Understanding the Key Gas Laws
Before delving into practice problems, let’s briefly review the essential gas laws that form the foundation of gas behavior:
1. Boyle's Law
Boyle's Law states that the pressure of a gas is inversely proportional to its volume when temperature remains constant. Mathematically, it can be expressed as:
\[ P_1V_1 = P_2V_2 \]
2. Charles's Law
Charles's Law posits that the volume of a gas is directly proportional to its absolute temperature (in Kelvin) when pressure is held constant. The equation is as follows:
\[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \]
3. Avogadro's Law
Avogadro's Law indicates that the volume of a gas is directly proportional to the number of moles of the gas when pressure and temperature are constant:
\[ V_1/n_1 = V_2/n_2 \]
4. Ideal Gas Law
The Ideal Gas Law combines the previous laws into one comprehensive equation:
\[ PV = nRT \]
Where:
- \( P \) = pressure
- \( V \) = volume
- \( n \) = number of moles
- \( R \) = ideal gas constant (0.0821 L·atm/(K·mol))
- \( T \) = temperature in Kelvin
Practice Problems
Now that we have reviewed the key gas laws, let's dive into some practice problems to solidify your understanding.
Problem 1: Boyle's Law
A gas occupies a volume of 4.0 L at a pressure of 2.0 atm. What will be the new volume of the gas if the pressure is changed to 1.0 atm, assuming the temperature remains constant?
Problem 2: Charles's Law
A sample of gas has a volume of 10.0 L at a temperature of 300 K. What will be the volume of the gas when the temperature is raised to 600 K, assuming pressure remains constant?
Problem 3: Avogadro's Law
A gas occupies 5.0 L at a certain pressure and temperature. If the amount of gas is increased from 2.0 moles to 4.0 moles, what will be the new volume of the gas at the same pressure and temperature?
Problem 4: Ideal Gas Law
Calculate the number of moles of an ideal gas that occupies a volume of 22.4 L at a pressure of 1.0 atm and a temperature of 273 K.
Problem 5: Combined Gas Law
A gas has an initial volume of 3.0 L, an initial pressure of 1.5 atm, and an initial temperature of 300 K. If the gas is compressed to a volume of 1.5 L and heated to a temperature of 600 K, what will be the new pressure of the gas?
Solutions to Practice Problems
Let’s now go through the solutions step by step.
Solution 1: Boyle's Law
Using Boyle's Law:
\[ P_1V_1 = P_2V_2 \]
Plugging in the values:
\[ (2.0 \, \text{atm})(4.0 \, \text{L}) = (1.0 \, \text{atm})(V_2) \]
Calculating:
\[ 8.0 = 1.0 \times V_2 \]
\[ V_2 = 8.0 \, \text{L} \]
Thus, the new volume is 8.0 L.
Solution 2: Charles's Law
Using Charles's Law:
\[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \]
Plugging in the values:
\[ \frac{10.0 \, \text{L}}{300 \, \text{K}} = \frac{V_2}{600 \, \text{K}} \]
Cross-multiplying:
\[ 10.0 \times 600 = 300 \times V_2 \]
\[ 6000 = 300 \times V_2 \]
\[ V_2 = 20.0 \, \text{L} \]
Therefore, the new volume is 20.0 L.
Solution 3: Avogadro's Law
Using Avogadro's Law:
\[ V_1/n_1 = V_2/n_2 \]
Substituting the known values:
\[ \frac{5.0 \, \text{L}}{2.0 \, \text{moles}} = \frac{V_2}{4.0 \, \text{moles}} \]
Cross-multiplying:
\[ 5.0 \times 4.0 = 2.0 \times V_2 \]
\[ 20.0 = 2.0 \times V_2 \]
\[ V_2 = 10.0 \, \text{L} \]
Thus, the new volume is 10.0 L.
Solution 4: Ideal Gas Law
Using the Ideal Gas Law:
\[ PV = nRT \]
Rearranging to solve for \( n \):
\[ n = \frac{PV}{RT} \]
Substituting the known values:
\[ n = \frac{(1.0 \, \text{atm})(22.4 \, \text{L})}{(0.0821 \, \text{L·atm/(K·mol)})(273 \, \text{K})} \]
Calculating:
\[ n = \frac{22.4}{22.414} \approx 1.00 \, \text{mol} \]
So, the number of moles is 1.00 mol.
Solution 5: Combined Gas Law
Using the combined gas law:
\[ \frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2} \]
Rearranging to find \( P_2 \):
\[ P_2 = P_1 \cdot \frac{V_1}{V_2} \cdot \frac{T_2}{T_1} \]
Substituting the known values:
\[ P_2 = 1.5 \, \text{atm} \cdot \frac{3.0 \, \text{L}}{1.5 \, \text{L}} \cdot \frac{600 \, \text{K}}{300 \, \text{K}} \]
Calculating:
\[ P_2 = 1.5 \cdot 2 \cdot 2 = 6.0 \, \text{atm} \]
Thus, the new pressure is 6.0 atm.
Conclusion
Understanding gas laws is crucial for anyone studying chemistry. By practicing problems like those presented above, you can develop a strong grasp of how gases behave under various conditions. It's also beneficial to recognize how these laws apply to real-world scenarios, from understanding how balloons expand in heat to calculating the behavior of gases in industrial processes. With consistent practice, you can become proficient in applying gas laws to solve a wide range of problems effectively.
Frequently Asked Questions
What is the ideal gas law equation and what do its variables represent?
The ideal gas law equation is PV = nRT, where P is pressure, V is volume, n is the number of moles of gas, R is the ideal gas constant, and T is temperature in Kelvin.
How can you calculate the volume of a gas at standard temperature and pressure (STP) using the ideal gas law?
At STP, the temperature is 273.15 K and pressure is 1 atm. You can rearrange the ideal gas law to find V: V = nRT/P. For 1 mole of gas at STP, V = (1 mol)(0.0821 L·atm/(mol·K))(273.15 K)/(1 atm) = 22.4 L.
If a gas occupies 10 L at 2 atm, what will its volume be if the pressure is changed to 1 atm, assuming temperature remains constant?
Using Boyle's Law (P1V1 = P2V2), we can rearrange to find V2: V2 = P1V1/P2 = (2 atm)(10 L)/(1 atm) = 20 L.
How does temperature affect the volume of a gas at constant pressure?
According to Charles's Law, at constant pressure, the volume of a gas is directly proportional to its temperature in Kelvin. V1/T1 = V2/T2.
What happens to the pressure of a gas if its volume is halved while keeping the temperature constant?
According to Boyle's Law, if the volume is halved, the pressure will double, assuming temperature remains constant.
If 3 moles of an ideal gas are at a temperature of 300 K and occupy a volume of 24 L, what is the pressure?
Using the ideal gas law PV = nRT, we rearrange to find P: P = nRT/V = (3 mol)(0.0821 L·atm/(mol·K))(300 K)/(24 L) = 3.1 atm.
What is the relationship between the pressure and temperature of a gas at constant volume?
According to Gay-Lussac's Law, the pressure of a gas is directly proportional to its absolute temperature when volume is held constant: P1/T1 = P2/T2.
How do gas laws apply to real gases versus ideal gases?
Real gases deviate from ideal gas behavior due to intermolecular forces and volume occupied by gas molecules, especially at high pressures and low temperatures, while ideal gases follow the gas laws perfectly under all conditions.
If a gas's volume increases from 5 L to 15 L, and its initial pressure was 1.5 atm, what will the new pressure be?
Using Boyle's Law, P1V1 = P2V2, we find P2: P2 = P1V1/V2 = (1.5 atm)(5 L)/(15 L) = 0.5 atm.
What is the significance of the gas constant R in the ideal gas law?
The gas constant R is a proportionality constant that relates the energy scale to the temperature scale in the ideal gas law. Its value depends on the units used; commonly, R = 0.0821 L·atm/(mol·K) or 8.314 J/(mol·K).
