9702 · 14.3Specific heat capacity and specific latent heat

Energy, temperature and molecules

0:00Fill a kettle with water from the tap and switch it on. In a few minutes, it's boiling. Now imagine the switch is broken and the kettle just keeps going. How long until all that water has boiled away? You might guess a few more minutes. In fact, it takes around six or seven times as long as it took to reach boiling in the first place. The whole time, the water sits at 100 degrees Celsius.

0:21The kettle is pouring energy in, and the temperature isn't going up at all. In this video, we'll meet specific heat capacity, the energy needed to change a material's temperature, and specific latent heat, the energy needed to change its state. And we'll see why boiling takes so much more energy than melting. Two quick reminders. A temperature change of one degree Celsius is the same size as a change of one kelvin, so for temperature changes it doesn't matter which you use.

0:47And molecules have two kinds of energy: kinetic energy, because they're moving, and potential energy, because of the forces between them, which depends on how far apart they are. Temperature is linked to the average kinetic energy of the molecules.

Specific heat capacity

1:00How much energy do you need to raise the temperature of something? It depends on three things. Twice the mass needs twice the energy. Twice the temperature rise needs twice the energy. And it depends on the material. The same energy that warms one kilogram of water by one degree would warm one kilogram of aluminium by more than four degrees. The specific heat capacity of a substance is the energy required per unit mass of the substance to raise its temperature by one degree.

1:26In other words, the energy needed to warm one kilogram of it by one kelvin. We give it the symbol c. Putting the three factors together, the energy transferred is: E equals m c delta theta, where delta theta is the change in temperature. Rearranged, c equals E divided by m delta theta. So the unit of c is joules per kilogram per kelvin. Joules per kilogram per degree Celsius means exactly the same thing.

1:52A few typical values. Water is about 4200 joules per kilogram per kelvin. Aluminium is about 900. Copper is about 390. A high value means a lot of energy is needed to warm the substance up, and a lot is given out as it cools, which is why water is used in hot water bottles. Heaters are the usual energy source in these questions. Power is energy per second, so the energy is power times time, or V times I times t.

Worked example: specific heat capacity

2:23Here's the question. A 1.0 kilogram aluminium block has an electrical heater and a thermometer placed in holes drilled into it. The heater runs on a potential difference of 12.0 volts with a current of 4.0 amps, for 5.0 minutes. The temperature of the block rises from 20.0 degrees Celsius to 35.5 degrees Celsius. Find the specific heat capacity of aluminium. Step one: find the energy supplied. The power is 12.0 times 4.0, which is 48 watts.

2:52The time must be in seconds, so 5.0 minutes is 300 seconds. So the energy is 48 times 300, which is 14,400 joules. Step two: find the temperature change. 35.5 minus 20.0 is 15.5 degrees, which is 15.5 kelvin. Step three: rearrange E equals m c delta theta for c. c equals E divided by m delta theta, which is 14,400 divided by 1.0 times 15.5. That's about 930 joules per kilogram per kelvin.

3:27That's a little above the accepted value of about 900, because some energy is lost to the surroundings, so the temperature rise is smaller and the calculated c is too big.

Specific latent heat and boiling

3:38Now back to the kettle puzzle. Take some ice, well below zero, and heat it at a steady rate until it has all turned to steam, plotting temperature against time. The ice warms up, then the line goes flat at zero degrees while it melts, even though energy is still going in at the same rate. The water warms to 100 degrees, then the line goes flat again while it boils, for much, much longer.

4:00Finally the steam warms up. So where does the energy go during the flat sections? If the temperature isn't changing, the average kinetic energy of the molecules isn't changing either. Instead, the energy is being used to pull the molecules further apart, against the attractive forces between them. That increases their potential energy. In a solid, the molecules are held in fixed positions. To melt it, you supply energy to break them out of that rigid arrangement so they can slide past one another.

4:26That energy goes into potential energy, not kinetic energy, so the temperature doesn't change. Hold on to this idea: during a change of state, energy raises the molecules' potential energy while their kinetic energy, and the temperature, stay the same. It comes back later, when we study internal energy. This energy is called latent heat, from the word for hidden, because it doesn't show up as a temperature rise. The specific latent heat of a substance is the energy required per unit mass to change its state, without any change in temperature.

4:57We give it the symbol L. The energy needed to change the state of a mass m is: E equals m L. Or rearranged, L equals E divided by m. So the unit is joules per kilogram. Notice there's no delta theta. During a change of state the temperature doesn't change, so the energy depends only on the mass and the substance. There are two specific latent heats, one for each change of state.

5:22The specific latent heat of fusion, L f, is the energy per unit mass needed to change a solid into a liquid at constant temperature. That's melting. The specific latent heat of vaporisation, L v, is the energy per unit mass needed to change a liquid into a gas at constant temperature. That's boiling. The same energy is released when the process runs backwards, as water freezes or steam condenses.

5:46That's why a scald from steam at 100 degrees is worse than one from water at 100 degrees: the steam releases its latent heat of vaporisation as it condenses on your skin. For water, the specific latent heat of fusion is about 3.3 times ten to the five joules per kilogram. The specific latent heat of vaporisation is about 2.26 times ten to the six joules per kilogram. That's nearly seven times bigger.

6:12It's exactly why the flat boiling section on our heating curve was so much longer than the flat melting section. Why such a big difference? Think about what happens to the molecules. When ice melts, the molecules break out of their fixed positions, but they stay very close together. Liquid water and ice take up almost the same volume. The molecules are only loosened from one another, and the increase in separation is small, so the increase in potential energy is fairly small.

6:37When water boils, the molecules are pulled completely apart. In steam, they're so far apart that the forces between them are almost zero. Every molecule has to be separated from all its neighbours, a much larger increase in separation and in potential energy. There's also a second, smaller effect. When water turns into steam, its volume increases enormously, well over a thousand times at atmospheric pressure. As the steam expands, it has to push the surrounding atmosphere back, and that means doing work against atmospheric pressure.

7:07That work also has to be supplied. Melting involves hardly any change in volume, so there's almost none of this.

Kettle and multi-stage calculations

7:14Now the opening question, with numbers. From here on, take the specific heat capacity of water as 4200 joules per kilogram per kelvin, and its specific latent heat of vaporisation as 2.26 times ten to the six joules per kilogram. A 2.0 kilowatt kettle contains 0.80 kilograms of water at 20 degrees Celsius. Assume all the energy from the kettle goes into the water. How long does it take to bring the water to the boil?

7:41And once it's boiling, how long does it take to boil away 0.10 kilograms of it? Part one is a temperature change, so we use E equals m c delta theta. The temperature rises from 20 to 100 degrees, a change of 80 kelvin. E equals 0.80 times 4200 times 80, which is 268,800 joules. The kettle supplies 2000 joules every second, because 2.0 kilowatts is 2000 watts. Time is energy divided by power, so t equals 268,800 divided by 2000, which is about 134 seconds.

8:18A little over two minutes. Part two is a change of state at constant temperature, so there's no delta theta. We use E equals m L. E equals 0.10 times 2.26 times ten to the six, which is 226,000 joules. Time equals 226,000 divided by 2000, which is 113 seconds. Boiling away just 0.10 kilograms takes almost as long as heating the whole 0.80 kilograms to boiling point. Boiling away all 0.80 kilograms would take about 900 seconds, around fifteen minutes, compared with just over two minutes to reach the boil.

8:56That's the six-to-seven times from the start of the video. Many problems chain several stages together, and the heating curve becomes your plan. How much energy is needed to turn 0.050 kilograms of ice at minus 10 degrees Celsius into steam at 100 degrees Celsius? Take the specific heat capacity of ice as 2100 joules per kilogram per kelvin, the specific latent heat of fusion of ice as 3.34 times ten to the five joules per kilogram, and use the same values for water as before.

9:26The trick is to take one stage at a time, and in each stage ask one question: is the temperature changing, or is the state changing? If the temperature is changing, use m c delta theta. If the state is changing, use m L. Stage one: warm the ice from minus 10 to zero. That's a temperature change of 10 kelvin, using the specific heat capacity of ice. E equals 0.050 times 2100 times 10, which is 1050 joules.

9:56Stage two: melt the ice at zero degrees. That's a change of state, so E equals m L f, which is 0.050 times 3.34 times ten to the five, giving 16,700 joules. Stage three: warm the water from zero to 100. Now it's water, so we use the specific heat capacity of water. E equals 0.050 times 4200 times 100, which is 21,000 joules. Stage four: boil the water at 100 degrees.

10:27E equals m L v, which is 0.050 times 2.26 times ten to the six, giving 113,000 joules. Add the four stages together. 1050 plus 16,700 plus 21,000 plus 113,000 gives 151,750 joules, which is about 1.5 times ten to the five joules. Notice that the boiling stage alone is about three quarters of the total. Pulling the molecules completely apart costs more than everything else put together. The heating curve shows this too.

11:03With constant power, a steeper slope means a smaller specific heat capacity, which is why the ice section is steeper than the water section: ice has about half the value. And a longer flat section means a bigger latent heat.

Ice cube energy balance and recap

11:16Here's a classic exam question that combines both ideas. A 0.030 kilogram ice cube at zero degrees Celsius is dropped into 0.250 kilograms of water at 25 degrees Celsius. Assuming no energy is exchanged with the container or the surroundings, what's the final temperature once all the ice has melted? Use the same data for ice and water as before. The key idea is conservation of energy. The energy lost by the warm water as it cools is equal to the energy gained by the ice.

11:47Call the final temperature T, in degrees Celsius. Now, what does the ice gain energy for? Two things. First, it has to melt at zero degrees, which needs m L f. Second, once it's melted, that cold water has to warm up from zero to T, which needs m c times T minus zero. And the warm water loses energy as it cools from 25 degrees down to T. That's m c times 25 minus T.

12:12Put the numbers in. On the left: 0.250 times 4200 times 25 minus T. That's 26,250 minus 1050 T. On the right: 0.030 times 3.34 times ten to the five, which is 10,020, plus 0.030 times 4200 times T, which is 126 T. So 26,250 minus 1050 T equals 10,020 plus 126 T. Collect the T terms on one side and the numbers on the other. 26,250 minus 10,020 is 16,230.

12:55And 1050 T plus 126 T is 1176 T. So T equals 16,230 divided by 1176, which is about 13.8 degrees Celsius. Sense-check: the answer must lie between zero and 25 degrees, and it does. It's a big drop for such a small ice cube, because melting the ice takes over 10,000 joules before the melted ice even starts warming up. That's why ice cools a drink much better than the same mass of water at zero degrees would.

13:26Four mistakes cost the most marks. Mistake one: using a temperature instead of a temperature change. Delta theta in m c delta theta is how much the temperature changed, not the final temperature. Heating water from 20 to 100 degrees is a delta theta of 80, not 100. Mistake two: using m c delta theta during a change of state. The temperature isn't changing, so delta theta is zero, and that would give zero energy.

13:55During a change of state, use m L. The energy is still going in, it's just raising the potential energy of the molecules instead of their kinetic energy. Mistake three: forgetting a stage. In the ice cube problem, students often melt the ice but forget that the melted ice then warms up to the final temperature. List the stages before you calculate. Mistake four: loose definitions. "The energy to heat something by one degree" is not enough for the specific heat capacity.

14:24You must say per unit mass. And for specific latent heat, you must say per unit mass and without a change in temperature. In exams, expect definitions, calculations with m c delta theta and m L, often with the energy from a heater as power times time, questions on heating curves, and this classic explanation. For that last one, build the answer in links, because each link usually earns a mark.

14:48First: during melting, the molecules stay close together, and their separation increases only slightly. Second: during boiling, the molecules are separated completely, to large distances apart, so there's a much greater increase in their potential energy. Third: during boiling, the volume increases a great deal, so work is done against the atmosphere as the vapour expands. Don't just say "boiling happens at a higher temperature". The reason is the change in separation of the molecules.

15:14Let's pull it all together. Specific heat capacity is the energy required per unit mass to raise the temperature by one degree. For a temperature change, E equals m c delta theta. Specific latent heat is the energy required per unit mass to change the state of a substance without any change in temperature. For a change of state, E equals m L. Fusion is solid to liquid, vaporisation is liquid to gas, and the same energy is released when the change runs backwards.

15:42During a change of state, the energy increases the potential energy of the molecules, not their kinetic energy, so the temperature stays constant. L v is much bigger than L f because boiling separates the molecules completely while melting barely changes their separation, and because the expanding vapour does work against the atmosphere. For any real problem, split it into stages, and for each one ask: is the temperature changing, or is the state changing?

16:07Then use m c delta theta or m L, and add them up. Next, we move on to gases, starting with the mole.