Understanding the Relationship Between Temperature and Gas Volume: Applying Charles' Law

Charles' Law is a fundamental concept in thermodynamics that helps us understand how the volume of a gas is directly proportional to its temperature, provided that the pressure remains constant. This law is crucial for various scientific and industrial applications, from everyday scenarios to advanced research. Let's delve into the application of Charles' Law to determine the increase in temperature necessary for a half-liter of gas to expand by 40 milliliters at a constant pressure.

Application of Charles' Law to Determine Temperature Increase

Charles' Law is mathematically represented by the formula:

(frac{V_1}{T_1} frac{V_2}{T_2})

Where:

V1 is the initial volume of the gas, T1 is the initial temperature in Kelvin, V2 is the final volume of the gas, T2 is the final temperature in Kelvin.

Given the scenario, we have:

Initial volume V1 0.5 L, Final volume increase V2 0.5 L 0.4 L 0.9 L (40 mL).

The initial temperature in Celsius is 25°C, which needs to be converted to Kelvin:

T1 25°C 273.15 298.15 K.

Now, let's apply the formula to find the final temperature T2:

(frac{0.5}{298.15} frac{0.9}{T_2})

By rearranging to solve for T2:

T2 (frac{0.9 times 298.15}{0.5})

Calculating T2:

T2 (frac{0.9 times 298.15}{0.5} frac{268.335}{0.5} 536.67 K

Therefore, the final temperature is approximately 536.67 K.

To find the temperature increase:

(Delta T T2 - T1 536.67 K - 298.15 K 238.52 K.

Conceptual Approach to Understanding the Temperature Increase

A conceptual approach to understanding this problem involves recognizing that when the pressure and the number of moles of a gas are constant, the volume is directly proportional to its Kelvin temperature. Given that a 40 mL increase in volume corresponds to a proportional increase in temperature:

(V_2 1.4 V_1)

Therefore, the final temperature must be:

(T_2 1.4 T_1 1.4 times 298 K 417.2 K)

Thus, the temperature increase is approximately 119.26 K, and the final temperature is 417.2 K.

Key Takeaways

Charles' Law ((frac{V_1}{T_1} frac{V_2}{T_2})) is useful for understanding the relationship between temperature and volume of a gas at constant pressure. The initial temperature must always be converted to Kelvin for accurate calculations. Temperature increases can be calculated to achieve specific volume changes, which is crucial in various scientific and industrial applications.

Understanding these principles ensures that we can effectively apply Charles' Law in practical scenarios, leading to a deeper comprehension of the behavior of gases under different conditions.