Heat: Energy Transfer Between the Universe

When you take out a cold can of cola from the fridge, place it under the heat of the sun and do something else because you told yourself to drink later on, by the moment you come back, I'm sure that once you come back for your drink, you're going to have a bad time. Why? Because the once cold can of cola has already heated up, and by doing so, it's temperature increased.

On the other hand, when you finally got the food you ordered but decided to eat it after your class (because you got your food just in time for class - good job!) and so placed it there in the corner right next to the air conditioning unit, by the time your class ends, do not expect that tasty hot fried chicken to be hot hot anymore. Because of the fact that you placed it next to the aircon, the temperature of that chicken decreased.

In both cases, we see that when you left the can of cola and food you ordered in their corresponding environments, its temperature changed because in these cases those objects of interest, or system, is trying to attain thermal equilibrium with its environment. This follows from the last blog wherein we talk about how objects can interact with one another to attain thermal equilibrium. However, what is different here is the fact that we are considering one system interacting not only with another system, but the whole environment.

While we are talking about the environment around the system, we are talking about the relevant environment. When you are outside under the sun with that can of cola, the temperature of the air inside the house does not affect the state of that can of cola. Similarly, that gust of wind in the air outside your classroom does not affect whatever happens to your food. Hence when we talk about the environment around the system, we talk about the environment immediately surrounding the system.

Furthermore, there are different types of systems. There are, a) Isolated, b) Open, and c) Closed Systems. Isolated systems are such that do not permit the transfer of matter and energy between the system and its surroundings. Open Systems, the opposite of Isolated Systems, permit the transfer of matter and energy. Closed Systems on the other hand, only permit energy to transfer between the system and its surroundings but not matter.

Figure 1: Example of an Open System.
Taste Testing Sinigang.

Forsooth, many of us interact, use, and exploit the properties of these types of systems. A typical example is one when we cook. When we taste test that sinigang we are cooking, we take off the lid of the pot and scoop up some soup from the concoction we are making. That is an example of an open system! Since the lid is open, clearly, energy can escape between the sinigang and air around it. Furthermore, the mere fact that we got some of the soup from it means that matter can be transferred between the sinigang and taste tester!

Figure 2: Example of a Closed System.
Without the straws puncturing the cover of the cup, matter
cannot be transferred from the cup to the
environment and vice versa.

Another example is when we go to a coffee shop and get Frappes. Upon getting it from the counter, we notice that it the cup and its cover entirely encapsulates the Frappe inside and the only way for you to be able to drink that ice-cold, refreshing coffee, is when you puncture a hole through the cover of that cup. In the moment the whole thing was covered, it was a closed system because initially the liquid inside cannot escape, however the energy can!

Isolated systems are little harder to come by, however one typical example of such is the thermos. We use thermos so that the temperature of the fluid or material inside is kept (on average) constant. Again, since energy and matter is not transferred between the thermos and its surroundings, it is isolated from its surroundings! (Hence the name!)

Now, the temperature of the system can change, and this is due to the change in the thermal of that system because energy is being transferred between the system and its environment. We can quantify the amount of energy transferred, and this is where the concept of heat $Q$ comes up.

When the heat is being absorbed by the system, or energy is transferred to the system from the environment, we denote heat to be positive. However, when heat is given off by the system, or energy is transferred to the environment from the system, we denote heat to be negative. Intuitively, when the temperature of the system is greater than or less than that of the environment surrounding it, then its temperature decreases or increases until its temperature is equal to the temperature of the environment. So from this, we can define heat to the energy transferred between the system and its environment because of the change in temperature between the two.

The unit of heat was once calorie, defined to be the amount of heat that would raise the temperature of 1 g of water from 14.5 $^\circ$C to 15.5 $^\circ$C. However, in the 1900s, it became clear to scientists that since heat involved the transfer of energy, its unit should be the unit of energy, or joule. Today we define 1 cal as 4.1868 J (exactly), and the nutritional calorie dietitians use, or Cal, is essentially 1 kilocalorie.

We now go to the concept of heat capacity and specific heat.

Heat capacity $C$ of a particular material is the amount of energy needed to raise the temperature of that material by 1 $^\circ$C. More specifically, it is the proportionality constant between the heat $Q$ of that material and the change in its temperature $\Delta T$. Explicitly:
\begin{equation}
\label{heatcap} Q=C\Delta T
\end{equation}
The specific heat, on the other hand, is the materials' heat capacity per unit mass $m$. Explicitly, this is expressed as:
\begin{equation}
c\equiv\dfrac{Q}{m\Delta T}
\end{equation}
Which makes sense as it takes twice the amount of heat to boil 2.00 L of water as such to boil 1.00 L, half the amount of heat to increase the temperature of 25 g of aluminum than that for 50 g of aluminum, etc. In particular, the specific heat is a measure of how thermally 'insensitive' a substance is for its temperature to increase. From here, we can generalize that heat $Q$ is:
\begin{equation}
\label{specheat} Q=mc\Delta T
\end{equation}

One technique to measure the specific heat of a particular substance is known as calorimetry, and this technique uses devices known as calorimeters. There are many types of calorimeters out there - the bomb calorimeter being one of it. The more commonly used, especially in the secondary and tertially level education, is the coffee-cup calorimeter.

To put it simply, in calorimetry, we heat up a sample to some known temperature, and place it in the calorimeter containing water of known mass and temperature less than the temperature of the sample. Then after some time, the temperature of the whole system is measured after it reached thermal equilibrium. Now, when the coffee-cup is ideal, one in which energy and matter is not transferred, then we can say that the heat lost by the sample is equal to the heat gained by the water. However, due to practicality and since we live in the real world, we have to assume that the heat lost by the sample $Q_\text{sample}$ is equal to the heat gained by the water $Q_\text{water}$ and the calorimeter $Q_\text{calorimeter}$! Explicitly:
\begin{equation}
\label{heat} -Q_\text{sample}=Q_\text{water}+Q_\text{calorimeter}\hspace{1cm}(Q_\text{sample}<0\ \text{and},Q_\text{water},Q_\text{calorimeter}>0)
\end{equation}
And this is the focus of the experiment.

The experiment is composed of two parts - the first part in which we obtain the heat capacity of the calorimeter, and the second, in which we obtain the heat capacity of aluminum, brass, and copper.

To obtain the heat capacity of the calorimeter, an amount of water of known mass is poured into a coffee cup and its temperature is measured. Then the temperature of an amount of boiling water of known mass was measured using a thermocouple and poured into the coffee cup as well. Immediately after all of the boiling water was poured, the timer started and the temperature of the mixture was measured in 20 s intervals for 5 minutes. This was done three times, after which the natural logarithm of the temperature was plotted as a function of time, and a linear fit of the form $T(t)=At+B$ was taken. Then from the slope $A$, the final temperature of the mixture $T_f$ was taken to be $\mathrm{e}^A$. Then using Equations \eqref{heat}, \eqref{heatcap}, and \eqref{specheat}, the heat capacity of the calorimeter was obtained and averaged from the three trials.

To get the heat capacity of the metal samples, essentially the method used in the first part was used, except this time the boiling water is replaced with the heated metal samples. Further, instead of monitoring its temperature for five minutes, the temperature at which the reading on the thermocouple became stable was measured, and again using Equations \eqref{heat}, \eqref{heatcap}, and \eqref{specheat}, the heat capacity of the metal samples were obtained.

Table 1: Obtained Specific Heats of Copper, Brass and Aluminum

	Copper	Brass	Aluminum
$c_\text{metal}$ ($\text{J}/\text{g}\cdot^\circ\text{C}$)	$0.4244$	$0.3792$	$1.2605$
$c_\text{metal,theo}$ ($\text{J}/\text{g}\cdot^\circ\text{C}$)	$0.3900$	$0.3800$	$0.9100$
$\%$ Deviation	$8.8161$	$0.2612$	$38.5160$

From the table, it is observed that the experimental specific heat of copper is $0.4244$ J/g$\cdot^\circ$C, brass is $0.3792$ J/g$\cdot^\circ$C, and $1.2605$ J/g$\cdot^\circ$C for aluminum, and further that these values are $8.8161\ \%$, $0.2612\ \%$ and $38.5160\ \%$ away from their theoretical values, respectively.

It is pretty interesting to find the percent deviations to be far apart from each other. It is speculated that the method of measuring the initial temperatures of the metals was flawed. This is due to the fact that their temperatures was taken while they were submerged in the boiling water. One knows that since they are conductors, once they are taken out of the water, they immediately interact with the environment, and effectively, their temperatures changes drastically. In my opinion, their temperatures should've been measured while exposed to air to decrease the error introduced into the system.

Another way to account for this is the fact that extra energy could've escaped from the coffee-cup calorimeter as it was constantly opened to ensure that the thermocouple sensor was not touching the coffee-cup itself. Furthermore, the placement of the thermocouple sensor could have introduced the error as it could have measured the temperature of the air or coffee-cup itself.

Despite the errors however, at the very least, we had fun from this experiment. To be honest, this was the most decent experiment so far. Hopefully the others will be the same.