# Biological Physics/Charles' Law

An important idea that can be directly derived from the Ideal Gas Law is known as Charles's Law. Charles's Law describes a relationship between temperature and volume of an ideal gas when other factors are held constant. It says that temperature and volume are directly proportional to each other $PV=Nk_{B}T$ . That means as one increases, the other also increases.

A simple experiment can be done to illustrate Charles's Law. Take a balloon, fill it with air and tie it. Next, subject the balloon to different temperatures and measure the circumference. By assuming that the balloon is spherical, we can easily calculate the volume and determine how volume is dependent on temperature.

$V=4\pi r^{3}$ $C=\pi r^{2}$ $r={\sqrt {\frac {C}{\pi }}}$ $V=4\pi {\Bigg (}{\sqrt {\frac {C}{\pi }}}{\Bigg )}^{3}$ You can see below the circumferences and subsequent volumes that we obtained and calculated of our balloon at different temperatures.

Temperature (K) Circumference (cm) Volume (m^3)
273 49 0.00199
296 51 0.00224
313 51.5 0.00231
338 53.5 0.00259

A graph illustrating how the balloons volume is directly proportional to temperature is shown below. There was one more temperature that was tested, but not formally recorded. When the balloon was submersed in liquid nitrogen (a temperature of about 77 K), it seemed to have completely deflated. What really happened was that the temperature was so low that the volume decreased dramatically. When the balloon heated back up, the volume again increased back to its prior size. This can be seen in a YouTube video.

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