91Ó°ÊÓ

Gas laws are a set of relationships that describe how gases behave under various conditions of pressure, volume, and temperature. One of the key equations combining these variables is the Combined Gas Law. The Combined Gas Law is particularly useful when dealing with a gas under two different sets of conditions. The expression for the Combined Gas Law is:\[ \frac{P_1 \times V_1}{T_1} = \frac{P_2 \times V_2}{T_2} \] In this equation, \(P\) represents the pressure, \(V\) is the volume, and \(T\) is the temperature measured in Kelvin. This law arises from a combination of Boyle's Law, Charles's Law, and Gay-Lussac's Law, which account for the interdependence of pressure, volume, and temperature.

• Boyle's Law: It states that the pressure of a gas is inversely proportional to its volume at constant temperature.
• Charles's Law: It states that the volume of a gas is directly proportional to its temperature at constant pressure.
• Gay-Lussac's Law: It states that the pressure of a gas is directly proportional to its temperature at constant volume.

These relationships allow us to predict how a gas will respond under changing conditions of pressure, volume, and temperature and answer practical problems such as predicting the volume of a balloon at different altitudes.

Temperature conversion is essential when working with gas laws, as gas laws require temperatures to be expressed in Kelvin. Celsius can be converted to Kelvin using the formula:\[ T(K) = T(°C) + 273.15 \]Kelvin is the SI unit of temperature and starts at absolute zero, the point at which molecular motion ceases. Because many scientific calculations, like those involving gas laws, rely on an absolute scale, Kelvin is preferred over Celsius.To illustrate:

• Convert an initial temperature of \(14^\circ C\): \[ T_1 = 14 + 273.15 = 287.15 \ K \]
• Convert a final temperature of \(20^\circ C\): \[ T_2 = 20 + 273.15 = 293.15 \ K \]

Converting temperatures from Celsius to Kelvin ensures that we accommodate the concept of absolute values in our calculations, thus preserving the mathematical integrity of gas law equations.

Calculating the volume of a gas under varying conditions is a practical application of the Combined Gas Law. This involves solving for the new volume \(V_2\) given the changes in pressure and temperature.Starting with the Combined Gas Law:\[ \frac{P_1 \times V_1}{T_1} = \frac{P_2 \times V_2}{T_2} \]To find \(V_2\), rearrange to:\[ V_2 = \frac{P_1 \times V_1 \times T_2}{P_2 \times T_1} \]

• Step 1: Insert the initial conditions: \(P_1 = 100.0 \ kPa\), \(V_1 = 5.0 \ dm^3\), \(T_1 = 287.15 \ K\).
• Step 2: Insert the final conditions: \(P_2 = 79.0 \ kPa\), \(T_2 = 293.15 \ K\).
• Step 3: Solve for \(V_2\): \[ V_2 = \frac{100.0 \times 5.0 \times 293.15}{79.0 \times 287.15} \]
• Result: \[ V_2 \approx 6.398 \ dm^3 \]

The calculation reveals the final volume of the balloon's gas at a new altitude and temperature. This example demonstrates how understanding pressure, volume, and temperature interrelations allows us to predict changes in a gas’s behavior.