Understanding Boyle’s Law and Charles’s Law in Ideal Gas Behavior: A Detailed Solution

When dealing with the behavior of gases, Boyle's Law and Charles's Law play crucial roles in understanding the relationships between pressure, temperature, and volume. In this article, we will explore how to solve a specific problem involving Charles’s Law to calculate the new pressure of a gas when the temperature is increased, while the volume remains constant.

Principles and Formulas

To begin with, let's revisit the key principles and formulas involved:

Boyle's Law: PV constant (if temperature stays constant) Charles's Law: (frac{P}{T} text{constant}) (if volume stays constant) Ideal Gas Law: (PV nRT) (full equation accounting for all variables)

Given the problem: An ideal gas has a pressure of 4.0 atm at 550°C. What is the pressure when the temperature is increased to 800°C?

Solution to the Problem

To solve this problem, we will use Charles's Law, which involves the relationship between pressure and temperature when volume is held constant. The formula for Charles's Law is:

Charles's Law: (frac{P_1}{T_1} frac{P_2}{T_2})

Where:

(P_1): Initial pressure (4.0 atm) (T_1): Initial temperature in Kelvin (550°C) (P_2): Final pressure (unknown, what we need to find) (T_2): Final temperature in Kelvin (800°C)

Step 1: Convert Temperatures to Kelvin

Remember that temperatures need to be in Kelvin (K) for the calculations. The conversion formula is:

Temperature in Kelvin (K) Temperature in Celsius (°C) 273.15

Calculating the temperatures in Kelvin:

(T_1 550 273.15 823.15 , text{K}) (T_2 800 273.15 1073.15 , text{K})

Step 2: Apply Charles's Law

Now, using the formula (frac{P_1}{T_1} frac{P_2}{T_2}), we can solve for (P_2):

(P_2 frac{P_1 times T_2}{T_1})

Substituting the values:

(P_2 frac{4.0 , text{atm} times 1073.15 , text{K}}{823.15 , text{K}})

(P_2 frac{4.0 times 1073.15}{823.15})

(P_2 approx 5.2 , text{atm})

Conclusion

Therefore, the pressure of the gas increases to approximately 5.2 atm when the temperature is increased from 550°C to 800°C, assuming the volume of the sample gas remains constant.

Further Insights

Understanding these principles is essential for solving problems in chemistry, physics, and engineering. Here are a few additional points to consider:

Boyle’s Law: This law is useful for calculating the pressure of a gas when the volume changes at a constant temperature. Charles’s Law: This law helps determine the effect of changes in temperature on the volume of a gas at constant pressure. Ideal Gas Law: The full equation (PV nRT) can be utilized to calculate any variable if the others are known.

Conclusion

In summary, Charles’s Law provides a straightforward way to solve problems involving changes in pressure and temperature when the volume remains constant. Applying the formula (frac{P_1}{T_1} frac{P_2}{T_2}) and converting temperatures to Kelvin is crucial for obtaining accurate results. For a thorough understanding of gas behavior, it is essential to familiarize oneself with both Boyle’s and Charles’s Laws.