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The pressure, volume, and temperature of a gas are interconnected in a fascinating way. This relationship is explained by the ideal gas law, which is a core concept in physics and chemistry. The law is typically expressed in the formula: \( PV = nRT \), where:
- \( P \) is the pressure of the gas,
- \( V \) is the volume,
- \( n \) is the number of moles of the gas,
- \( R \) is the gas constant,
- \( T \) is the temperature in Kelvin.

In a closed system, where the amount of gas doesn't change, the ideal gas law can also be modified to compare two states of the same gas. It becomes: \( \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \). This equation helps us predict how changes in one condition (such as temperature or volume) will affect the others, assuming the amount of gas remains constant.
Understanding this relationship is vital in solving problems like the one presented in the exercise, where we want to find the new pressure after altering both temperature and volume while maintaining the amount of gas.

Temperature is a crucial factor in gas behavior, but it’s vital to use the correct scale for calculations. The Kelvin scale is the standard unit used in gas equations because it starts at absolute zero (the point where particles have minimal vibrational motion), ensuring that all calculations remain mathematically consistent.

To convert Celsius to Kelvin, you simply add 273.15 to the Celsius temperature. This is crucial because the Celsius scale doesn't align with the principles of thermodynamics used in calculations like those in gas laws:

• Initial temperature: \( 27^{\circ} \text{C} \) converts to \( 300.15 \text{ K} \)


• Final temperature: \( 157^{\circ} \text{C} \) converts to \( 430.15 \text{ K} \)

Using Kelvin ensures that our calculations within the gas laws reflect the true thermodynamic temperatures, allowing for accurate predictions of gas behavior.

Gas laws collectively describe how gases behave under different conditions of pressure, volume, and temperature. The ideal gas law is a combination of several simpler laws that each describe how gases respond to a change in one variable while keeping others constant:

• Boyle’s Law: Describes the inverse relationship between volume and pressure. As volume decreases, pressure increases (assuming constant temperature).


• Charles’ Law: Explains that volume and temperature are directly proportional. As temperature increases, so does volume, if pressure remains constant.


• Gay-Lussac’s Law: Shows that pressure and temperature are directly proportional. Increasing the temperature increases pressure, if volume remains constant.

The ideal gas law synthesizes these observations into a single expression, useful for analyzing real-world scenarios where more than one variable changes simultaneously. In the exercise provided, understanding these gas laws allows you to track how the compressed gas reacts to both changes in volume and increased temperature, resulting in a distinct change in pressure.