A sample of a gas occupies a volume of \(1025 \mathrm{~mL}\) at \(75^{\circ} \mathrm{C}\) and \(0.75 \mathrm{~atm}\). What will be the new volume if temperature decreases to \(35^{\circ} \mathrm{C}\) and pressure increases to \(1.25\) atm?

The Ideal Gas Law is a fundamental principle in chemistry that relates the pressure, volume, temperature, and amount of gas present. It is expressed by the equation:
\( PV = nRT \)
Where:

• \(P\) stands for pressure.
• \(V\) represents volume.
• \(n\) is the number of moles of gas.
• \(R\) is the universal gas constant (approximately 0.0821 atm·L/(mol·K)).
• \(T\) represents temperature in Kelvin.

This equation assumes ideal conditions where gas molecules do not interact with each other, and the volume occupied by the gas molecules themselves is negligible compared to the container. Real gases approximate ideal behavior under many conditions, but the ideal gas law provides a good starting point for understanding gas behavior.
In practical use, solving problems with the Ideal Gas Law often involves rearranging the equation to solve for one of the variables if the other three are known. For example, if you know the pressure, volume, and temperature, you can find the number of moles of gas by rearranging the equation:
\( n = \frac{PV}{RT} \).

The Combined Gas Law provides a way to predict how a change in one property of a gas (pressure, volume, or temperature) affects the other properties when the amount of gas remains constant. The formula is:
\( \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \)
Where:

• \(P_1\) and \(P_2\) represent the initial and final pressures
• \(V_1\) and \(V_2\) represent the initial and final volumes
• \(T_1\) and \(T_2\) represent the initial and final temperatures in Kelvin

This equation comes in handy when dealing with scenarios where both the temperature and pressure of the gas are changing, as seen in our example problem.
To use the Combined Gas Law effectively:
1. **Convert temperatures to Kelvin:** Always convert Celsius to Kelvin before using the equation.
2. **Plug in known values:** Substitute the given values into the equation.
3. **Solve for the unknown:** Rearrange the equation to solve for the unknown variable.
By applying these steps meticulously, you can find the new state of the gas, such as the new volume when temperature decreases and pressure increases.

Understanding temperature conversion is essential when using gas laws. The Kelvin scale is the standard unit for temperature in gas law equations because it starts at absolute zero, where particle motion ceases. To convert Celsius to Kelvin, you use the formula:
\( T(K) = T(°C) + 273.15 \).
For example:

• If you have a temperature of \(75^\text{°C}\), it converts to Kelvin as follows:
  \( 75 + 273.15 = 348.15 \text{ K} \).
• Similarly, a temperature of \(35^\text{°C}\) converts to Kelvin as:
  \( 35 + 273.15 = 308.15 \text{ K} \).

Kelvin conversion is crucial because gas laws rely on temperature values that reflect the relative thermal energy of the particles accurately.
Whenever solving gas law problems, ensure that all temperature values are in Kelvin. Neglecting to convert temperatures can lead to incorrect results, as seen in gas laws where temperature directly affects the pressure and volume calculations.