass. First, solve the Ideal Gas Law for n/V: \(n/V= \dfrac{P}{RT}\) Plug in the values for P, R, and T for ethylene at S.T.P.: \(n/V = \dfrac{1 \mathrm{~atm}}{0.0821 \mathrm{~L~atm/mol~K} \times 273.15 \mathrm{~K}} = 0.0446 \mathrm{~mol/L}\) We are given the density of ethylene as \(1.25 \mathrm{~g}\)/liter. We can now solve for molar mass by dividing the given density by the moles per liter: Molar Mass = \(\dfrac{1.25 \mathrm{~g/L}}{0.0446 \mathrm{~mol/L}} = 28.02 \mathrm{~g/mol}\)

Calculate the empirical formula of ethylene

We are given the ratio of carbon to hydrogen atoms as 1:2, which means that for every one carbon atom, there are two hydrogen atoms present. Carbon has an atomic mass of 12.01 g/mol, and hydrogen has an atomic mass of 1.01 g/mol. We can now calculate the empirical formula mass by adding the atomic masses of the elements present according to their proportion: Empirical Formula Mass = (1 × 12.01 g/mol [C]) + (2 × 1.01 g/mol [H]) = 14.03 g/mol

Determine the molecular formula of ethylene

To find the molecular formula, we need to compare the molar mass of ethylene, which we calculated in Step 1, with the empirical formula mass calculated in Step 2: \(n\) = \(\dfrac{\text{Molar Mass}}{\text{Empirical Formula Mass}} = \dfrac{28.02 \mathrm{~g/mol}}{14.03 \mathrm{~g/mol}} = 2\) This result indicates that the molecular formula of ethylene is two times the empirical formula. The empirical formula is CH2, so the molecular formula of ethylene is \(C_2H_4\). Therefore, the molecular weight of ethylene is 28.02 g/mol, and its molecular formula is C2H4.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry that relates the pressure, volume, temperature, and number of moles of a gas. The law is often expressed as PV = nRT.

In the context of this exercise, we used the Ideal Gas Law to find out the molar mass of ethylene gas at Standard Temperature and Pressure (STP). By rearranging the law and solving for the relationship between the number of moles per unit volume (/Vdfrac{P}{RT}), we can determine the molar mass when provided with the gas's density at STP conditions. Remember, STP implies that the pressure () is 1 atmosphere and the temperature () is 273.15 Kelvin, which greatly simplifies calculations.

Empirical Formula

An empirical formula represents the simplest whole-number ratio of atoms of each element in a compound. Unlike the molecular formula, it does not necessarily reflect the actual numbers of atoms within a molecule but does give you the proportionate number of each type of atom.

In the case of ethylene, the ratio of carbon to hydrogen atoms was given as 1:2. This means the empirical formula reflects one carbon atom for every two hydrogen atoms. The empirical formula is especially useful because it's a stepping stone to determining the molecular formula of a compound.

Molecular Formula

The molecular formula provides the actual number of atoms of each element in a molecule of a compound. It's a multiple of the empirical formula. To deduce the molecular formula, as we did for ethylene, we compare the molar mass of the compound to the mass of the empirical formula.

For ethylene, with a molar mass of 28.02 g/mol and an empirical formula mass of 14.03 g/mol, the ratio was exactly 2, indicating that the molecular formula is a double of the empirical formula, CH₂, thereby giving us C₂H₄.

Molar Mass Calculation

Molar mass is the mass of one mole of a substance and it's usually expressed in grams per mole (g/mol). The calculation of molar mass is crucial for converting between mass and moles of a substance.

In our example, the molar mass of ethylene is derived by dividing the gas's density by its moles per liter, which was determined using the Ideal Gas Law. Understanding this process is vital for students as it's a foundational skill in chemistry that applies not only to gases but to any substance when working with chemical equations and reactions.

STP Conditions

STP stands for Standard Temperature and Pressure, which are conditions commonly used to simplify gas calculations. At STP, the temperature is set at 0°C or 273.15 K and the pressure at 1 atm.

When a question mentions STP conditions, it lets you know immediately the values of two variables in the Ideal Gas Law, reducing complexity and making it easier to solve for unknowns such as molar mass. Remember, STP is a standard reference point used in the study of gasses, so becoming familiar with these conditions is beneficial for understanding gas behavior and properties.