Charles' Law (Temperature - Volume Relationship)

Discovering the volume-temperature relationship

Charles and Gay Lussac, working independently while trying to improve hot-air-balloon technology, ran a series of experiments on gases. They found that for a fixed mass of gas at constant pressure, volume increases on heating and decreases on cooling. More precisely, for each degree rise in temperature, a gas's volume grows by $\dfrac{1}{273.15}$ of its volume at $0\,^{\circ}\text{C}$. If $V_0$ and $V_t$ are the volumes at $0\,^{\circ}\text{C}$ and $t\,^{\circ}\text{C}$ respectively:

Introducing the Kelvin scale

This pattern motivates a new temperature scale: define $T = 273.15 + t$, so that $0\,^{\circ}\text{C}$ corresponds to $T_0 = 273.15$. This is the Kelvin (absolute) temperature scale, also called the thermodynamic scale, and it is the scale used throughout scientific work — note that no degree sign is written with a Kelvin value. Substituting $T_t = 273.15+t$ and $T_0 = 273.15$ gives:

This final relationship — at constant pressure, the volume of a fixed mass of gas is directly proportional to its absolute temperature — is Charles' Law. The constant $k_2$ depends on the gas's pressure, its amount, and the units used for $V$.

Isobars and absolute zero

Charles found that, for any given pressure, a plot of volume against temperature (in $^{\circ}\text{C}$) is a straight line; extending each such line — called an isobar — to zero volume, every line meets the temperature axis at the same point, $-273.15\,^{\circ}\text{C}$ (Fig. 5.6), regardless of the pressure used. …