Sound travels slower in colder air. why does the speed of sound depend on air temperature?

The speed of sound depends on air temp. b/c sound travels through molecules. So that would lead to sound traveling slower in the colder air compared to hotter air. This is because, in hotter air, the molecules are moving faster (the hotter the temperature, the faster the molecules vibrate) which leads to these molecules to bump into each other more often, leading to the transmission of sound through each molecule. In colder air, the molecules are not vibrating as fast as in hot air.