Question

A gas at ${27^ \circ }C$ and a pressure of $30atm$. is allowed to expand to atmospheric pressure and volume $1.5$ times larger. The final temperature of the gas is
A) $- {123^ \circ }C$
B) ${123^ \circ }C$
C) ${273^ \circ }C$
D) ${373^ \circ }C$

Answer 1

Atmospheric pressure is also called barometric pressure. The combined law of gases will be used here. It is t be noted that the combined law of gases is a combination of three gas laws- Charles Law, Boyle’s Law, and Gay-Lussac’s Law.

Complete step by step answer:
According to the combined law of gases, the ratio of the product of pressure and volume and the absolute temperature is always equal to a constant.
According to the combined law of gases, the relation between pressure, volume, and temperature can be written as:
$\dfrac{{PV}}{T} = k$
Here, $P$ is the pressure, $V$ is the volume, $T$ is the temperature, and $k$ is constant.
Given that the gas is allowed to expand and its volume increases, therefore it can be written that
$\dfrac{{{P_{_1}}{V_1}}}{{{T_1}}} = \dfrac{{{P_2}{V_2}}}{{{T_2}}}$ ​​
​$​\Rightarrow {T_2} = \dfrac{{{P_2}{V_2}{T_1}}}{{{P_1}{V_1}}}$
Here, ${P_1} = 30atm,{P_2} = 1atm$
Let ${V_1} = V,{V_2} = 1.5V$
$\Rightarrow {T_1} = {27^ \circ }C = 27 + 273 = {300^ \circ }K$
${T_2} = ?$
substituting these values, we get
$\Rightarrow {T_2} = \dfrac{{1 \times 1.5V \times 300}}{{V \times 30}}$
$\Rightarrow {T_2} = 150$
On simplification,
$\Rightarrow {T_2} = 150 - 273$
$\Rightarrow {T_2} = - 123K$

$\therefore$ The final temperature of the gas will be $- {123^ \circ }K$. Hence option (A) is the right answer.

Note:
It is to be noted that the temperature, pressure, and volume are dependent on each other. If the temperature and pressure are kept constant, then the volume of the gas is directly proportional to the number of molecules of the gas. If the temperature and volume remain constant, the change in pressure of the gas is directly proportional to the number of molecules present in the gas. It is to be taken care of that the combined gas law is not to be mixed with the ideal gas law. Though they both are used to determine the pressure, volume, or temperature at given conditions, but they are both different. This is because the ideal law of gases can be used to calculate the density, but the combined law of gases can not calculate the density.