Calorimetry

Transcript

0:00Introduction

Hello everyone! Okay, a small puzzle to start. You drop fifty grams of ice, at zero degrees Celsius, into one hundred grams of water at twenty degrees Celsius, and you wait. The water is warm, so all the ice melts, right? Hmm, are you sure? Hold on to your answer; we will settle it in this video. Our topic is calorimetry, the physics of heat lost and heat gained. Questions on mixing hot and cold things come up often in JEE, so let's begin.

0:34Heat to change temperature

So, heat can do two jobs. The first job: raise the temperature. The heat needed is Q equals m times s times delta T, okay? Q is the heat, m is the mass, and delta T, with the Greek capital delta in front of capital T, is the change in temperature. With me? And s is the specific heat of the material: the heat needed for each unit of mass, for each degree of rise. The product m times s is the heat capacity of the whole body. Got it? Now a quick question. Delta T is only a change in temperature. Is a change of one degree Celsius the same as a change of one kelvin? Yes, exactly! One kelvin step is the same size as one Celsius degree, so a change is the same number on both scales. That is why, for water, s equals four thousand two hundred joules per kilogram per kelvin, which is the same as one calorie per gram per degree Celsius. The small minus one powers on screen simply mean per. Right? And to switch between the units: one calorie equals four point two joules. This is called the mechanical equivalent of heat. Clear?

1:48Heat to change phase: Q=mL

Now look at this graph. We heat ice at a steady rate, and plot its temperature, in degrees Celsius, against the heat supplied. The temperature rises, and then, look, it goes flat at zero while the ice is melting. Heat is still going in. So where is that heat going? Into changing the phase! While ice turns into water at zero degrees, and later while water boils into steam at one hundred degrees, the heat goes into the change of phase, and the temperature stays constant. On the sloping parts, the graph writes Q equals m s delta theta. Theta, the Greek letter, is just another name for temperature here, so that is our same formula. Okay? On the flat parts, the heat is Q equals m times capital L, where capital L is the latent heat: the heat for each unit of mass to change the phase, at constant temperature. With me? For ice, we write L with a small f below it, f for fusion, which means melting. L f equals three hundred and thirty six kilojoules per kilogram, or, in brackets, eighty calories per gram. Got it? For water boiling, L with a small v, v for vaporisation: two thousand two hundred and fifty six kilojoules per kilogram, about five hundred and forty calories per gram. Oh no, careful here! Q equals m L has no delta T at all, because the temperature does not change. Clear?

3:19The principle of calorimetry

Now the main idea. Put something hot and something cold together in an insulated container, so no heat leaks in or out. The hot one cools down, and the cold one warms up. So where does the heat from the hot one go? Into the cold one, all of it! So, heat lost by the hot body equals heat gained by the cold body. This is the principle of calorimetry. But only when the system is insulated, okay? Now, this mixing is done in a vessel called a calorimeter, and the calorimeter takes some heat too. So we use its water equivalent: m c times s c, over s w. The small c below means calorimeter, and the small w means water. It is the mass of water with the same heat capacity as the calorimeter. Right? And now, the most important rule of today. Whenever ice is involved, first check: is there enough heat to melt all of it? Got it? If not, and the ice starts at zero degrees Celsius, as in our puzzle, the final temperature is exactly zero degrees Celsius, with some ice left over. Remember our puzzle? Keep this rule ready for Example two. Clear?

4:33Example 1: final temperature when hot and cold water mix

Now, the first example. One hundred grams of water at eighty degrees Celsius is mixed with two hundred grams of water at twenty degrees Celsius. Find the final temperature. Pause the video, and try it first. Done? Okay. The quick answer: halfway between eighty and twenty is fifty, so fifty degrees Celsius, right? No, no, wait! There is twice as much cold water, so the answer must be closer to twenty. Let us do it properly. Call the final temperature capital T, okay? The hot water cools from eighty down to capital T, so the heat lost is one hundred times s times, eighty minus T, in brackets. The cold water warms from twenty up to capital T, so the heat gained is two hundred times s times, T minus twenty. With me? Now, heat lost equals heat gained. Both are water, so the same s sits on both sides and cancels. Perfect! One hundred times, eighty minus T, equals two hundred times, T minus twenty. Right? Solve it: T equals one hundred times eighty, plus two hundred times twenty, all over three hundred. Eight thousand plus four thousand is twelve thousand, over three hundred: forty degrees Celsius! Closer to twenty, just as we said. Clear? And here is a topper's trick. When water is mixed with water, the final temperature is simply the mass weighted average. Got it?

6:10Example 2: 50 g ice in 100 g water, does all of it melt?

Now, remember our puzzle from the start? This is Example two. Fifty grams of ice at zero degrees Celsius is put into one hundred grams of water at twenty degrees Celsius. Find the final state. Pause and try. Hint: use our rule for ice. Alright. The usual rush: write heat lost equals heat gained, assume all the ice melts, and solve for a final temperature. Wait! What did our rule say we must check first? Yes, exactly: is there enough heat? Let us work in grams, so for water, s is four point two joules per gram per kelvin. The most heat the water can give is when it cools all the way down to zero degrees Celsius: one hundred times four point two times twenty, that is eight thousand four hundred joules. Okay? Now, to melt all fifty grams of ice: L f is three hundred and thirty six joules per gram, so we need fifty times three hundred and thirty six, that is sixteen thousand eight hundred joules. With me? Compare them. Eight thousand four hundred is less than sixteen thousand eight hundred; the sign on screen means less than. The water has only half the heat needed. So, not all the ice melts! Right? So how much does melt? Each gram of ice takes three hundred and thirty six joules, and the water gives eight thousand four hundred. Eight thousand four hundred over three hundred and thirty six: twenty five grams. Clear? So, the final state: zero degrees Celsius, with twenty five grams of ice left, and one hundred plus twenty five, that is one hundred and twenty five grams of water. So the answer to our puzzle is no! Only half the ice melts. See how the check saved us?

7:59Quick revision: Q = ms delta T, Q = mL, heat lost = gained

So, quick revision. One: Q equals m s delta T to change the temperature, and Q equals m L to change the phase, at constant temperature. Two: in an insulated system, heat lost equals heat gained. Three: with ice, first check if there is enough heat to melt it all. Now go and try the practice problems on this page. All the best!