Problem 4 - Chemistry: An Atoms-Focused Approach

Question

Suppose the temperature of the gas in Figure \(\mathrm{P} 10.2(\mathrm{b})\) increases from \(200 \mathrm{K}\) to \(400 \mathrm{K}\) but that the pressure on the piston remains constant. Does the position of the piston change? If so, by how much does the volume of the gas change?

Step-by-Step Solution

Answer

Answer: Yes, the position of the piston changes. The volume of the gas increases by an amount equal to its initial volume.

Step 1: Identify the given variables

We are given the initial temperature (T1 = 200 K), the final temperature (T2 = 400 K), and the constant pressure. We need to find the change in volume (ΔV).

Step 2: Use the ideal gas law with constant pressure

Since the pressure is constant, we can use the relation V1/T1 = V2/T2 for our calculations. We need to find the change in volume, which can be expressed as ΔV = V2 - V1.

Step 3: Solve for the change in volume

Rearrange the equation V1/T1 = V2/T2 to find the relationship between V2 and V1: V2 = V1 * (T2/T1). Then, solve for the change in volume: ΔV = V1 * (T2/T1 - 1).

Step 4: Substitute the given values and calculate ΔV

Now, substitute the given values of T1 and T2 into the equation: ΔV = V1 * (400 K/200 K - 1). Simplify the expression: ΔV = V1 * (2 - 1). Therefore, ΔV = V1. #Conclusion#The change in volume (ΔV) for the gas is equal to the initial volume of the gas (V1). Thus, the position of the piston does change, and the volume of the gas increases by an amount equal to its initial volume.

Key Concepts

Gas Volume Change

When discussing the concept of gas volume change, it's essential to refer to the behavior of gases under varying conditions. In the context of the Ideal Gas Law, the change in volume of a gas is influenced by temperature, pressure, and the number of moles of the gas.

For a gas contained within a piston as described in the original exercise, an increase in temperature while maintaining constant pressure will result in a change in volume. This is because as the temperature of the gas rises, the gas molecules move faster and need more space, causing the volume to expand. To quantify this change using the Ideal Gas Law, you would take the initial and final states of the gas and apply the relationship between pressure, volume, and temperature.

In our example, after increasing the temperature from 200 K to 400 K, while keeping pressure constant, the volume of the gas doubles, which means the piston will move. This change signifies the responsiveness of gas volume to temperature changes under constant pressure conditions.

Constant Pressure

Maintaining constant pressure is a crucial part of this exercise, especially when using the Ideal Gas Law. When pressure stays the same, we can focus on the relationship between temperature and volume for a given amount of gas.

The principle of constant pressure is visible in the Ideal Gas Law equation modified for volume and temperature:

• When the pressure is constant, the formula simplifies from \( PV = nRT \) to \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \).

This relationship highlights how volume changes proportionally with temperature, as long as pressure is not a variable in the equation.

In the exercise, this allows us to predict that doubling the temperature will double the volume, provided the amount of gas remains constant. The effect of constant pressure ensures that the piston in a closed system must move to accommodate the increased volume.

Temperature and Volume Relationship

The relationship between temperature and volume in gases is directly proportional, according to Charles's Law, which is a part of the broader Ideal Gas Law. It states that for a fixed amount of gas at a constant pressure, the volume is directly proportional to its temperature in Kelvin.

Mathematically, Charles's Law can be expressed as:

• \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \)

This equation shows that if you increase the temperature (\( T_2 > T_1 \)), the volume (\( V_2 \)) must increase, assuming pressure and the amount of gas remain unchanged.

In our scenario, raising the gas temperature from 200 K to 400 K results in the volume increasing by a factor of two. This expansion illustrates the volume's sensitivity to changes in temperature, and ultimately affects the position of the piston. Understanding this relationship is essential for solving problems where temperature and volume interactions are crucial, especially in thermodynamics and physical chemistry applications.