Problem 100
Question
Calculate A toy manufacturer uses tetrafluoroethane (\(\mathrm{C}_{2} \mathrm{H}_{2} \mathrm{F}_{4}\)) at high temperatures to fill plastic molds for toys. a. What is the density \((\operatorname{in} g / \mathrm{L})\)of \(\mathrm{C}_{2} \mathrm{H}_{2} \mathrm{F}_{4}\) at \(\mathrm{STP}\) ? \right. b. Find the molecules per liter of \(\mathrm{C}_{2} \mathrm{H}_{2} \mathrm{F}_{4}\) at \(220^{\circ} \mathrm{C}\) and 1.0 atm.
Step-by-Step Solution
Verified Answer
a. Density at STP is approximately 4.55 g/L. b. There are about 1.49 x 10^{22} molecules/L at 220°C and 1 atm.
1Step 1: Find Molar Mass
Calculate the molar mass of \(\text{C}_2\text{H}_2\text{F}_4\).- Carbon (C) has an atomic mass of approximately 12 g/mol. For 2 carbons: \(2 \times 12 = 24\) g/mol.- Hydrogen (H) has an atomic mass of approximately 1 g/mol. For 2 hydrogens: \(2 \times 1 = 2\) g/mol.- Fluorine (F) has an atomic mass of approximately 19 g/mol. For 4 fluorines: \(4 \times 19 = 76\) g/mol.The total molar mass is: \(24 + 2 + 76 = 102\) g/mol.
2Step 2: Density at STP
At STP, 1 mole of any ideal gas occupies 22.4 L.Using the molar mass from Step 1, the density is calculated as density \( = \frac{\text{molar mass}}{\text{volume at STP}} = \frac{102 \, \text{g/mol}}{22.4 \, \text{L/mol}} \approx 4.55 \, \text{g/L}\).
3Step 3: Calculate the number of molecules per liter at given conditions
First, use the Ideal Gas Law: \(PV = nRT\). - \(P = 1 \text{ atm}\)- \(T = 220^\circ\text{C} = 493\text{ K}\) - \(R = 0.0821 \, \text{L atm mol}^{-1} \text{K}^{-1}\) - Solve for \(n/V\) which is equal to \(\frac{P}{RT}\)Calculate: \(\frac{n}{V} = \frac{1}{0.0821 \times 493} \approx 0.0247 \, \text{mol/L}\).
4Step 4: Convert mol/L to molecules per liter
Use Avogadro's number to convert moles to molecules. Avogadro's number \(= 6.022 \times 10^{23}\) molecules/mol.Number of molecules per liter \( = 0.0247 \, \text{mol/L} \times 6.022 \times 10^{23}\, \text{molecules/mol} \approx 1.49 \times 10^{22} \, \text{molecules/L}\).
Key Concepts
Molar Mass CalculationDensity at STPAvogadro's Number
Molar Mass Calculation
Molar mass is an essential concept when working with chemical substances. It refers to the mass of one mole of a given element or compound. To find the molar mass, you need to know the atomic masses of the elements, usually found on the periodic table.
For example, let's calculate the molar mass of tetrafluoroethane, \( \mathrm{C}_{2} \mathrm{H}_{2} \mathrm{F}_{4} \). This molecule consists of:
Understanding molar mass is crucial for various calculations in chemistry, including determining the number of moles of a substance in a given mass and converting between mass and volume.
For example, let's calculate the molar mass of tetrafluoroethane, \( \mathrm{C}_{2} \mathrm{H}_{2} \mathrm{F}_{4} \). This molecule consists of:
- 2 carbon atoms, each with an atomic mass of approximately 12 g/mol. Therefore, for carbon it is \(12 \times 2 = 24\) g/mol.
- 2 hydrogen atoms, each with an atomic mass of approximately 1 g/mol. Thus, for hydrogen, it's \(1 \times 2 = 2\) g/mol.
- 4 fluorine atoms, each with an atomic mass of approximately 19 g/mol, giving \(19 \times 4 = 76\) g/mol for fluorine.
Understanding molar mass is crucial for various calculations in chemistry, including determining the number of moles of a substance in a given mass and converting between mass and volume.
Density at STP
Density at standard temperature and pressure (STP) involves knowing how much mass a gas occupies in a standard volume. STP is defined with a temperature of 0°C or 273.15K and a pressure of 1 atm, where 1 mole of an ideal gas occupies 22.4 liters.
To find the density of a gas at STP, you take the molar mass of the gas molecule and divide it by the volume a mole occupies at STP:\[\text{Density} = \frac{\text{Molar Mass}}{\text{Volume at STP}} = \frac{102 \, \text{g/mol}}{22.4 \, \text{L/mol}} \approx 4.55 \, \text{g/L}\]This tells us how much 1 liter of the gas weighs at standard conditions.
Calculating density at STP is important for comparing gases or working with gas mixtures. It helps chemists and engineers predict how different gases behave and interact under the same conditions, allowing them to design processes and products accordingly.
To find the density of a gas at STP, you take the molar mass of the gas molecule and divide it by the volume a mole occupies at STP:\[\text{Density} = \frac{\text{Molar Mass}}{\text{Volume at STP}} = \frac{102 \, \text{g/mol}}{22.4 \, \text{L/mol}} \approx 4.55 \, \text{g/L}\]This tells us how much 1 liter of the gas weighs at standard conditions.
Calculating density at STP is important for comparing gases or working with gas mixtures. It helps chemists and engineers predict how different gases behave and interact under the same conditions, allowing them to design processes and products accordingly.
Avogadro's Number
Avogadro's number is one of the most important constants in chemistry. It's \(6.022 \times 10^{23}\), which represents the number of molecules or atoms in one mole of a substance.
This number helps bridge the gap between the macroscopic world we can see and the microscopic worlds of atoms and molecules. For example, if you have a mole of a substance, using Avogadro's number, you can determine the exact number of molecules you have.
In practical calculations, if you know the amount of substance in moles, you multiply by Avogadro's number to find how many molecules are in that amount. For instance, at given conditions, if you have \(0.0247 \, \text{mol/L}\) of a gas, you can find the molecules per liter by multiplying with Avogadro’s number:\[\text{Number of molecules per liter} = 0.0247 \, \text{mol/L} \times 6.022 \times 10^{23}\, \text{molecules/mol} = 1.49 \times 10^{22} \, \text{molecules/L}\]Avogadro’s number allows us to comprehend and manipulate the vast quantities of tiny particles in routine chemical calculations.
This number helps bridge the gap between the macroscopic world we can see and the microscopic worlds of atoms and molecules. For example, if you have a mole of a substance, using Avogadro's number, you can determine the exact number of molecules you have.
In practical calculations, if you know the amount of substance in moles, you multiply by Avogadro's number to find how many molecules are in that amount. For instance, at given conditions, if you have \(0.0247 \, \text{mol/L}\) of a gas, you can find the molecules per liter by multiplying with Avogadro’s number:\[\text{Number of molecules per liter} = 0.0247 \, \text{mol/L} \times 6.022 \times 10^{23}\, \text{molecules/mol} = 1.49 \times 10^{22} \, \text{molecules/L}\]Avogadro’s number allows us to comprehend and manipulate the vast quantities of tiny particles in routine chemical calculations.
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