At what Kelvin temperature will \(25.2 \mathrm{~mol}\) of Xe occupy a volume of \(645 \mathrm{~L}\) at a pressure of 732 torr?

In chemistry, temperature is often measured in Kelvin instead of Celsius or Fahrenheit. The Kelvin scale starts at absolute zero. When doing calculations with the Ideal Gas Law, it's essential to use Kelvin. This scale helps make the math simpler and more accurate.

To convert Celsius to Kelvin, simply add 273.15 to the Celsius temperature. For example, if a temperature is 25°C, it would be 298.15 K when converted. Similarly, if it were -10°C, it would be 263.15 K in Kelvin.

Understanding and accurately using the Kelvin scale is crucial, especially in gas law calculations where temperature must be in absolute units.

Pressure in gas law problems is often given in various units like torr, atmospheres (atm), or Pascals (Pa). To make calculations consistent and correct, it is vital to convert these pressures into a common unit, usually atmospheres.

For example, in our problem, the pressure is given as 732 torr. To convert to atmospheres, use the conversion factor: 1 atm = 760 torr. Therefore, \( P = \frac{732 \text{ torr}}{760 \text{ torr/atm}} = 0.963 \text{ atm} \).

This conversion ensures you can accurately use the Ideal Gas Law, which typically uses atmospheres for pressure.

The amount of gas (in moles) and its volume have a direct relationship, as described by the Ideal Gas Law (PV = nRT). The term 'mole' refers to a specific quantity of molecules. One mole contains Avogadro's number of molecules (approximately 6.022 x 10^23).

In our exercise, we're given that 25.2 moles (n) of Xe gas occupies a volume of 645 L. This relationship is crucial because we use this value of moles and volume along with the other variables in the Ideal Gas Law equation.

Remember: More moles of gas will generally result in a higher volume if the pressure and temperature are kept constant, highlighting the directly proportional relationship.

The Ideal Gas Law, represented by the equation \( PV = nRT \), is fundamental for understanding the behavior of gases. This equation combines the physical properties of pressure (P), volume (V), amount of gas in moles (n), and temperature (T) using the Ideal Gas Constant (R).

To find an unknown, rearrange the equation. In our example, we're solving for temperature (T). So, rearrange to \( T = \frac{PV}{nR} \).

Insert the known values: \( P = 0.963 \text{ atm}, V = 645 \text{ L}, n = 25.2 \text{ mol}, R = 0.0821 \text{ L atm K}^{-1} \text{ mol}^{-1} \). This results in \( T = \frac{(0.963 \text{ atm}) (645 \text{ L})}{(25.2 \text{ mol}) (0.0821 \text{ L atm K}^{-1} \text{ mol}^{-1})} = 300.4 \text{ K} \).

Using the Ideal Gas Law helps predict how changing one variable affects the others, making it an essential tool in chemistry.