# Calculating formula prep using thermodynamics [closed]

I suspect there is a very simple mathematical formula for preparing formula (forgive the pun) that simply mixes a combo of freshly boiled water and room-temp water.

To be specific, if I need to prepare an n mL feed, I would like to:

1. Mix x mL room temp (r) water + y mL boiling water to get 70°C water
2. Mix in the scoops of powder
3. Mix in z mL of room temp water to bring it down to 45°C.

x + y + z should equal n. But I'm struggling to figure out what x, y and z are given r.

Background:

Infant formula requires initially mixing with water at 70°C to kill cronobacter, then bringing it to about 45°C so baby can drink it. Prep guidelines say to boil water, wait for 30 mins till it gets to approx 70°C, add the powder, then rinse the bottle under tap water until it reaches about 45°C. Very imprecise and inefficient with a screaming baby.

As this post says, the heat capacity of water stays almost constant between 0°C to 100°C, therefore (for example) you can mix 30% of 0°C water and 70% of 100°C water and you'll end up with a liquid that is at 70°C, up to a small error. So I'm hoping this kind of calculation could be applied to my problem, albeit with room temp (r) water instead of 0°C.

• The amount of water in the first step should be maximized so the powder can be adequately dissolved.
• Approximate results are ok, though we should err on the side of slightly higher than 70°C for the first step and slightly under 45°C for the third, if anything.
• Room temperature is a variable to account for different climates.
• I guess the addition of powder would cool the result down slightly, but as I mentioned, slightly under 45°C for the final result is ok.

My baby brain can't figure this out. Can anyone help? The goals is to create a spreadsheet so that I can simply look up all the required measurements for a quick and easy (and safe) midnight feed. You would be helping out a very stressed mum :)

• Very imprecise and inefficient with a screaming baby. The objective is to be safe by a reasonable margin, not efficient. – StephenG Mar 30 at 8:44
• Is the goal to kill bacteria from the milk powder, or from the water? In the prep guidelines, all of the water is boiled, so it should kill bacteria in the water. With this other method, the second batch of water doesn't get heated much at all. – Subhaneil Lahiri Mar 30 at 9:32
• I was also concerned about the use of tap water, so you should edit that info about it being pre-boiled into your question. However, your question might be closed, due to this site's policy on homework-like questions. Also, many people are not comfortable giving what amounts to medical advice on Stack Exchange sites. – PM 2Ring Mar 30 at 10:04
• I'm voting to close this question as off-topic because this question needs a medically safe procedure which is best advised on by appropriate medical professionals, not a general physics forum. – StephenG Mar 30 at 10:25
• What @StephenG is saying is correct: there's almost certainly a very good reason the instructions don't say "add cooler water to bring the mixture to temp". Those instructions are specific because that's what they tested to be safe. To speculate that one can do so without increasing the risk to the baby is dangerous, this question should be deleted. – eps Mar 30 at 17:03

If we mix 2 substances the energy of the warmer substance gives off equals the energy the colder substance recieves. This gives us our first equation ($$T_M$$ being our mixing teperature):

$$E_{H1} = E_{H2}\\ m_1c_1\Delta T_1 = m_2c_2\Delta T_2\\ m_1c_1(T_1-T_M) = m_2c_2(T_M-T_2)$$

We can now plug in our specific variables, divide by $$c$$ (the heat capacity of both sides is the same since we are mixing water with water) and get 1 equation for each mixing process:

$$m_{100}(T_{100} - T_{70}) = m_{Room1}(T_{70} - T_{Room})\\ m_{70}(T_{70} - T_{45}) = m_{Room2}(T_{45} - T_{Room})$$

Lastly we know that the resulting mass should be the sum of all these individual masses (our target amount of feed) and that $$m_{70}$$ is the result of the first mixing process:

$$n = m_{feed} = m_{100} + m_{Room} + m_{Room2}\\ m_{70} = m_{100} + m_{Room}$$

We can now do some shuffeling around of terms and solve these equations until we end up with

$$x = m_{Room} = m_{feed}[(\frac{T_{70} - T_{Room}}{T_{100}-T_{70}} + 1)( \frac{T_{70} - T_{45}}{T_{45 - T_{Room}}} + 1)]^{-1}\\ y = m_{100} = m_{Room}\frac{T_{70} - T_{Room}}{T_{100}-T_{70}}\\ z = m_{Room2} = m_{70} \frac{T_{70} - T_{45}}{T_{45 - T_{Room}}}$$

All of these are now solvable from just the room temperature and the target amount of feed.

If we put all of this in a spreadsheet, we get something like this (I used 100 feed, so the numbers represent percentages):

Notes:

• I have no idea how baby formula works in detail, so I can't give any advice about what's "more save" or to what degree of accuracy those temperatures need to be hit in practice. The formulas above attempt to give an answer to the physics of the problem, nothing more.

• In this specific case we can use temperatures in Celcius, because only the difference in temperature counts, but just in case someone finds these equations and uses them for something different, I stuck to generic variables since most of the time using temepratures in Kelvin is the better idea.

• The boiling temperature $$T_{100}$$ might differ depending on surrounding air pressure.

• I have ignored the baby formula itself (the powder). This might - depending on it's amount (mass), temperature and heat capacity - require adding an additional mixing step along the lines of the first two.

The problem gives us three equations: $$z \, T_r + (x + y) \, 70^\circ \text{C} = n \, 45^\circ \text{C}$$ $$y \, T_r + x \, 100^\circ \text{C} = (x + y) \, 70^\circ \text{C}$$ $$x + y + z = n$$

With $$T_r$$ beeing room temperature.

From this I get:

$$x = n \, \frac{\frac{45^\circ \text{C}}{T_r} - 1}{(1 + \frac{30^\circ \text{C}}{70^\circ \text{C} - T_r}) \, (\frac{70^\circ \text{C}}{T_r} - 1)}$$ $$y = x \, \frac{30^\circ \text{C}}{70^\circ \text{C} - T_r} = n \, \frac{\frac{30^\circ \text{C}}{70^\circ \text{C} - T_r} \, (\frac{45^\circ \text{C}}{T_r} - 1)}{(1 + \frac{30^\circ \text{C}}{70^\circ \text{C} - T_r}) \, (\frac{70^\circ \text{C}}{T_r} - 1)}$$ $$z = n - x - y = n \, \frac{25^\circ \text{C}}{70^\circ \text{C} - T_r}$$

No guarantees that there are no errors though...

For $$T_r = 20^\circ \text{C}$$ one gets: $$x = \frac{5}{16} \, n = 0.3125 \, n$$ $$y = \frac{3}{16} \, n = 0.1875 \, n$$ $$z = 0.5 \, n$$