Problem 60 - Chemistry: An Atoms-Focused Approach

Question

Which of the following actions would produce the greater increase in the volume of a gas sample? a. Doubling the amount of gas in the sample at constant temperature and pressure b. Raising the temperature from \(244^{\circ} \mathrm{C}\) to \(1100^{\circ} \mathrm{C}\)

Step-by-Step Solution

Answer

Answer: Action (b) produces a greater increase in the volume of the gas sample, because the volume increases by a factor of 2.6549 compared to a factor of 2 in action (a).

Step 1: Understand the Ideal Gas Law

The Ideal Gas Law relates the volume (V), pressure (P), amount of gas (n), and temperature (T) of a gas sample. The Ideal Gas Law is given by: \(PV = nRT\) where R is the Ideal Gas constant.

Step 2: Calculate the change in volume for doubling the amount of gas

Since we are doubling the amount of gas (n) at constant temperature (T) and pressure (P), we can write the Ideal Gas Law equation for the initial state (n) and final state (2n) as follows: \(PV = nRT\) (initial state) \(P(2V) = (2n)RT\) (final state) The final volume (new V) will be twice the initial volume.

Step 3: Calculate the change in volume for raising the temperature

First, convert the given temperatures to Kelvin: \(244^{\circ} \mathrm{C} = 244 + 273 = 517 K\) \(1100^{\circ} \mathrm{C} = 1100 + 273 = 1373 K\) Now let's analyze the change in volume, keeping the pressure (P) and amount of gas (n) constant: \(PV = nRT_1\) (initial state with temperature \(T_1 = 517 K\)) \(PV_2 = nRT_2\) (final state with temperature \(T_2 = 1373 K\)) To find the new volume, divide the final state equation by the initial state equation: \(\frac{V_2}{V} = \frac{nRT_2}{nRT_1}\) Simplify the equation and plug in the temperatures: \(\frac{V_2}{V} = \frac{T_2}{T_1} = \frac{1373}{517} = 2.6549\)

Step 4: Compare the changes in volume for both actions

For action (a), doubling the amount of gas, the volume increased by a factor of 2. For action (b), raising the temperature from \(244^{\circ} \mathrm{C}\) to \(1100^{\circ} \mathrm{C}\), the volume increased by a factor of 2.6549. Since 2.6549 > 2, action (b) would produce a greater increase in the volume of the gas sample.

Key Concepts

Volume of Gas

The volume of a gas is an important concept in understanding how gases behave under different conditions. According to the Ideal Gas Law, the volume ( V ) of a gas is directly proportional to the amount of gas ( n ) and temperature ( T ), and inversely proportional to the pressure ( P ). To put it simply, if you increase the amount of gas or its temperature while keeping pressure constant, the gas will occupy more space:

• Doubling the amount of gas in a sample will result in a doubling of the volume, if both temperature and pressure remain unchanged. This is because the molecules need more space to move around.
• When pressure is constant, volume changes in direct proportion with the temperature when measured in Kelvin.

When handling gas problems, ensure you're using consistent units, especially converting Celsius to Kelvin for temperature changes. This understanding is crucial as it helps predict the behavior of gas in various physical and chemical processes.

Temperature Change

Temperature change has a significant impact on how gases behave according to the Ideal Gas Law. When you increase the temperature of a gas while keeping pressure and the amount of gas constant, the gas molecules move more vigorously, which leads to an increase in the volume:

• The degree of temperature change is usually measured in Kelvin. Note that an increase in Kelvin temperature will result in a proportional increase in gas volume.
• For example, increasing the temperature from 244 °C (517 K) to 1100 °C (1373 K) results in an increase in the gas's volume by a factor of 2.6549, illustrating how sensitive gas volume is to temperature changes.

It's important to recognize that even small changes in temperature can lead to noticeable changes in gas volume due to the direct proportionality dictated by the Ideal Gas Law.

Gas Expansion

Gas expansion is a core concept when considering how gases respond to changes in temperature, amount, or pressure. Using the Ideal Gas Law, we understand that gases expand or contract when subject to changes among these variables:

• Gas expansion occurs when the amount of gas increases. For instance, doubling the number of gas particles while maintaining the same temperature and pressure will double the gas volume, as the gas molecules spread out to occupy more space.
• Similarly, gas expansion occurs with increased temperature (again, with constant pressure and amount of gas). This is because higher temperatures give the gas molecules more energy, causing them to move faster and spread out more, thus increasing the volume.

Understanding gas expansion is critical in many real-world applications, such as in engines, refrigerators, and even balloon science. Knowing how gas volume changes in response to various conditions empowers us to predict and control gas behavior in practical situations.