91Ó°ÊÓ

A fundamental concept in gas laws is proportionality, which describes the relationship between two quantities where one changes uniformly with respect to the other. In the context of gases, this often involves variables like volume and temperature.
The problem highlights a specific case of proportionality: the volume of a gas is directly proportional to its temperature in Kelvin. This means if you increase the temperature, the volume will increase, assuming the pressure remains constant.
Mathematically, this relationship is represented by the formula:

• \( V = kT \)
  • \( V \) stands for volume of the gas.
  • \( T \) is the temperature in Kelvin.
  • \( k \) is a constant specific to the given conditions of the gas.

To solve changes in volume or temperature using this principle, we first need to determine the constant \( k \) using known conditions, then apply the formula to new conditions.

In physics and chemistry, temperature conversion is crucial because different scales are used in different applications. For gas law problems, it's important to work in Kelvin, the absolute temperature scale.
The Kelvin scale is directly used in the formula \( V = kT \) because it starts at absolute zero, eliminating negative temperature values, which is essential for calculating gas volumes proportionally. Conversions between Celsius and Kelvin are simple:

• To convert Celsius to Kelvin, add 273.15: \( T_{K} = T_{°C} + 273.15 \)
• To convert Kelvin to Celsius, subtract 273.15: \( T_{°C} = T_{K} - 273.15 \)

In this gas problem, the temperatures given and calculated (280 K, 330 K, and 180 K) are already in Kelvin. This allows you to apply the proportionality formula without additional conversion steps, simplifying the calculations.

Volume calculation involves determining the amount of space occupied by a gas at different conditions of temperature while maintaining proportionality. Volume is an important parameter in gas laws as it helps to predict the behavior of gases under varying temperatures.
In our problem:

• We start with an initial volume of \( 3.50 \, \text{L} \) at a temperature of \( 280 \, \text{K} \).
• Using the formula \( V = kT \), and the calculated constant \( k = 0.0125 \, \text{L/K} \), we can find new volumes at different temperatures.

For a new temperature of \( 330 \, \text{K} \):

• Apply \( V = 0.0125 \, \times \, 330 \). This calculates to a volume of \( 4.125 \, \text{L} \).

If the volume changes to \( 2.25 \, \text{L} \), and according to \( V = kT \), solving for \( T \) gives:

• \( 2.25 \, \text{L} = 0.0125 \, T \)
  Solve for \( T \): \( T = \frac{2.25}{0.0125} = 180 \, \text{K} \).

These calculations show how using direct proportionality and a known constant, you can find the new state variables for gases efficiently.