 Hello, welcome to another video of understanding thermodynamics. In this video we will introduce the concept of internal energy. The contents of this video include an introduction to the concept of internal energy. We will calculate the internal energy of gases for examples of real gases such as steam, ideal gases such as air and perfect gases such as helium. Now let us consider a crystal. The molecules in a crystal are tightly packed and they do not move around much. They may vibrate a bit and all molecules at temperatures higher than zero does but that is essentially all that they're doing. In contrast the molecules in a gas do move around at high velocities. If we heat up water some individual molecules will gain enough energy to break free of the bonds that keep them in the liquid phase and they will vaporize. The heat that was added to the liquid water increased the energy of the individual water molecules. The internal energy of the molecules in the vapor phase are therefore higher than the internal energy of the molecules in the liquid phase. We consider three types of kinetic energy associated with individual molecules. You get translational kinetic energy, rotational kinetic energy and vibrational kinetic energy. It is also possible that there is potential energy associated with inter-particle interaction. Particles may attract or repulse each other and energy may be required to bring them closer together or pull them apart. Internal energy is thus the sum of the total of all the microscopic kinetic and potential forms of energy. Now we will consider the determination of the internal energy of three types of gases, ideal gases, perfect gases and real gases. Let's start with real gases. Superheated steam is characterized as a real gas. The symbol for internal energy is U and its units is kilojoules per kilogram. The values of internal energy for steam can be found in the steam tables either in your handbook or on the internet. Now for this example the question asks us to determine U or the internal energy for steam at 10 kPa and 300 degrees Celsius. First we need to determine the phase. The saturated temperature of saturated steam at 10 kPa is 45.81 degrees Celsius. 300 degrees which is given to us in the question is thus bigger than 45 degrees Celsius and we can say that the steam is thus superheated and we can use the superheated steam tables. Now we can read the internal energy from the tables of superheated water. For pressure of 10 kPa and 300 degrees Celsius we get an internal energy of 2812.3. Now before we go on to the next example it can be shown that internal energy is an intensive variable. For a single phase substance if we specify the values of two intensive variables such as pressure and temperature we can determine the values of the others. Now let's have a look at another example showing this. The question asks determine the temperature of steam at 10 kPa with an internal energy of 2700 kilojoules per kilogram. First we need to determine the phase of the water. Steam at 10 kPa have a saturation internal energy of 2437.2 kilojoules per kilogram. The value given of 2700 kilojoules per kilogram is greater than 2437.2 thus we can say that the steam is superheated. Now if we look at the value given for internal energy of 2700 kilojoules per kilogram and we have a look at the tables you will see that the value falls between the internal energy for temperatures of 200 and 250 degrees Celsius. It is thus necessary that we need to interpolate to get the temperature between 200 and 250 degrees Celsius. So for an unknown value for temperature between 200 degrees and 250 degrees Celsius we get corresponding internal energies for at 200 degrees Celsius and for 250 degrees Celsius. The question has given us the value of internal energy and it wants us to determine the temperature which is unknown. We can thus determine the value of A which is the difference between our unknown value T and 250 degrees and we get an answer of 24.16 degrees Celsius. To get the value of T we take 250 and subtract A from it to get a final value of 225.8 degrees Celsius. Now let's look at an example where pressure is the unknown. Calculate the pressure of water at 200 degrees Celsius with an internal energy of 2616 kilojoules per kilogram. We know that the saturated internal energy of water at 200 degrees Celsius is 2594.2 kilojoules per kilogram and that is less than the given value for the internal energy of 2616 kilojoules per kilogram and we can thus assume that the water is a superheated vapor. So at 1000 kilo Pascal the internal energy at 200 degrees Celsius is 2622.3 kilojoules per kilogram. We also know that the internal energy at 1200 kilo Pascal and 200 degrees Celsius is 2612.9 kilojoules per kilogram. I will show in a later video how I got these values. We thus need to interpolate again and we get a value for pressure of 1134 kilo Pascal for water at 200 degrees Celsius and 2616 kilojoules per kilogram of internal energy. Now let's have a look at an ideal gas example. An ideal gas consists of hard spheres that collide elastically with each other and the walls of the container in which they are in. There is no interparticle attraction or repulsion, therefore there is no energy required to bring the particles closer together or pull them further apart. The internal energy of an ideal gas is dependent solely on the kinetic energy of the individual particles and internal energy is therefore dependent only on the temperature of the gas. The hotter the gas the higher the internal energy. Let's show this with an example. The value for internal energy of an ideal gas like air is found in tables such as the one shown here. You will note that internal energy is only a function of temperature. We can therefore determine the internal energy of air at 300 Kelvin which is 214.07 kilojoules per kilogram. We can now use the ideal gas law to determine the specific volume of the air using the value of pressure given to us as 200 kilo Pascal and we get a value of 0.4306 cubic meters per kilogram. As air is a single phase system it means if I specify the value of two independent intensive variables I can determine the value of the others as we just did above. Note however that internal energy and temperature for an ideal gas are not independent from each other. If I were asked to determine the specific volume of air at 300 Kelvin and internal energy of 214.07 kilojoules per kilogram I would not have been able to do it as internal energy and temperature are not independent. They are a function of each other. Now let's look at an example of a perfect gas. We can assume that the specific heats of mono atomic gases such as helium are constant and does not depend on temperature. We therefore can use the equation internal energy at temperature 2 minus the internal energy at temperature 1 equals the specific heat times the difference in temperature and this will give us the change in internal energy. Let's do an example to illustrate this. The temperature of argon changes from 300 Kelvin to 100 degrees Celsius. How much does the internal energy change? The answer is 22.84 kilojoules per kilogram. The temperature changes by 73.15 degrees Celsius or Kelvin. Remember the value of temperature difference expressed in Celsius or Kelvin is the same. We can also assume that these gases will obey the ideal gas law. We call ideal gases with constant specific heats perfect gases. So in summary we introduce the concept of internal energy as the sum total of all the microscopic forms of energy. Internal energy as an intensive variable its value can therefore be used with another intensive variable to fix the state and calculate the values of the other intensive variables. For an ideal gas internal energy is a function of temperature only and fixing the value of t and u does not fix the state as they are interdependent. For mono atomic gases where the specific heats is not a function of temperature we can use the equation delta u equals the specific heat value times delta t and lastly perfect gases are ideal gases whose specific heats are constant and that's it. The course notes which this video is based on is available on my website odromsblock.com. I'm also on Twitter. My Twitter handle is at asv90 and you can connect with me and ask any questions and I will gladly answer them. Thank you very much for watching and I will see you in the next video. Bye.