Keeping It Hot

According to Newton's Law of cooling, fall in temperature is directly proportional to the temperature difference between an object and its surroundings. So, less the temperature difference, slower will the object cool. Adding cream cools down the coffee and reduces the temperature difference between the coffee and its surroundings.

I tried to solve some equations a long time ago and I'm pretty sure that you'll have the same final temperature if you do either:

• take the creamer out of the fridge, pour it into the coffee and then wait for time T;
• take the creamer out of the fridge, let it warm up for the same time T (while coffee is cooling down) and then pour it into the coffee.

Then, it's quite obvious that if you let the creamer in the fridge during this time, you'll add cooler creamer and the coffee would be cooler than if you take the creamer out of the fridge and let it warm up (which gives the same final temperature as pouring it right after coffee).

This shows also that if the creamer is always at room temperature, it won't change if you pour it right after the coffee ou just before drinking it.

More important, it also shows that the given answer here could be wrong: if the outside ambient temperature is cooler than your fridge, you better leave the creamer the longest time possible in the fridge if you want the warmest coffee.

The sooner we add the creamer, the higher the equilibrium temperature. Otherwise, the coffee will spend time cooling.

Matt Parker did nice video about this some years ago: Milk first or last?

Just though someone may be interested of seeing physics in action :-)

You are not supposed to drink coffee in the lab at all. That's what the tea-room is for.

I agree with Avineil Jain. Some have asked for the math, so here's what I came up with.

First, our assumptions:

1.) The mass of the coffee and the mass of the creamer must always add to one. I.e. the cup has a finite volume, and he can only pour so much milk into the coffee before it overflows, and our calculations become meaningless. This is represented by the equation $m=m_c + m_m$ , and allows us to talk about fractions of milk to coffee easily, no matter the total volume.

2.) The milk is always added at the cold temperature, and therefore has no time-dependence. In the equations, $T_m$ is always the same temperature.

3.) To first order, the coffee follows Newton's Law of cooling. Other effects, like evaporative cooling may amplify the cooling, but it turns out that doesn't change the overall SHAPE of the temperature vs. time curve. (That's the important bit here!) So as soon as it's poured, the coffee begins to lose heat according to the equation $T(t)=T_a - (T_{c0} - T_a)e^{-kt}$ , where $T_a$ is the ambient temperature, $T_{c0}$ is the initial temperature, and $k$ is the coefficient of cooling for the given substance. In this case, coffee has a very similar coefficient of cooling to water, and it varies with volume, but a good average value is $k=0.0447$ . (For reference, some experimental data is here ).

4.) Lastly, we assume that the milk and the coffee, when mixed, follow basic laws of Thermodynamics, and the heat lost by the coffee is gained by the milk. Mathematically: $Q_{lost} = -Q_{gained}$ , or $m_c C_c \Delta T_{c0}=-m_m C_m \Delta T_m$ . Expanding the $\Delta T_c$ and $\Delta T_m$ and rearranging to find $T_f$ , we arrive at: $T(t)=\frac{m_c C_c T_{c}(t)+m_m C_m T_m}{m_c C_c +m_m C_m}$

The graph of this function is here: DESMOS!

This is an exponential decay, showing the final temperature of the coffee/milk mixture for any time that the milk is added. In order to get the hottest mixture when our hero sits down at his desk, he should pour the milk in as soon as possible. In other words, because the milk stays at the same temperature, it's the temperature of the coffee at the time of mixing that determines the final temperature. Keep that temperature as hot as you can, and you'll have a slightly warmer cup when you sit down at your desk.

The larger coffee cup will loose more heat to the air than the small creamer container will gain. Because of the larger surface area of the cup, the higher temp difference of the coffee/air vs air/creamer.

Assumptions: - (1) While walking the ambient temperature is room temperature (20C). - (2) Coffee is close to 100C (boiling point), so let's say 90C as it will cool down a bit when poured into the cup. - (3) Creamer is at fridge temperature of about 4C. - (4) Creamer volume is small enough relative to the volume of coffee that the heating up of the creamer during this time is not material, so the impact of adding the creamer right away (when it is colder) vs later (when it warms up a bit) is not material.

Loss of heat and drop in temperature is proportional to temperature difference, so the hot coffee without the creamer mixed in loses heat faster than the colder coffee with the creamer mixed in.

Technically while adding the creamer later results in a greater loss of heat from the hot coffee, it is offset slightly by the creamer warming up a bit because when the creamer is added later, it does not have as much of a cooling effect. This is hard to estimate qualitatively.

Nature likes everything in it's lower energy state ASAP

I might be doing this the wrong way, but if the creamer is in the fridge for longer it might be colder. I know this is the wrong way of answering the question, but it gives the right answer: in order for the coffee to be hotter, the creamer has to be in the fridge shorter, thus Josh has to pour the creamer immediately.

The closer the mix to the ambient temperature, the slower it will lose energy (heat) and longer it will keep warm. So, the best solution among the given is to add the creamer right after he pours the coffee. From that point on, the loses of heat are minimized.

He shouldn't be eating or drinking in the lab (OK, that depends on what kind of lab... greatest danger in bio or chem) :)

If the creamer weren't kept in a magic refrigerator running alongside Josh, it would be a different problem. However, there is usually less creamer so it would have to be very cold indeed to gain more heat from the room than the coffee loses.

The larger the volume the longer it will take to cool. The coffee ex the cream will cool at a slightly faster temperature than with the cream added. With the cream added, the volume will be more and less cooling over the same period of time.

This may be the logic to the solution. Walking with the cup produces an air flow over the top of the cup cooling the coffee. Refrigerated creamer is added "at some point" If the creamer is added immediately after pouring the coffee and left floating on the top of the coffee it would produce an insulating layer keeping airflow off the coffee and so keeping the coffee hot. I'm not familiar with the properties of soy creamer

Let's assume that walking time is long enough to make the coffee at lab=way=station temperature. If he puts the cold creamer at the lab, coffee will be colder than lab temperature while if he puts it right after pouring hot coffee, it will be at lab temperature when he arrives at his lab which is higher (hotter).

Creamer being less dense than coffee, will float on the coffee. Also it is a bad conductor of heat and so it will insulate the coffee. Thus the coffee will lose lesser heat compared to if it was left exposed

it is a daily practice question.while mixing cool water to hot water before bath we mix them, immediately to save heat and reduce rate of cooling