 The problem reads, the bulb and capillary of a mercury thermometer is completely filled with mercury when the temperature at a pressure of one bar reaches 50 degrees Celsius. Estimate how much pressure is established inside the thermometer when the temperature rises to 52 degrees Celsius. The thermal expansion coefficient of mercury is 1.8 10 to the minus 4 1 over Kelvin and its isothermal compressibility is 3.9 10 to the minus 6 1 over bar. So first let's write down what we have. At the beginning, level one, we have P1 equals 1 bar and T1 equals 50 degrees Celsius and we have V1 is equal to some V. Why do we write this down? The key part is that it's completely filled at 50 and so the same volume is present at 52 degrees. So that's why we write V1 equal to V because on level two we have P2 equals, we're looking for that. We have T2 equals 52 degrees Celsius and we have V2 equals the same V. So this is the same. That makes delta V equal to zero. That makes delta V equal to zero. Now the key to doing this problem is to know what these are. The thermal expansion coefficient, usually denoted by alpha, the thermal expansion coefficient. Here we have that it's 1.8 10 to the minus 4 times 1 over Kelvin. Its formula is 1 over V delta V delta T. Notice that the V units cancel and the T is like that. So this is under a constant pressure and the unit is 1 over K. On the other hand, this isothermal compressibility is usually denoted by beta. That's 3.9 10 to the minus 6 1 over bar and you have to remember the minus here. 1 over V delta V and now delta P. The temperature is constant and that's 1 over bar. So the idea is how do you relate this? You can know the formula that says 4 delta V equal to zero alpha over beta equals delta P over delta T. We would have delta P is P2 minus 1 and delta T because it's a difference and the degrees are the same in Kelvin and Celsius would be 2. So we would substitute alpha beta P2 minus 1 and 2 and get P2. Let's see how this came about just for a few minutes. Notice that this has delta V delta T delta V delta P. So it's part of a total differential. The total differential would be dV equals delta V delta T at a constant pressure times dt plus delta V delta P at a constant temperature dP. This is the total differential. It says just as a standard use of differentials. And the idea is that you move this into the delta notation so that we would have delta V equals dV dt P delta T plus dV dt T delta P. And we already decided that this is equal to zero. So then we can multiply through by anything we want to. Obviously we're going to multiply through by 1 over V. So equals 1 over V plus 1 over V. And what we get is zero equals alpha dt minus beta dP. And if we solve this we will get exactly this. So that's where this formula comes from. Now let's use that formula. So what we have is delta P equals P2 minus 1 and delta T and delta T equals 2 degrees. It doesn't matter that it's Celsius or Kelvin. And alpha and beta are the right units. So we have, here we have 1.8 times 10 to the minus 4 divided by 3.9 10 to the minus 6 equals P2 minus 1 over 2. So what is P2 equal? So P2 equals 2 times 1.8 times 10 to the minus 4 divided by 3.9 10 to the minus 6. That's that. And then we just add that 1 plus 1. Get our calculator out. So we have 2 times 1.8 ee minus 4 divided by 3.9 ee minus 6. I'm going to hit Enter and then I add the 1. So 93.3. So P2 equals 93.3 and we had bar. This is the answer.