Question

A vessel contains 1 mole of $O_2$ (molar mass 32 gm) at a temperature T. The pressure is P. An identical vessel containing 1 mole of He (molar mass 4 gm) at a temperature 2T has a pressure:
$\begin{align}
  & \text{A}\text{. P} \\
 & \text{B}\text{.}\dfrac{\text{P}}{\text{8}} \\
 & \text{C}\text{. 2P} \\
 & \text{D}\text{. 8P} \\
\end{align}$

Answer 1

Hint: The given gases can be assumed as an ideal gas and apply the formula $PV=nRT$, for each gas. Then divide both equations. Then using the given information get the required ratio (for example the two vessels are equal so the volume will be same)

Complete step by step answer:
For an ideal gas, the volume of the gas is inversely proportional to the pressure applied on it. This is called the Boyle’s law. The pressure is directly proportional to the temperature of the gas and also on the amount of gas present. The volume of the gas is also proportional to temperature. It is called Charles’ law. It is also dependent on the amount of gas present.
These are related through the ideal gas equation given by,
$PV=nRT$
where, P is the pressure, V is the volume, n is the amount of gas (moles) present, R is the universal gas constant and T is the temperature of the gas. For two gases, the properties can be compared as,
$\dfrac{{{P}_{1}}{{V}_{1}}}{{{n}_{1}}{{T}_{1}}}=\dfrac{{{P}_{2}}{{V}_{2}}}{{{n}_{2}}{{T}_{2}}}$
where subscript 1 is for $O_2$ and 2 for He.
In the given question, both the gases are in identical vessels, and therefore occupy the same volume, hence $V_1 = V_2$. Thus, the process is isochoric. Also, since they are both in 1 mole, $n_1 = n_2 = 1$. Also, $T_1 = T$, $T_2 = 2T$and $P_1 = P$. Therefore, we may write the above equations as,
\[\dfrac{P}{T}=\dfrac{{{P}_{2}}}{2T}\]
\[{{P}_{2}}=2P\]

Therefore, the correct option is C.

Note:
Under the assumption of ideal gas, the molecules of different gases are considered identical and occupy negligible amounts of space. What matters is the number of molecules, and so the weight of the molecule is insignificant. The ideal gas equation holds true only for gases at low pressure and high temperature such that the gas molecules are far apart from each other.