Problem 134 - General Chemistry: Principles and Modern Applications

Question

For a fixed amount of gas at a fixed pressure, changing the temperature from \(100.0^{\circ} \mathrm{C}\) to \(200 \mathrm{K}\) causes the gas volume to (a) double; (b) increase, but not to twice its original value; (c) decrease; (d) stay the same.

Step-by-Step Solution

Answer

The answer is (c) decrease. The volume of the gas will decrease because, for a fixed amount of gas at a fixed pressure, the volume is directly proportional to the temperature (Gay-Lussac's Law). So if the temperature decreases, the volume will also decrease.

Step 1: Conversion of temperature from Celsius to Kelvin

First, we need to convert the initial temperature given in Celsius degrees to Kelvin, since Kelvin is the standard measurement for temperature in this context. The formula for changing Celsius to Kelvin is \( K = C + 273.15 \). So the starting temperature in Kelvin is \( 100.0^{\circ} \mathrm{C} + 273.15 = 373.15K \)

Step 2: Compare the initial and final temperatures

We then compare the initial and final temperatures. As stated, the final temperature is 200K and the initial temperature is 373.15K. Notice that the final temperature is less than the initial temperature.

Step 3: Apply Gay-Lussac’s Law

Next, we apply Gay-Lussac's law which states that the volume of a gas is directly proportional to its temperature, provided that the pressure remains the same. Since our pressure is constant, a decrease in temperature would cause a corresponding decrease in volume as per this law. This is true because as the temperature decreases, the gas particles slow down and occupy less space.

Key Concepts

Gay-Lussac's Law and Its Implications

In understanding the behavior of gases, Gay-Lussac's Law is a key concept. This law states that if the pressure of a given mass of gas is constant, the volume of the gas is directly proportional to its absolute temperature, measured in Kelvin. Essentially, it means that if the temperature increases, the volume increases proportionally, and vice versa.

Think of it as speaking the language of gas particles. When temperature rises, gas particles gain energy and move faster, requiring more space, which results in increased volume. Conversely, cooling down these particles results in less movement and decreased volume.

In the exercise, as the initial temperature of the gas changes from 373.15 K to 200 K, Gay-Lussac's Law helps us understand that the gas volume will decrease. Such understanding allows students to predict changes in gas behavior efficiently without complex calculations. Remember: more heat, more space; less heat, less space.

Understanding Temperature Conversion

For any scientific calculations involving gases, temperature Conversion is crucial. Generally, gas laws operate with temperatures in Kelvin rather than Celsius or Fahrenheit. This is because Kelvin is the absolute temperature scale, and it ensures temperatures are always positive, crucial for accurate calculations.

To convert Celsius to Kelvin, you simply add 273.15 to the Celsius temperature, as shown in the example. So, for a temperature of \(100.0^{\circ} \mathrm{C}\), add 273.15 to get 373.15 K. This conversion allows you to engage with gas calculations using the Ideal Gas Law more accurately.

Remember:

• The Kelvin scale starts at absolute zero, where theoretically, all molecular motion stops.
• Unlike Celsius, Kelvin doesn’t use degree symbols (°) for temperature.
• Positive values in Kelvin are non-negative, facilitating calculations without fallacies such as division by zero."},

Ensuring you're working in Kelvin saves you from pitfalls common in complex calculations.

Gas Volume and Temperature Relationship Simplified

The gas volume and temperature relationship is a fundamental concept expressed by Gay-Lussac's Law. This relationship signifies that at a constant pressure, changes in a gas's temperature directly impact its volume. Let’s simplify it further.

When the temperature of a gas increases, its molecules move faster, colliding more frequently and with more energy. This increased movement requires more space, causing the gas to expand. Therefore, the volume increases. However, when the temperature drops, the opposite occurs — the particles slow down and require less space, contracting the volume.

In the exercise, we're observing a temperature decrease from 373.15 K to 200 K. According to the gas volume-temperature relationship, the volume will decrease. This is because the particles at a lower temperature have less kinetic energy, take up less space, and thus, the volume is reduced.

This understanding assists in predicting how gases will behave when factors like temperature shift — vital for fields requiring precise gas management such as chemistry, meteorology, and engineering.