Mixing Warm and Cold Water to Achieve a Desired Temperature
Imagine a scenario where your father fills a bathtub with 50 liters of warm water at 60°C, and you need to find out how much cold water at 10°C should be added to get the bathtub water to a comfortable 40°C for a relaxing bath. This example will not only provide the answer but also explore the science and the process behind temperature calculations in water mixing.
The Short Answer
By setting up the equation for temperature equilibrium, we can effectively calculate the quantity of cold water needed. The formula is simple: the heat lost by the warm water equals the heat gained by the cold water. Here, we denote the required amount of cold water as x liters.
The mathematical setup is:
50 × 60°C (50 x) × 40°C
Solving for x, we get:
3000 2000 4 - 2000
x 33.33 liters
Therefore, to achieve a 40°C bath, 33.33 liters of cold water at 10°C must be added to the 50 liters of warm water at 60°C.
The Energy Calculation
The process of cooling the warm water from 60°C to 40°C involves a significant amount of energy transfer. To understand this, let's delve into the energy calculations:
Heating Cold Water to a Target Temperature
For a 10°C water, to increase the temperature from 10°C to 40°C, a 30°C difference must be overcome. The energy required to heat 1 kg of water by 1°C is 1 calorie. For 30°C, the energy is:
30 calories/kg of water
Since the added cold water must be heated to 40°C, we calculate the mass of water as follows:
1000 kcal / 30 33.3 kg of 10°C water (33.3 liters of water)
This confirms our initial calculation using the temperature equilibrium method.
The Science Behind the Process
The physics behind temperature changes in water involves several constants, including the specific heat capacity of water. At standard conditions, the specific heat capacity of water is 1 calorie/gram°C or 1 kcal/kg°C. This means to change the temperature of 1 kilogram of water by 1 degree Celsius requires 1 kilocalorie.
It is important to note that while the specific heat capacity is constant, the mass of 1 liter of water is not exactly 1 kg at all temperatures. The density of water at 10°C is approximately 999.65 g/L, and at 69°C, it is about 983.19 g/L. These slight variations are immaterial when measuring volume with a common bucket, as the difference is negligible for practical temperature changes.
The question posed here might stem from a math problem created by a non-engineering teacher, who may not be fully aware of these subtle factors. In practice, however, the key is to understand the principles of heat transfer and the use of the specific heat capacity of water to achieve the desired temperature.
Understanding these concepts can be particularly useful in various fields, from domestic tasks like filling bathtubs to more complex industrial applications involving the cooling and heating of fluids in a multitude of processes.
By ensuring you have the correct amount of cold water added to the warm water, you can not only achieve a comfortable bathing temperature but also optimize the energy efficiency of the process.