Problem 48 - Physics

Question

A frictionless gas-filled cylinder is fitted with a movable piston, as the drawing shows. The block resting on the top of the piston determines the constant pressure that the gas has. The height \(h\) is \(0.120 \mathrm{~m}\) when the temperature is \(273 \mathrm{~K}\) and increases as the temperature increases. What is the value of \(h\) when the temperature reaches \(318 \mathrm{~K}\) ?

Step-by-Step Solution

Answer

The height \( h \) is approximately \( 0.140 \ m \) when the temperature is \( 318 \ K \).

Step 1: Understanding the Problem

We are given a gas-filled cylinder with a piston, and the height \( h \) of the gas is \( 0.120 \ m \) when the temperature is \( 273 \ K \). We need to find the new height when the temperature is \( 318 \ K \), assuming the pressure is constant.

Step 2: Applying Charles's Law

Charles's Law states that the volume of a gas is directly proportional to its temperature at constant pressure. Therefore, \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \), where \( V_1 \) and \( V_2 \) are the initial and final volumes, and \( T_1 \) and \( T_2 \) are the initial and final temperatures, respectively.

Step 3: Expressing Volume in Terms of Height

Since the volume \( V \) of the gas is proportional to its height \( h \) in the cylinder, we can write \( V_1 = k \cdot h_1 \) and \( V_2 = k \cdot h_2 \), where \( k \) is a proportionality constant. Therefore, \( \frac{h_1}{T_1} = \frac{h_2}{T_2} \).

Step 4: Calculating the New Height

We know \( h_1 = 0.120 \ m \), \( T_1 = 273 \ K \), and \( T_2 = 318 \ K \). Substitute these values into the equation: \( \frac{0.120}{273} = \frac{h_2}{318} \). Solve for \( h_2 \) by cross-multiplying and dividing: \[h_2 = \frac{0.120 \times 318}{273} \approx 0.140 \ m.\]

Step 5: Verifying the Solution

Recheck the calculations to ensure they are correct and consistent with Charles's Law. The calculation confirms the result: when the temperature increases to \( 318 \ K \), the height of the piston increases to approximately \( 0.140 \ m \).

Key Concepts

Proportionality Constant

A proportionality constant is a fixed value that relates two variables in a proportional relationship. In our example of a gas-filled cylinder, this constant helps relate the volume of the gas to the height of the piston. Think of it like a "glue" that holds two related quantities together.

When the gas is under constant pressure inside the cylinder, the relationship between the volume and the height stays consistent. This is where the proportionality constant, denoted as "\( k \)", becomes crucial. If any changes occur in one variable, the constant allows us to understand how the other variable will change.

In Charles's Law, the volume of a gas is directly proportional to the temperature. Therefore, we can say \( V = k \cdot T \) or, if we're considering height instead, \( h = k \cdot T \). The proportionality constant "\( k \)" remains unchanged as long as the external conditions, such as pressure, stay the same.

Gas Laws

Gas laws are scientific principles that describe the behavior of gases. They connect various properties such as pressure, volume, temperature, and the quantity of gas. Charles's Law is one of the most fundamental gas laws.

Charles's Law specifically focuses on the relationship between volume and temperature while keeping pressure constant. As per Charles's Law, the volume of a gas increases as the temperature increases, assuming the pressure and the amount of gas are constant. This can be represented mathematically as \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \).

These crucial gas laws are not only vital in theoretical physics but also in practical applications. For instance, understanding how gas laws function helps in predicting how gases will behave under different conditions, making them essential in chemical processes and engineering applications.

Temperature-Volume Relationship

The temperature-volume relationship described by Charles's Law highlights a key characteristic of gases: they expand when heated. This happens because as the temperature rises, the gas particles gain more energy, causing them to move faster and exert more force on the walls of their container.

In our exercise, we observe this relationship as we see the height of the gas-filled cylinder increase from 0.120 m to 0.140 m when the temperature rises from 273 K to 318 K. By keeping the pressure constant, the change in height directly reflects the expansion of the gas due to temperature changes.

For students learning about gas laws, it is intuitive to think of the volume of any contained gas as adjustable based on temperature. The importance of this concept is shown experimentally, and it helps explain many phenomena, such as why balloons expand more in the warmth.