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Chapter 5: Problem 85

The empirical formula of a compound is CH. At \(200^{\circ} \mathrm{C}, 0.145 \mathrm{~g}\) of this compound occupies \(97.2 \mathrm{~mL}\) at a pressure of 0.74 atm. What is the molecular formula of the compound?

Short Answer

Expert verified
The molecular formula is \( \text{C}_6\text{H}_6 \).

Step by step solution

01

Determine Molar Volume at STP

First, calculate the volume at STP using the ideal gas law. Standard Temperature and Pressure (STP) is 0掳C and 1 atm, where 1 mole of gas occupies 22.4 L. Use the formula \[ V_2 = V_1 \frac{P_1}{P_2} \frac{T_2}{T_1} \] where \(P_1 = 0.74\) atm, \(P_2 = 1\) atm, \(T_1 = 473\) K (200掳C), and \(T_2 = 273\) K. Calculate \[ V_2 = 0.0972 \times \frac{0.74}{1} \times \frac{273}{473} \approx 0.0412 \text{ L} \]
02

Calculate Moles of Gas

Using the volume at STP, calculate the moles of gas. Since 1 mole at STP occupies 22.4 L, the number of moles is \( n = \frac{0.0412}{22.4} \approx 0.00184 \text{ moles} \).
03

Calculate Molar Mass

With the mass of the compound provided, calculate the molar mass using the formula \[ M = \frac{m}{n} \] where \( m = 0.145 \) g and \( n = 0.00184 \text{ mol} \). Thus, the molar mass \( M \approx \frac{0.145}{0.00184} \approx 78.80 \text{ g/mol} \).
04

Determine Molecular Formula

Given the empirical formula CH, calculate the molar mass of the empirical unit: \( 12 + 1 = 13 \text{ g/mol} \). The ratio of the molecular mass to the empirical mass is \( \frac{78.80}{13} \approx 6 \), indicating that the molecular formula is 6 times the empirical formula. Therefore, the molecular formula is \( \text{C}_6\text{H}_6 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Empirical Formula
Understanding the empirical formula is crucial in chemistry. It represents the simplest whole-number ratio of atoms in a compound. For instance, if we consider a compound with an empirical formula of CH, this means there is one carbon atom for every hydrogen atom. The empirical formula does not reflect the actual number of atoms in a molecule, only their proportions.

When determining a molecular formula, we start with the empirical formula as a stepping stone. In many exercises, finding the molecular formula entails comparing the given molar mass with the empirical formula mass to locate a multiplying factor.
Ideal Gas Law
The ideal gas law is a fundamental equation that connects multiple properties of a gas: pressure (P), volume (V), temperature (T), and the number of moles (n). The law is typically expressed as \( PV = nRT \), where \( R \) stands for the ideal gas constant. This formula is very versatile for many calculations.

By using the ideal gas law, we can find unknown values if enough information is provided. In the current exercise, the ideal gas law helps convert between conditions at different temperatures and pressures. Particularly, it is useful in translating gas volumes to standard temperature and pressure for easier comparison and calculation.
Molar Mass Calculation
Molar mass is defined as the mass of one mole of a substance, usually expressed in grams per mole (g/mol). Calculating the molar mass is often necessary in exercises involving chemical reactions or gas laws. Given mass and calculated moles, the molar mass is calculated by dividing mass by the number of moles: \( M = \frac{m}{n} \).

In our exercise, knowing the molar mass lets us crosscheck with the empirical formula mass. This comparison lets us discover the factor by which the empirical formula must be multiplied to obtain the correct molecular formula.
Moles of Gas
When considering gases, the concept of moles links to several important properties. The number of moles tells us about the quantity of a substance in terms of its basic units, atoms, or molecules. To calculate moles from the conditions given at standard temperature and pressure, divide the corrected volume by the standard molar volume of a gas, 22.4 L/mol.

In our exercise, calculating the moles from given volume and mass is an essential step. It provides the necessary link to determine the molar mass, which then helps in verifying and computing the molecular formula.
Standard Temperature and Pressure (STP)
Standard Temperature and Pressure, abbreviated as STP, is a conventional reference point for expressing properties of gases. At STP, the temperature is set to 0掳C (273 K), and the pressure is 1 atmosphere. Under these conditions, one mole of an ideal gas occupies a volume of 22.4 liters.

STP conditions simplify the relationships between volume, pressure, and moles, making it easier to compare results across different conditions. In our problem, converting the initial conditions of temperature and pressure to STP helps calculate the volume and number of moles under more familiar standard conditions, facilitating further calculations.

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Most popular questions from this chapter

At \(27^{\circ} \mathrm{C}, 10.0\) moles of a gas in a 1.50 - \(\mathrm{L}\) container exert a pressure of 130 atm. Is this an ideal gas?

A mixture of gases contains \(\mathrm{CH}_{4}, \mathrm{C}_{2} \mathrm{H}_{6},\) and \(\mathrm{C}_{3} \mathrm{H}_{8} .\) If the total pressure is \(1.50 \mathrm{~atm}\) and the numbers of moles of the gases present are 0.31 mole for \(\mathrm{CH}_{4}\), 0.25 mole for \(\mathrm{C}_{2} \mathrm{H}_{6}\), and 0.29 mole for \(\mathrm{C}_{3} \mathrm{H}_{8}\), calculate the partial pressures of the gases.

Some commercial drain cleaners contain two components: sodium hydroxide and aluminum powder. When the mixture is poured down a clogged drain, the following reaction occurs: $$\begin{aligned}2 \mathrm{NaOH}(a q)+2 \mathrm{Al}(s)+& 6 \mathrm{H}_{2} \mathrm{O}(l) \longrightarrow \\\& 2 \mathrm{NaAl}(\mathrm{OH})_{4}(a q)+3 \mathrm{H}_{2}(g) \end{aligned}$$ The heat generated in this reaction helps melt away obstructions such as grease, and the hydrogen gas released stirs up the solids clogging the drain. Calculate the volume of \(\mathrm{H}_{2}\) formed at STP if \(3.12 \mathrm{~g}\) of \(\mathrm{Al}\) is treated with an excess of \(\mathrm{NaOH}\).

Describe how a barometer and a manometer are used to measure gas pressure.

Helium is mixed with oxygen gas for deep sea divers. Calculate the percent by volume of oxygen gas in the mixture if the diver has to submerge to a depth where the total pressure is \(4.2 \mathrm{~atm} .\) The partial pressure of oxygen is maintained at 0.20 atm at this depth.
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