Chapter 11: Problem 83
How many atoms of nitrogen are there in a \(230.0-\mathrm{mL}\) sample of \(\mathrm{N}_{2} \mathrm{O}_{4}\) gas that has a pressure of \(745.0 \mathrm{~mm} \mathrm{Hg}\) at \(34.0^{\circ} \mathrm{C}\) ?
Short Answer
Expert verified
In a 230.0 mL sample of N₂O₄ gas with a pressure of 745.0 mmHg at 34.0°C, there are approximately \(1.105 \times 10^{23}\) atoms of nitrogen.
Step by step solution
01
Convert pressure from mmHg to atm
First, we need to convert the given pressure from mmHg to atm using the following conversion factor: 1 atm = 760 mmHg.
Given pressure = 745.0 mmHg
To convert to atm:
Pressure (atm) = (745.0 mmHg) / (760 mmHg/atm)
Pressure (atm) = 0.98026 atm
02
Convert temperature from Celsius to Kelvin
Next, let's convert the given temperature from Celsius to Kelvin using the formula:
Temperature (K) = Temperature (°C) + 273.15
Temperature (K) = 34.0° C + 273.15
Temperature (K) = 307.15 K
03
Use Ideal Gas Law to determine moles of N₂O₄
Now, we will use the Ideal Gas Law equation (PV = nRT) to find the number of moles (n) of N₂O₄:
P = 0.98026 atm (pressure)
V = 230.0 mL (volume) = 0.230 L (convert to liters)
R = 0.0821 L atm / (mol K) (the ideal gas constant)
T = 307.15 K (temperature)
n = PV / (RT)
n = (0.98026 atm * 0.230 L) / (0.0821 L atm/mol K * 307.15 K)
n = 0.00918 mol
04
Calculate the number of nitrogen atoms
Now that we have the number of moles of N₂O₄, we can find the number of nitrogen atoms (each molecule of N₂O₄ contains 2 nitrogen atoms) using Avogadro's number, 6.022 x 10²³ atoms/mol.
Total moles of nitrogen atoms = 0.00918 mol of N₂O₄ * 2 = 0.01836 mol of nitrogen atoms
Number of nitrogen atoms = 0.01836 mol of nitrogen atoms * 6.022 x 10²³ atoms/mol
Number of nitrogen atoms = 1.105 x 10²³ atoms
In conclusion, there are approximately 1.105 x 10²³ atoms of nitrogen in the 230.0 mL sample of N₂O₄ gas.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Pressure Conversion
Pressure conversion is crucial when dealing with gas laws because pressure can be measured in different units. One common unit is millimeters of mercury (mmHg), and another is atmospheres (atm). To convert pressure from mmHg to atm, remember that 1 atm is equivalent to 760 mmHg.
To perform the conversion:
For example, if you have a pressure of 745 mmHg, the conversion would be:\[\text{Pressure (atm)} = \frac{745.0 \text{ mmHg}}{760 \text{ mmHg/atm}} = 0.98026 \text{ atm}\]Understanding this conversion is essential for using the Ideal Gas Law effectively.
To perform the conversion:
- Take the given pressure in mmHg.
- Divide it by 760 mmHg/atm.
For example, if you have a pressure of 745 mmHg, the conversion would be:\[\text{Pressure (atm)} = \frac{745.0 \text{ mmHg}}{760 \text{ mmHg/atm}} = 0.98026 \text{ atm}\]Understanding this conversion is essential for using the Ideal Gas Law effectively.
Temperature Conversion
In gas calculations, temperatures must be in Kelvin because it is an absolute scale that starts at zero, which aligns with the laws of thermodynamics. Converting from Celsius to Kelvin is straightforward and is done using the formula:
Let's say you have a temperature of 34.0°C. Using the conversion formula:\[\text{Temperature (K)} = 34.0^\circ \text{C} + 273.15 = 307.15 \text{ K}\]With this Kelvin value, you can confidently use it in any gas law calculations, like the Ideal Gas Law.
- Temperature (K) = Temperature (°C) + 273.15
Let's say you have a temperature of 34.0°C. Using the conversion formula:\[\text{Temperature (K)} = 34.0^\circ \text{C} + 273.15 = 307.15 \text{ K}\]With this Kelvin value, you can confidently use it in any gas law calculations, like the Ideal Gas Law.
Avogadro's Number
Avogadro's number is crucial when dealing with moles and particles. It tells us exactly how many particles (atoms, molecules, etc.) are in one mole of a substance. This number, 6.022 x 10²³, is constant and used extensively in chemistry.
Why is it important? Because it allows us to translate between moles, which we can calculate and measure, and actual amounts of atoms and molecules, which are too small to count individually.
For example, if we need to find the number of nitrogen atoms in a sample, we start by finding the moles of nitrogen, then multiply by Avogadro's number to find the actual number of atoms:\[\text{Number of nitrogen atoms} = \text{moles of nitrogen atoms} \times 6.022 \times 10^{23}\]Grasping this concept is key to understanding molecular measurements.
Why is it important? Because it allows us to translate between moles, which we can calculate and measure, and actual amounts of atoms and molecules, which are too small to count individually.
For example, if we need to find the number of nitrogen atoms in a sample, we start by finding the moles of nitrogen, then multiply by Avogadro's number to find the actual number of atoms:\[\text{Number of nitrogen atoms} = \text{moles of nitrogen atoms} \times 6.022 \times 10^{23}\]Grasping this concept is key to understanding molecular measurements.
Mole Calculations
Mole calculations are central to chemistry, especially when using the Ideal Gas Law. A mole is a measure of quantity that is universally used to count molecules or atoms.
When using the Ideal Gas Law, which is expressed as \( PV = nRT \), you need to solve for \( n \), the number of moles, based on your known values of pressure (P), volume (V), and temperature (T). The constant \( R \) depends on the units of pressure and volume and is typically 0.0821 L atm/mol K for gases.
Let's move through an example:\[n = \frac{PV}{RT}\]Plug the values of pressure, volume, and temperature converted to the correct units (atm, L, K) into this formula to find the moles of substance.
Once you know the moles, you can further explore how many molecules or atoms are present by using Avogadro's number. This calculation lets you transition between measurable quantities and molecular quantities.
When using the Ideal Gas Law, which is expressed as \( PV = nRT \), you need to solve for \( n \), the number of moles, based on your known values of pressure (P), volume (V), and temperature (T). The constant \( R \) depends on the units of pressure and volume and is typically 0.0821 L atm/mol K for gases.
Let's move through an example:\[n = \frac{PV}{RT}\]Plug the values of pressure, volume, and temperature converted to the correct units (atm, L, K) into this formula to find the moles of substance.
Once you know the moles, you can further explore how many molecules or atoms are present by using Avogadro's number. This calculation lets you transition between measurable quantities and molecular quantities.