Heat Capacity

There have been a few questions about heat capacity…

Heat capacity is a measure of the amount of heat required to change the temperature of a given amount of a substance by some amount. A common unit for heat capacity is "calories / (gram)(°C)". For water, heat capacity is 1 calorie / (gram)(°C), so if I have 1 gram of water and I want to increase its temperature by 1°C, I have to add 1 calorie of energy to the water. What if I have more than 1 gram or I want to increase the temperature by more than 1°C? Multiply!

The reverse of this problem is really the same problem, it just requires some different math. What if I have 18.00mL of water that is initially at 12.6°C and I add 49calories of energy to that water? First part… the density of pure water is 1 g/mL so 18.00mL of water has a mass of 18.00g. Now, if the heat capacity of water is 1 calorie / (gram)(°C), and we have 18.00g of water, we can again multiply to get:

{1 calorie / (gram)(°C)} x 18.00g = 18.00 calories per °C

So for every 18.00 calories of energy we add to this specific sample, we will increase the temperature by 1°C. We are adding 49 calories to this specific sample so:

49 calories / 18.00 calories per °C = 2.7°C

This is how much the temperature changes when we add this amount of energy. Since the sample was initially at 12.6°C and we added energy, the new final temperature must be 2.7°C higher than the initial temperature, 12.6°C + 2.7°C = 15.3°C.

There are some assumptions in this description (like the density of water) that simplify the problem… if you want to get the absolutely perfectly correct answer, you'd have to take some of those assumption into account, but this is close enough for our purposes.

Microwave Heating Experiment (Fall 2015)

There are quite a few questions about the Microwave Heating experiment (procedure here), so let me try to address them all in one place.

The most common and persistent questions are related to the data analysis. The hands-on procedure and collection of the data seems to be going well, so let's not dwell on that part of the experiment. Once you have collected your data, you'll have a sea of numbers that you, as a scientific investigator, will have to process and interpret so you can get some useful information out of your experiment. The data you are collecting will look something like this:

The procedure tells you to prepare a graph of this data. Whenever you are collecting data (or just observing something interesting in your daily life) where one variable is under your control (in the case of this specific example, time), and another variable is simply a quantity you are measuring (in this case, temperature), it is ALWAYS interesting to consider what the graph of that information looks like. In this case, the graph of the above data looks like:

Hmm, interesting, although in this case not too surprising: as more time in the microwave passes, the sample gets hotter. But how much hotter? And how fast does it get hotter? If I want to heat a cup of water to make tea, should I put it in the microwave for 30 seconds or 5 minutes? The rate of heating is an interesting and important piece of information we can extract from this data!

How do we calculate a rate? Rate is the change in some observable quantity divided by the change in time. If you can eat a cheeseburger in 5 minutes, your rate of eating cheeseburgers is:

Change in the number of cheeseburgers / Change in time

1 cheeseburger / 5 minutes = 0.2 cheeseburgers per minute

In this experiment, you weren't eating cheeseburgers (well, OK, I guess you might have been eating cheeseburgers while you were watching water heat up…), but you were observing the change in temperature as time passed, so we should be able to calculate a rate in a similar way. Looking back at the data above, we can pick any two points and look at the change in temperature divided by the change in time. For example, from 90 seconds to 135 seconds the temperature changed from 116.04°C to 184.70°C, so the rate over that time period was:

(184.70°C - 116.04°C) / (135 seconds - 90 seconds)

(68.66°C) / (45 seconds)

1.5°C per second

We could pick any two points out of our data and calculate a rate, and they would all be pretty similar, but because there is some variability in our experiment they would not all be identical. But wait, if that's all we are going to do, then why did we make a graph? Graphs are pretty, but making a graph for the sake of making a graph doesn't seem like a great use of your time. Can we use the graph to determine the rate of heating? Hmm, change in temperature… the vertical axis (y-axis) is temperature, so that one should track changes in temperature… and the horizontal axis (x-axis) is time… WAIT! The data points in the graph look pretty close to linear, and the slope of a line is "rise over run"… if "rise" is the change in the y-axis variable and "run" is the change in the x-axis variable, then we should be able to get the slope from the graph, and the slope should be the rate of heating! If we draw a line that looks like it fits the data pretty well, we get:

From the line (we're not looking at specific points now), it looks like over the time period from 0 seconds to 300 seconds the temperature rises from about 5°C to about 390°C. So the rate based on the fit line is:

(390°C) / (300 seconds)

1.3°C per second

By using a fit line, we can even out some of the variability associated with any two points we might choose and we should get a more reliable answer. If we look at the rate for any two adjacent points (any 15 second period in our experiment) and calculate the rate using only those two points, we would get answers as low as 0.75°C per second and as high as 1.9°C per second, so using the fit line seems like a pretty good way to get a reliable single answer for the experiment.

Good luck on your data collection and analysis.