Gas Laws

# Charles’ Law

​At constant pressure, volume of certain mass of a gas changes with changing temperature. The effect of temperature and volume with constant pressure was studied by Jaccqus Charles in 1787. He gave his law based on this effect.
Charles’ Law states that, “At constant pressure, the volume of given mass of gas increases or decreases by $1/273$ parts of its volume at $0°C$ for each $1°C$ rise or fall in temperature.”

​Let $V_0$ be the volume of gas at $0°C$.

For $1°C$ rise in temperature, $V_1 = V_0 + \frac{1}{273} \text{ of }V_0$ $V_1 = \left(1+\frac{1}{273}\right)V_0$

For $2°C$ rise in temperature, $V_1 = \left(1+\frac{2}{273}\right)V_0$

For $273°C$ rise in temperature, $V_1 = \left(1+\frac{273}{273}\right)V_0$

For $1°C$ fall in temperature, $V_1 = \left(1-\frac{1}{273}\right)V_0$ For $2°C$ fall in temperature, $V_1 = \left(1-\frac{2}{273}\right)V_0$

For $273°C$ fall in temperature, $V_1 = \left(1-\frac{273}{273}\right)V_0$ $V_1 = 0 × V_0$

This $-273°C$ is known as absolute scale of temperature. This is only theoretically possible because at this temperature, the molecules are in solid or liquid state and cannot obey gaseous law.

Similarly, For $t°C$ change in temperature, $V_t = \left(1+\frac{t}{273}\right)V_0$

Let $V_1$ and $t_1$ be the initial volume and temperature of the gas respectively. Then, $V_1 = \left(1+\frac{t_1}{273}\right)V_0$ $V_1 = \left({273+t_1}{273}\right)V_0\text{ __(1)}$

Similarly, Let $V_2$ and $t_2$ be the final volume and temperature of the gas respectively. Then, $V_2 = \left(1+\frac{t_2}{273}\right)V_0$ $V_2 = \left({273+t_2}{273}\right)V_0\text{ __(2)}$

Dividing $(1)$ by $(2)$, $\frac{V_1}{V_2}=\frac{273+t_1}{273+t_2}\text{ __(3)}$

Let $T_1$ and $T_2$ be the initial and final absolute temperature respectively. Then, $T_1 = 273 + t_1$ $T_2 = 273 + t_2$ From $(3)$, $\frac{V_1}{V_2}=\frac{T_1}{T_2}$ $\frac{V_1}{T_1}=\frac{V_2}{T_2}$ $V ∝ T$

Hence, Charles’ law may also be defined as, “at constant pressure, the volume of given mass of gas is directly proportional to its absolute temperature.”

## Graphical Representation of Charles’ Law

When volume of given mass of gas is plotted against different temperature, an inclined straight line is obtained. When this line is extra potted in backward direction, it meets at $-273°C$ temperature which is known as absolute scale or Kelvin scale temperature.