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

$\frac{1}{273}$

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B

fall in temperature

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Solution

## Step 1: State Charles lawIt states that “At constant pressure, the volume of a fixed mass of gas is directly proportional to absolute temperature”.$V\propto T\phantom{\rule{0ex}{0ex}}V=kT$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 formAt 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 temperaturesLet ${T}_{1}=0°\mathrm{C}=273\mathrm{K}$ and ${T}_{2}=100°\mathrm{C}=373\mathrm{K}$. The temperatures are chosen arbitrarily. Then,$\begin{array}{cc}& \frac{{V}_{1}}{{V}_{2}}=\frac{373}{273}\\ ⇒& {V}_{1}={V}_{2}\left(1+\frac{100}{273}\right)\\ ⇒& {V}_{1}={V}_{2}+\frac{100{V}_{2}}{273}\end{array}$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 $\frac{100}{273}}{100}=\frac{1}{273}$.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.

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