by sevendust62 » Nov 6th, '05, 11:57
I imagine there is an equation for this buried somewhere in thermodynamics....but I haven't touched general chemistry for some years!
You don't need thermodynamics. Since both the hot water and cold water are the exact same chemically (they come from the same tap), you can just average the two temperatures together, without worrying about what their compositions are. You need to take into account the two different amounts of hot and cold, but you don't need specific heat (How much heat, in Joules, it takes to raise the temperature of a substance by a certain number of degrees Celsius) or anything else like that. Here's a formula:
Final temperature = [(amount of hot * temperature of hot) + (amount of cold * temperature of cold)] / (total amount of hot and cold water)
You know your final temperature, the total amount of water, the two individual temperatures, and you know that the two individual amounts summed equal the total amount. So just solve for your two individual amounts.
This is simply averaging the two temperatures together, taking into account the different amounts of water. In technical terms, it is a "weighted average".
Specific heat (again, Joules need to raise temp a certain number of deg C) is only needed when combining two different substances. This is because you need to use each of their individual specific heats with their individual temperatures (degrees Celsisus) to find their individual Joules (heat). Then, you sum up the two different numbers of Joules to find total heat, and then you have to find the specific heat of the two combined substances, and then find the total temperature from that. All the while, you have to keep in mind the amounts of the two substances. This is complex thermodynamics.
But since the hot water and cold water have exactly the same specific heat (because they are the same substance), their temperatures are proportional to their heat in exactly the same proportions, so averaging Joules will yield the same thing as averaging degrees. So you can use the afforementioned formula to just do a weighted average of temperature!