Temperature causes molecules (the tiny bits that make up everything) to move faster. The higher the temperature, the faster they move. This helps them to dislodge anything water soluble (ie: can dissolve in water) in the clothes.
Oils are not water soluble. This is where detergent comes in. It acts like a hook, pulling the oil out from the clothes. The faster the water molecules agitate – ie: the higher the temperature – the faster that works.
Many detergents are biological now days. They work via enzymes, which act kind of similar to detergents (this is a massive oversimplification). Enzyme-based detergents don’t need as high temperatures to be effective, but they do need some temperature. Between 30-40 degrees celsius is optimal for them usually. Much higher and the enzymes are denatured – ie: destroyed – and lower they don’t really work.
Result: A cold wash isn’t going to reliably wash your clothes properly. With modern bio detergent, 30 degrees Celsius should generally be fine, 40 would be more inefficient but more effective sometimes, and above that you don’t want bio detergent.
Very high temperatures can also be used to sterilize clothes. Not really relevant to most people’s situation, but a 90 degree Celsius wash will kill most germs. It will also destroy the enzymes in biological detergent, so you’d need to use non-bio detergent, and it’s really energy inefficient. If you don’t specifically need sterilized clothes (eg: from working with bodily fluids in a hospital), then it’s really not necessary.
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