Problem 33 - College Physics

Question

The pressure on a low-density gas in a cylinder is kept constant as its temperature is increased. (a) Does (2) decrease, or the volume of the gas (1) increase, (3) remain the same? Why? (b) If the temperature is increased from \(10^{\circ} \mathrm{C}\) to \(40{ }^{\circ} \mathrm{C},\) what is the percentage change in the volume of the gas?

Step-by-Step Solution

Answer

(a) The volume increases. (b) The percentage change is approximately 10.6%.

Step 1: Understanding the Relationship between Pressure, Volume, and Temperature

Given that the pressure is constant, we can apply Charles's Law, which states that the volume of a gas is directly proportional to its temperature (in Kelvin) when pressure is held constant. The formula is \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \), where \(V\) represents volume and \(T\) represents temperature in Kelvin.

Step 2: Checking the Effects of Increasing Temperature

As the temperature increases, based on Charles's Law, the volume must also increase to maintain the proportional relationship under constant pressure. Therefore, the volume of the gas increases.

Step 3: Converting Celsius to Kelvin

Convert the temperatures from Celsius to Kelvin by adding 273.15. Thus, \(10^{\circ}C\) becomes \(283.15\, K\), and \(40^{\circ}C\) becomes \(313.15\, K\).

Step 4: Applying Charles's Law to Calculate Volume Change

Using Charles's Law \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \), rearrange to find \(V_2\) (the final volume): \( V_2 = V_1 \times \frac{T_2}{T_1} \). Since \(V_2\) depends directly on the temperatures, the initial volume \(V_1\) cancels out for the percentage change calculation.

Step 5: Calculating the Percentage Change

Substitute values into the equation: \( \frac{V_2}{V_1} = \frac{313.15}{283.15} = 1.106 \). The percentage change in volume is given by \((1.106 - 1) \times 100\%\), which calculates to approximately \(10.6\%\).

Key Concepts

Gas Volume

When we talk about gas volume, we refer to the space that a gas occupies. Gases have the particular property of being compressible and expandable due to the large spaces between their particles. This means their volume can change easily with changes in temperature and pressure.
One important principle related to gas volume is Charles's Law. It tells us there is a direct relationship between the volume of a gas and its temperature when the pressure remains constant. If the temperature goes up, the gas particles move faster and spread out, increasing the volume. If the temperature decreases, they slow down and come closer together, reducing the volume.
Understanding how a gas's volume changes helps in various applications like weather forecasting and even cooking. For instance, in a hot air balloon, as the air inside the balloon is heated, it expands, causing the balloon to rise.

Temperature Conversion

Sometimes, temperatures are given in Celsius, but for calculations involving gas laws, we need them in Kelvin. The Kelvin scale is preferred because it is an absolute temperature scale where 0 K corresponds to absolute zero, the point where particles theoretically have no kinetic energy.
To convert from Celsius to Kelvin, a simple step is needed: just add 273.15 to the Celsius temperature. For example:

• For 10°C, the conversion to Kelvin is 10 + 273.15 = 283.15 K.
• For 40°C, it becomes 40 + 273.15 = 313.15 K.

This conversion is essential for applying gas law equations correctly, as they require temperatures to be in Kelvin to ensure that the proportionality remains valid.

Kelvin Scale

The Kelvin scale is vital for understanding and working with gas laws, specifically Charles's Law. Unlike the Celsius scale, the Kelvin scale does not have negative numbers, which aligns perfectly with the concept of absolute zero.
Absolute zero, 0 K, is theorized to be the lowest possible temperature where particles have minimal vibrational motion. This makes Kelvin a natural choice for scientific calculations involving temperature because it provides a true ratio between temperatures concerning motion and energy.
In any equation derived from Charles's Law, ensuring that temperature is in Kelvin ensures accurate proportionality between the variables, which is crucial for maintaining consistent results. Thus, Kelvin is a preferred scale in physics and chemistry when dealing with gas behaviors.

Ideal Gas Laws

The ideal gas laws are a series of relationships that describe the behaviors of an ideal gas. These laws include:

• Boyle's Law (pressure-volume relationship),
• Charles's Law (volume-temperature relationship), and
• Avogadro's Law (volume-amount relationship).

Charles's Law specifically tells us that at constant pressure, the volume of a gas is directly proportional to its temperature in Kelvin.
These laws are combined in the Ideal Gas Law formula: \[ PV = nRT \]where

• \( P \) is the pressure,
• \( V \) is the volume,
• \( n \) is the number of moles of gas,
• \( R \) is the ideal gas constant, and
• \( T \) is the temperature in Kelvin.

Understanding these relationships helps in many real-world applications and experiments, providing predictions for how gases will behave under varying conditions.