Problem 57 - Chemistry Matter and Change

Question

Use Boyles, Charless, or Gay-Lussac's law to calculate the missing value in each of the following. a. \(V_{1}=2.0 \mathrm{L}, P_{1}=0.82 \mathrm{atm}, V_{2}=1.0 \mathrm{L}, P_{2}=?\) b. \(V_{1}=250 \mathrm{mL}, T_{1}=?, V_{2}=400 \mathrm{mL}, T_{2}=298 \mathrm{K}\) c. \(V_{1}=0.55 \mathrm{L}, P_{1}=740 \mathrm{mm} \mathrm{Hg}, V_{2}=0.80 \mathrm{L}, P_{2}=?\)

Step-by-Step Solution

Answer

a. \(P_2 = 1.64\) atm; b. \(T_1 = 186.25\) K; c. \(P_2 = 508.75\) mm Hg.

Step 1: Identify the Law for Part a

Part (a) involves changes in pressure and volume while temperature is constant. Therefore, we use Boyle's Law, which states that the product of the initial pressure and volume equals the product of the final pressure and volume: \( P_1 V_1 = P_2 V_2 \).

Step 2: Solve for P2 in Part a

Substitute the given values into Boyle's Law from step 1: \( 0.82 \; \text{atm} \times 2.0 \; \text{L} = P_2 \times 1.0 \; \text{L} \). Solve for \( P_2 \) by dividing both sides by \( 1.0 \; \text{L} \): \( P_2 = 0.82 \times 2.0 = 1.64 \; \text{atm} \).

Step 3: Identify the Law for Part b

Part (b) involves changes in temperature and volume. Therefore, we use Charles's Law, which states that the volume is proportional to the absolute temperature when pressure is constant: \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \).

Step 4: Solve for T1 in Part b

Substitute the given values into Charles's Law: \( \frac{250 \; \text{mL}}{T_1} = \frac{400 \; \text{mL}}{298 \; \text{K}} \). Solve for \( T_1 \) by cross-multiplying and dividing: \( T_1 = \frac{250 \times 298}{400} = 186.25 \; \text{K} \).

Step 5: Identify the Law for Part c

Part (c) involves changes in pressure and volume while temperature is constant; thus, again we use Boyle's Law: \( P_1 V_1 = P_2 V_2 \).

Step 6: Solve for P2 in Part c

Substitute the given values into Boyle's Law: \( 740 \; \text{mm Hg} \times 0.55 \; \text{L} = P_2 \times 0.80 \; \text{L} \). Solve for \( P_2 \) by dividing both sides by \( 0.80 \; \text{L} \): \( P_2 = \frac{740 \times 0.55}{0.80} = 508.75 \; \text{mm Hg} \).

Key Concepts

Boyle's Law

Boyle's Law connects the pressure and volume of a gas at a constant temperature. This law is crucial for understanding how gases behave when confined in a container where temperature is kept steady.
It states, in simple terms, that the pressure of a gas is inversely proportional to its volume. This means if you increase the volume of the gas's container, the pressure will decrease, assuming the temperature is unchanged, and vice versa.
We express Boyle's Law mathematically as:

• \[ P_1 V_1 = P_2 V_2 \]

In this equation, \( P_1 \) and \( V_1 \) are the initial pressure and volume, while \( P_2 \) and \( V_2 \) are the final conditions of the gas. So, for example, if a gas initially at a pressure of 0.82 atm is compressed from a volume of 2.0 L to 1.0 L, we can find the final pressure by rearranging the formula to solve for \( P_2 \), like so:

• \[ P_2 = \frac{P_1 V_1}{V_2} \]

This law is a fundamental part of the ideal gas law and helps us understand how gases compress or expand.

Charles's Law

When dealing with gases, Charles's Law provides insight into how the volume of a gas changes with temperature at constant pressure. Charles's Law tells us that the volume of a gas is directly proportional to its absolute temperature. In other words, as you heat a gas, it expands if the pressure remains constant. Conversely, cooling the gas will cause it to contract.
This relationship is expressed with the equation:

• \[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \]

Here, \( V_1 \) and \( T_1 \) are the initial volume and temperature, while \( V_2 \) and \( T_2 \) are the final volume and temperature. A practical example is a balloon expanding in heat. If a balloon fills 250 mL at an unknown temperature, and later occupies 400 mL at 298 K, Charles's Law lets us calculate the initial temperature by rearranging to:

• \[ T_1 = \frac{V_1 \times T_2}{V_2} \]

Remember always to measure temperatures in Kelvin for Charles's Law.

Gay-Lussac's Law

While Charles's Law works with volume changes, Gay-Lussac's Law deals with pressure changes in gases at constant volume. It speaks to how the pressure of a gas depends on its absolute temperature if the volume does not change.
This law is compellingly straightforward:

• \[ \frac{P_1}{T_1} = \frac{P_2}{T_2} \]

In this formula, \( P_1 \) and \( T_1 \) indicate the initial pressure and temperature, while \( P_2 \) and \( T_2 \) depict the final state. When you heat a gas within a rigid container, such as a sealed can, the pressure will increase due to the lack of expansion space. Similarly, cooling the gas will cause the pressure to drop. This can be easily remembered as: "Temperature goes up, pressure goes up."
Thus, if you know three of these variables, you can always solve for the fourth, ensuring the behavior of gases can be precisely predicted under different thermal conditions.

Pressure and Volume Relationship

The relationship between pressure and volume in gases comes directly from Boyle's Law, as it explores how one impacts the other. Since the two are inversely proportional, their behaviors contrast each other:

• As volume increases, pressure decreases
• As volume decreases, pressure increases

This is because a gas's molecules, when compressed into a smaller volume, will collide more frequently within the container's walls, amplifying the pressure. So, if we double the volume of a container without changing the temperature, the gas pressure inside it drops to half its original value. This relationship underscores the importance of keeping conditions consistent when comparing gas behaviors.
It proves crucial in situations like scuba diving, where ascents and descents change the pressure and can significantly affect the gas volume the diver is breathing, potentially leading to serious health risks if not carefully managed.

Temperature and Volume Relationship

Understanding how temperature and volume interlink is key in many real-world applications, particularly those described by Charles's Law. This law illustrates that temperature and volume share a direct relationship:

• Increase in temperature leads to increase in volume
• Decrease in temperature causes decrease in volume

For instance, if you heat a balloon, it will expand. This happens because heating increases the kinetic energy of the gas particles inside, causing them to move faster and occupy more space. Similarly, cooling the gas would slow those particles and condense the gas, lowering the volume.
This relationship is essential for applications like hot air balloons, where heating the air inside the balloon makes it less dense than the cooler air outside, causing it to rise. Remember: when dealing with gas laws and their relationships, always use absolute temperatures measured in Kelvin to ensure accuracy and consistency.