Question

At what temperature will the volume of a gas be twice the volume at ${27^ \circ }{\text{ }}C$ at a given pressure.
A. ${327^ \circ }{\text{ }}C$
B. ${54^ \circ }{\text{ }}C$
C. ${127^ \circ }{\text{ }}C$
D. ${100^ \circ }{\text{ }}C$

Answer 1

Hint: In the given question we have to find the temperature of a gas when its temperature is changed. We have to use Charles’ Law to formulate an equation. Then by comparing both equations we will find our answer.

Complete step by step answer:
The pressure of the gas is constant. Let the initial volume of the gas be $V$. According to the question the gas is at a temperature of ${27^ \circ }{\text{ }}C$.Converting degree Celsius scale into Kelvin scale we get,
$K = C + 273$
where $K$ is in Kelvin and $C$ is in degree Celsius.
So, we get ${27^ \circ }{\text{ }}C$ in Kelvin as $\left( {27 + 273} \right) = 300{\text{ }}K$
According to Charles’ Law we get,
$V = kT$
where $V$ is the volume of the gas, $T$ is the temperature and $k$ is the constant of proportionality.

Now by substituting the values in Charles’ Law we get,
$V = 300k$
Arranging it we get,
$\dfrac{V}{{300}} = k - - - - - \left( 1 \right)$
Now, when the volume is doubled means the final volume is then, $2V$.
Let the temperature at which the volume becomes doble be $T$.
Again, by substituting the values in Charles’ Law we get,
$2V = Tk$
Hence, arranging the equation we get,
$\dfrac{{2V}}{T} = k - - - - - - \left( 2 \right)$

Comparing equation $\left( 1 \right)$ and equation $\left( 2 \right)$ we get,
$\dfrac{V}{{300}} = \dfrac{{2V}}{T}$
By cross-multiplication we get,
$T = 600$
Thus, the temperature is found to be $600{\text{ }}K$.
Now as the options are in degree-Celsius we have to convert the value from Kelvin to degree-Celsius. Converting Kelvin scale into degree-Celsius scale we get,
$C = K - 273$
where $K$ is in Kelvin and $C$ is in degree Celsius.
$C = 600 - 273 \\
\therefore C= 327$
So, the final temperature is ${327^ \circ }{\text{ }}C$.

Hence the correct option is A.

Note: It must be noted that Charles’ Law is only applicable when the pressure of the given gas is constant. When the temperature of a gas is constant, we can follow Boyle’s Law and in case the volume is constant then we will use Gay-Lussac’s Law. It is very important that the mass of the gas in all cases must be constant also.