Question

According to Charles' Law, the volume of a gas increases or decreases by ____ of its original volume at $0°\mathrm{C}$ for each degree centigrade rise or ____

• A. $\frac{1}{273}$
• B. fall in temperature

Answer 1

Step 1: State Charles law

It states that “At constant pressure, the volume of a fixed mass of gas is directly proportional to absolute temperature”.

$$\begin{gathered} V\propto T \\ V=kT \end{gathered}$$

Where $V$ is the volume of the gas, $T$ is the temperature and $k$ is a constant.

Step 2: Express Charle's law in relational form

At a particular temperature $T_1$, the volume is $V_1$. By the above law,
$V_1=k{T}_1$ $...\left(1\right)$

Similarly, at another temperature $T_2$, the volume is $V_2$ and,
$V_2=k{T}_2$ $...\left(2\right)$

Dividing $\left(1\right)$ by $\left(2\right)$,

$\frac{V_1}{V_2}=\frac{T_1}{T_2}$

Step 3: Consider specific values for the two temperatures

Let $T_1=0°\mathrm{C}=273\mathrm{K}$ and $T_2=100°\mathrm{C}=373\mathrm{K}$. The temperatures are chosen arbitrarily. Then,

$\begin{array}{cc}& \frac{V_1}{V_2}=\frac{373}{273}\\ \Rightarrow & V_1=V_2\left(1+\frac{100}{273}\right)\\ \Rightarrow & V_1=V_2+\frac{100V_2}{273}\end{array}$

We see that for every $100°\mathrm{C}$ rise in temperature, volume increases by a factor of $\frac{100}{273}$.

So for every $1°\mathrm{C}$ rise in rise in temperature, volume increases by a factor of $\frac{{\displaystyle \raisebox{1ex}{$100$}\!\left/ \!\raisebox{-1ex}{$273$}\right.}}{100}=\frac{1}{273}$.

Similarly, it can be shown that for every $1°\mathrm{C}$ fall in rise in temperature, volume decreases by a factor of $\frac{1}{273}$.

Therefore, the volume of a gas increases or decreases by $\frac{\mathbf{1}}{\mathbf{273}}$ of its original volume at $0°\mathrm{C}$ for each degree centigrade rise or fall in temperature.