Question

An open vessel at 300K is heated till $\dfrac{2}{5}$ of the air is expelled. Assuming that the volume of the vessel remains constant, the temperature to which the vessel is heated is:
(a) 750K
(b) 400K
(c) 500K
(d) 1500K

Answer 1

Hint: The parameters given in the question are pressure, volume and the gas constant. Since they remain constant we have to relate the number of moles and temperature using the ideal gas equation.

Complete step by step answer:
Given in the question,
Temperature = $300 \ K$
Two fifth of the air escaped. If air is escaped it means that there is change in the number of moles of gas. So let’s assume the initial number of moles of gases be n1 and the final number of moles of gases be n2.
Let us assume that $n1 = 1$ mole
So the final number of moles of gases $n2$ will be = $n1-\dfrac{2}{5}=1-\dfrac{2}{5}$
Final number of moles of gases $n2$ = $\dfrac{3}{5}$ moles
As per the question, the volume remains constant
According to the ideal gas equation, $PV = nRT$; where P is pressure, R is the ideal gas constant, V is the volume of the vessel.
We can relate the number of moles with temperature, $nT$ = constant.
${{n}_{1}}{{T}_{1}}={{n}_{2}}{{T}_{2}}$ = constant
Now put the value of number of moles and temperature in the above equation
\[\left( 1 \right) \left( 300 \right) =\dfrac{3}{5}{{T}_{2}}\]
So,
\[{{T}_{2}}=\dfrac{300X5}{3} \ K\]
\[{{T}_{2}}=500 \ K\]
Hence the correct option is (C).
The temperature to which the vessel is heated is 500K.

Note: 1 mole of any gas at the standard temperature and pressure condition it occupies a volume of 22.4L. The ideal gas equation is the equation of state of an ideal gas. This law tells about the approximation of the behavior of gases under an ideal condition. Ideal gas law is a combination of empirical laws such as Boyle’s law, Charles law and avogadro's law.