Question

An open flask contains air at 27. To what temperature it must be heated to expel one-fourth of the air?

Answer 1

Hint: When gas heated the intermolecular attraction forces of the gas molecules decreases. as a result, the volume of the gas increases. For the ideal gas, the attraction forces of the gas molecules are zero. When the temperature of the gas increases the molecules of the gas become excited and volume will be increased.

Formula used: \[{n_1}{T_2} = {n_2}{T_1}\]

Complete step by step answer:
Let's consider this is an ideal gas. Therefore, the ideal gas equation is. pV=nRT.where, p is the pressure, V is the volume, R is the gas constant, n is the number of the mole, T is the temperature.
 Now at constant temperature and constant pressure. When the temperature increases the number of mole of the gas decreases and vice-versa. The formula is
 \[{n_1}{T_2} = {n_2}{T_1}\]
Where, \[{n_1}\] and \[{n_2}\] are the number of moles, \[{T_1}\] and \[{T_2}\] are temperatures.
Therefore, when the air in the open flask is heated the temperature increases, and when one-fourth of the air will be expelled, the remaining number of the mole is,
 \[
  n = 1 - \dfrac{1}{4} \\
  n = \dfrac{3}{4} \\
 \]
Now the temperature would be,
 \[
  \dfrac{{{T_2}}}{{{T_1}}} = \dfrac{{{n_1}}}{{{n_2}}} \\
  \dfrac{{{T_2}}}{{300}} = \dfrac{1}{{\dfrac{3}{4}}} \\
  {T_2} = \dfrac{4}{3} \times 300 \\
  {T_2} = 400 \\
 \]
So, the temperature would be 400k or \[{127^0}C\] .

Addition information:
Let's discuss some terms which indicate the deviation from the ideal gas.
1.Van der Waals constant: The Van der Waals equation of state is a modified version of the ideal gas law, which takes into consideration the interactive forces of the molecules and the size of the molecules. This equation can be represented as:
  \[[P + a{(\dfrac{n}{v})^2}](\dfrac{V}{n} - b)\, = \,RT\] ; where a and b are constants.
Here the constant value ‘a’ is positive and corrects the intermolecular forces.
2.Boyle Point: Boyle point is the temperature at which a real gas begins to exhibit the properties of an ideal gas. It is a characteristic property and is dependent on the nature of the real gas. And that temperature is known as Boyles temperature.
3.Compressibility factor: Compressibility factor or compression factor can be explained as the ratio of molar volume of an ideal gas to the molar volume of a real gas. In simpler terms, it measures how far-off are the values from ideal gases to real gases.
4.Critical temperature: Critical temperature can be explained as the temperature above which a gas cannot be liquefied. This means that it is the highest at which the given gas can be liquefied.

Note:Vander Waals equation of state for real gases is obtained from the modification of ideal gas equation or law. According to an ideal gas equation, pV=nRT where P is the pressure, V is the volume, T is the temperature, R is the universal gas constant and n is the number of moles of an ideal gas. The Vander Waals equation considers the molecular interaction forces i.e. both the attractive and repulsive forces and the molecular size. There comes a volume correction in the Vander Waal equation. As the particles have a definite volume, the volume available for the movement of molecules is not the entire volume of the container but less than that.