 Hi, I'm Zor. Welcome to a new Zor education. I would like to solve a couple of problems, actually four problems related to measurements of the heat and comparison to whatever other knowledge about physics we have between the heat and the energy and how it's measured, etc. Now, these are very simple problems. And this is basically a continuation of this chapter related to measuring heat, which is part of the course called Physics for Teens, presented on Unisor.com. This is the website where it's presented. Now, this is the course, which means if you have found this lecture somewhere separately, like on YouTube for instance, I just want you to know that this is just one of many lectures which are related to each other and they're put into some logical order as a course. So basically you have to really start from the beginning because there is definitely a dependency. And on Unisor not only you have references to these videos, but also the textual material for each lecture, which is basically like a textbook. You also have certain problems, you have exams for those who want to challenge themselves. The site is completely free, there are no advertisements at all, no financial strings attached. And you don't even have to sign in if you don't want to. But if you do, it will enable you to study the course with a supervision, which means that somebody like a teacher or a parent gives you an assignment, you do the assignment, you do the test, you have the results, and then you basically continue going under this supervision after you successfully pass the exam. Okay, so problems on heat and its measurement. Okay, first problem is extremely easy. You have let's say one liter of water, which is one kilogram in mass. So liter is a volume of water and we are assuming this is a pure water under good condition. It's one kilogram. Okay, so one kilogram of water. We would like to heat it up from 20 degree Celsius to boiling point, 100 Celsius. Question is how much heat do you need for this? Well, by the way, in calories, it's very easy because we know that to heat one kilogram of water to heat by one degree is one calorie. One kilogram is one kilo calorie. You have 80 degrees to heat from 20 to 100, so it's 80 kilo calories. But I would like actually to use the system, which is basically always used in physics, which is C, System Internationality, and the amount of energy and heat is an energy. So its amount of heat will be measured in joules. Now for this, we have to really know something which is called specific heat capacity of the water. Because if we know this specific heat capacity, then we can basically do the proper multiplication. It's just one algebraic operation of multiplication and you will get the answer. So again, specific heat capacity expressed in units of C. So what is it? Let me just repeat it. It's amount of energy needed in joules, needed to heat up unit of mass by one degree of Celsius, or Kelvin doesn't really matter. I mean usually in physics we are using Kelvin's degree, but since one degree of Celsius is equal to one degree of Kelvin, that's the same thing. The unit of measurement is the same. So this is the unit in which we are measuring specific heat capacity of any substance. In case of water, it's 400, 180, I think 4,000, 183, depends. It's just a matter of precision and a matter of purity of the water, etc. Consider everything is approximate in this case. So I will assume that this is exactly the specific heat capacity. So I need 4,184 joules to heat one kilogram of water by one degree of Celsius. Now we have to heat up one kilogram of water, so we have to multiply by one kilogram. And we have to heat it up by the difference between these two, 100 minus 20, which is... So one is one, so this is 80, so it's 80 times 4,000, whatever. And the result is 3, 3, 4, 7, 20 joules, right? Kilogram of Celsius. This is denominator, this is numerator, so we have only joules which remain. So this is amount of heat necessary to heat up the water from 20 to 100, if it's one kilogram of water. That's very simple problem. All we needed basically is this number. And this number is given as part of the condition of this problem. Okay, next. Okay, now assume the following experiment. We have certain amount of, let's say, water, some kind of a liquid. And we know about this water, everything, whatever is needed, which means it's mass, it's temperature, and specific heat capacity. So this is the reservoir of this amount of water, which is kept under this temperature. And we know about water, what's its specific heat capacity. Now, we have certain, let's say, a piece of metal, right? Now, we know about its mass and we know about its temperature. But we have no idea about what its heat capacity. Now, let's assume that we do have a table of heat capacity of each particular metal. And we don't know what kind of a metal we're dealing with. So maybe if we can determine the heat capacity, I'll just check it again, some table, and that's how I can identify my metal. Because obviously different metals have different heat capacities, so whatever matches, that's my metal. So, how can I determine the heat capacity of this? Okay, so let's make this experiment. We have this amount of water under this temperature with this specific heat capacity. Now, I have this piece of metal, I know what its temperature, etc. So what I will do, I will put this piece of metal into the water. Now, let's just assume that metal is not too hot, so water doesn't really boil and evaporate. And it's not too cold, so water doesn't really freeze. So water will just change its temperature. So let's assume that the water is somewhere around room temperature, more or less. And so we know this temperature. And this one may be a little bit hotter, but not by much. So we don't really have any kind of boiling or whatever. So no change of state of matter, right? So what happens? Well, if this is a little more than this, the temperature, this is warmer than this one, Well, eventually, because of all these processes which are happening, we will achieve thermodynamic equilibrium. Thermal equilibrium means that both the water and the metal will come to the same temperature. Water will be heated a little bit, metal will be cooled down a little bit, and the temperature will be the same. And then we measure this temperature. This is T. That's actually sufficient amount of information to determine heat capacity. And here is why. Whatever amount of heat metal gave to the water, water has obtained from the metal. Now, knowing the difference between beginning temperature and the ending temperature of the water and its heat capacity and mass, we can determine how much heat water has consumed. Its multiplication of heat capacity times mass of the water times difference in temperature. Now, it's exactly the same amount of heat which this particular piece of metal gave to the water, right? Now, we have just assumed that the metal is warmer than the water, so final temperature is definitely greater. So, whatever... I can write it in two different ways. I can write it as a sum plus capacity of the metal times mass of the metal times difference between temperature, ending temperature and beginning temperature. And this is supposed to be equal to zero, right? Because whenever I'm giving away heat, it's with one sign, let's say it's plus. And whenever I'm giving, it's another sign, obviously an opposite sign, which is minus. So, because it's just amount of heat going from one to another. One is giving, another is receiving, or vice versa. So, basically, this is the main equation. Or I can write it basically slightly differently, knowing what exactly is positive and what is negative, so I can put this as equal to and change the sign here. This is exactly the same thing, right? So, from the original one, I'll just change the position of one of the members and change the sign, same thing. Now, T is greater than TW, because the final temperature of the water is greater than its beginning temperature. But the final temperature of the metal is less than the original temperature. This is positive and this is positive. So, this is basically enough to determine this, because everything else we know. So, what is this capacity of the metal? Well, it's this times this times this. So, knowing capacity, specific heat capacity of the water, knowing masses of the water and the metal and knowing initial and final temperature of components of this experiment, we can determine what's my specific heat capacity for the piece of metal and that's how, for instance, we can determine what kind of a metal this is. So, this is my second problem. Again, very simple, it's basically solved in one algebraic equation. You have the amount of heat which is given, you have amount of heat which is received, either you combine them together with proper signs or you change the sign and put the equal sign between them, it doesn't really matter, but this is just one little equation from which everything can be obtained. By the way, from the same equation, we can do different things. For instance, we don't know the mass, but we do know what kind of a metal this is and what's the specific heat capacity of the metal. Well, this is the way how we can determine the mass, if you want to, because it can be resolved for the mass of the metal. So anyway, whatever unknown, if there is one unknown, then from this equation we can basically get this value knowing all others. So, we can have the mass unknown, we can have this unknown, we can have the final temperature unknown, for instance, because it can be resolved for T. So, whatever it is, we can always find it out. And by the way, I'll probably do something like exams for this chapter of the course and in the exams I'll probably put some problems like this, resolving this relative to different components. Now, next one. Next one we will consider a burger. Now, I don't know how it's done. However, food companies put specific amount of calories into each product, whatever they're making. I read somewhere that the burger has 300 big calories, which means basically kilo calories in physical terms. Okay, so this is amount of energy in this particular burger. My question is, knowing certain things which I will talk about, I would like to determine if I have eaten a burger, how much energy I have to spend in terms of climbing the stairs, how many floors I have to climb up to basically spend this energy. That's my problem, because people are exercising and they have to know how much they have to exercise to burn this amount of calories. Well, let's just calculate it using certain information which I will just provide. So, first of all, let's consider that we're talking about the person whose mass is 75 kilograms, which is kind of an average. Now, we also talk about climbing the flights, and let's say between the floors I have three meters distance from the first to the second three meters, from the second to the third three meters, etc. Whatever number of steps, I mean usually it's like 12 or 14 steps, or whatever, 15 steps. Whatever number of steps, I just assume that this is a difference in height between the floors. So, this is how much I have to really climb. Now, question is, how many floors I have to climb up. I would like to also introduce another important component into this problem. You see, whenever I'm climbing, I'm not only spending the energy of these bargers to climb the steps. I'm also maintaining basically my body, because the body is functioning and not only my legs are working, so the whole body is functioning and for this we need energy, because we are each to maintain the energy needed for our body to exist. Now, only certain amount of this energy is spent towards real climbing. The rest is spent to maintain our existence, to maintain body in a living condition. So, let's assume that only 25% of these calories we are using to climb the steps. The rest is basically maintaining our life. Now, okay, that's enough actually. That's enough information to calculate the number of steps, number of floors which we have to climb. Well, first of all, let's find out in the same system of units in joules basically how much energy is here. So, we know that 1 kcal is equal to 4183 or 84 joules. So, I can find out basically from this how much energy is in my 300. So, 300 kcal is 300 times 4184 which is equal to 1, 2, 5, 5, 200 joules. So, that's amount of energy in one burger, right? Seems to be a big number by the way. Now, I can spend only 25% of this to climb the floors, right? So, I have to multiply this by 0.25 which gives me 313800 joules. So, this is amount of energy I have to spend climbing. Okay, now, let's count how much energy needed to climb, let's say, one floor. Now, I have the mass, but if I have the mass, I have the gravity because I'm going against the gravity, right? So, my force which is directed upwards should be equal to my weight actually which is 75 kg times 9.8 m2 which is acceleration of the free fall. M times A or G actually in this case, M times G, that's my weight. This is the force which earth attracts the body. And now, since I know the force, I have to multiply it by distance to get the work which is needed, right? So, I have to multiply it by 3 m and it's equal to amount of newton meters which is joules and the answer is 2205 joules. So, that's how much energy I need to climb one floor. And this is amount of energy which I have to spend. So, how many floors should I climb if I have this amount of energy and one floor requires this much? Well, you have to divide this by this and the result will be approximately 142 floors. Could you imagine 142 floors? That's higher than any building in the United States. I think the highest building is around 100 floors more or less. So, it's higher than the tower in Chicago, than the World Trade Center in New York. That's a lot and that's for one burger only. So, numbers, by the way, are relatively, I would say reasonable numbers including the 25% and including 300. So, amount of food which we are eating is really very rich in energy because we know how to extract this energy properly. And well, basically, we have to be very careful with eating because it's not easy to spend so much energy. You have to really climb like for one burger, if you have extra one burger you have to really climb 142 floors to get rid of it. That's not easy. So, well, be careful with your food. Next problem is about melting ice. Now, this problem is different from the previous ones because we will also talk about change of the state of the matter from solid ice to liquid water. So, here is the problem. Let's say you have 100 gram, 0.1 kilogram of ice at temperature minus 10 degree Celsius. Minus 10, which means it's frozen, right? It's frozen to a temperature minus 10. Now, my question is how much water, minimum amount of water you need to melt this ice if the water which you take is of room temperature. So, you need certain amount of room temperature to convert this amount of ice. Let's say you take this amount of ice, 100 gram, and put it in reservoir. I mean, if reservoir is large enough and it's of a room temperature, obviously the ice will melt. My question is, what's the minimum amount of water in this reservoir under this temperature to melt completely ice? Because if it's just a small amount of water, well, part of the ice might actually temporarily melt and then the rest will probably freeze this amount of water and it will be still kind of a, it will be still ice, but maybe with slightly less temperature like minus 9, but it will be still ice. So, the only need I have to bring temperature of the ice down to zero. That's the melting point. So, I have to bring it down to zero. And by the way, zero is very specific number. Zero of Celsius is when the water is water and the ice is ice and both can have this zero temperature. And there is a special amount of extra energy which is needed to convert from ice to water. Or if you want to freeze the water, then the water should actually be, the energy should be taken from the water. So, either the water will receive the energy, I mean ice will receive the energy to be converted into water under the same temperature of zero degrees or the water should be taken certain amount of heat if it's already under zero, but it's supposed to be converted into ice. You have to take some amount of water, of energy. Alright, so in this particular case, we are melting ice, which means we have to bring temperature down to zero while ice is still ice with its own specific heat capacity. Now, what's the specific heat capacity of the ice? It's twenty ninety. Jows per kilogram degree, degree of Celsius. And specific water, heat is forty one eighty. I think in this case forty three. Sometimes I put forty one eighty three, sometimes eighty four. It's all approximations, so it depends in the same units. Jows by kilogram and degree of Celsius. So, my water, whatever the amount of water I have, let's say I have M, should be used under this temperature. And it should be used to heat up the ice first by ten degrees from minus ten to zero, and it's still ice. And then to melt it. And again, this is a completely different amount of energy if ice is already under zero temperature degree and converted into water under the same zero degrees Celsius. You still have to spend certain amount of heat. And this is a known amount, it's a melting amount of energy and it's measured in Jows per kilogram, right? So, the melting point amount of heat for the melting point is equal to three three three zero zero zero Jows per kilogram. So, that's how much energy you need to melt one kilogram at zero degrees Celsius of ice into zero degrees of Celsius water. Just to change the state from ice to water, you need that much energy. Now, this energy also should be contained in our mass of water. So, what's the total amount of energy which I have to have to convert this ice into the water at zero degrees? Well, first I have to have sea ice times one kilogram of mass times number of degrees which is from zero. I have to subtract minus ten, so it's plus. So, that's my amount of energy. So, it's twenty ninety, basically times ten. So, it's twenty ninety ten, right? That's the amount of heat Jows. That's how much I have to spend. Now, this is only part of the problem. Now, the next part of the problem is I have to spend this amount of energy. Now, my ice is at zero degrees Celsius. But this is still ice. I need to convert it into water. It's still one kilogram of ice. I would like to convert it into one kilogram of water without changing the temperature. Just to convert from one state, from the solid state into the liquid. So, I have to add this amount to this. So, it will be three three three thousand plus twenty ninety. And that would be three five three nine zero zero Jows. That's how much energy I need to first warm up the ice to zero and then to convert it from ice at zero to water at zero degrees. All this energy must be released by the water which is initially at twenty degrees. And it has certain mass which I don't know. I would like to actually find out. So, what's the minimum amount is when my sea water, the specific heat capacity of the water times its mass, times what's the difference in degrees. Water will be brought into zero as well. We are bringing ice to zero and the water to zero which is melting ice. So, that would be zero minus twenty. So, it will be twenty degrees. Obviously, this will be with a minus sign because it's zero minus twenty. So, minus this one plus this one should be equal to zero. So, if I would like to find out really my M. Sea water, I know what it is. It's four, one, eight, three. So, if I'm equating this to this, I will have my mass equals to, okay, I didn't really calculate it, but you can calculate it yourself. I put it into a textual description. That would be the answer. It's four, one, sorry, it would be three, five, three, nine hundred divided by four, one, eight, three times twenty. So, that's what it is. So, that's amount of water needed. Okay? I think I made a little mistake here. Zero point one kilogram. So, it's this. So, it's this and it's different. It's three, three, five. Something like this, right? So, three, five, zero, nine, three, three, five, zero. So, whatever it is. This is the amount of water at twenty degrees, which I need to melt my one hundred grams of ice. Well, that's it. These are my very simple problems. And next, probably I will spend some time and I will prepare the exams for this particular topic and similar problems will be part of the exams. I do encourage you, first of all, to go to this website and read the text for this lecture. And the next after the problems, there will be exam and I do encourage you to take exam. That's just another little challenge for you. Thanks very much. So, that's it for today. Thank you.