Dimensions in Heat, Waves and Modern Physics

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. Think of a big bucket of water on a stove. Its heat capacity tells you the heat it takes per kelvin. Now think of Boltzmann's constant, k B, the constant that fixes the kelvin itself. One belongs in a kitchen, the other in the definition of a base unit. Can they have exactly the same dimensions? Sounds impossible, right? By the end of this class, you will see why they do. And look, in JEE, dimension questions love these constants. So, let's begin.

0:37New base symbols: kelvin and mole

So, in mechanics we needed only three base quantities: mass, length and time, written M, L and T, okay? These square brackets on screen mean, the dimensions of. But today we go into heat, waves and modern physics. Tell me, the moment we talk about heat, which new base quantity shows up? Temperature, yes! Exactly. Temperature gets its own symbol, K, for kelvin. And the amount of substance, counted in moles, gets its own symbol too, mol. Both are base quantities, so they never break into M, L and T, clear? Next, heat itself. Heat is just energy, so its dimensions are M L squared T to the power minus two. Write this down. You will use it again and again today. And the old rule still holds: pure numbers, like two, pi, or three by two, have no dimensions. So just drop them, okay?

1:31Expansion: the lengths cancel, the kelvin stays

Now, let's try it on a metal rod that grows longer when heated. Delta l equals l, into alpha, into delta T, okay? Delta, the little triangle, means change, and alpha, this Greek letter, is the coefficient of linear expansion. Now you tell me. Alpha is delta l, divided by l into delta T, okay? Length on top, length at the bottom. So what is left? Only temperature, yes! Length cancels length, and alpha is K to the power minus one, simply per kelvin. So any expansion coefficient, per kelvin, always gives K to the minus one. Really simple, right? One more from the table: thermal conductivity, written capital K, okay? Its dimensions are MLT to the power minus three, K to the power minus one. But look, a sneaky trap. The quantity is capital K, and the kelvin inside the bracket is written K too! Same letter, two meanings. Read every bracket carefully.

2:33Radiation: Stefan's and Wien's constants

Alright, now radiation. The energy radiated per unit area, per unit time, equals sigma into T to the power four. Sigma, this Greek letter, is Stefan's constant. So sigma is energy, divided by area, divided by time, divided by T to the four. T to the four at the bottom, so the answer has T to the power minus four, right? No, no, careful! This is the JEE trap. In this formula, capital T is temperature, not time! So the bottom is area, then time, then K to the power four. L squared cancels, the extra time makes T to the minus three, and sigma is M T to the power minus three, K to the power minus four. Got it? And Wien's constant: lambda m, a wavelength, into temperature, equals b. This arrow means, so. So b is length into temperature, L K, okay? Easiest one in the table!

3:32Waves and modern physics

Next up, waves. Intensity is power per unit area. Power is M L squared T to the power minus three; divide by L squared, and intensity is M T to the power minus three, okay? Wave number is one by lambda, one over the wavelength, so simply L to the power minus one. Clear? Now the Rydberg constant, capital R, okay? One by lambda equals R, into one by n one squared, minus one by n two squared. What are the dimensions of that bracket? None, exactly! n one and n two are plain numbers, so the bracket is a pure number. So R has the dimensions of one by lambda: L to the power minus one, just like wave number. Simple as that. And modern physics: Planck's constant, small h, is M L squared T to the power minus one. Now think back to the last class. Which quantity had exactly these dimensions? Angular momentum, yes! Absolutely right. JEE loves to pair these two in match the column questions, okay? Quick check with de Broglie: wavelength lambda equals h by p, where p is momentum. M L squared T to the power minus one, divided by MLT to the power minus one, leaves just L, a length! Exactly what a wavelength must be. So dimensions can also check a formula.

4:55Example 1: specific heat, latent heat and heat capacity

Time for Example one. Find the dimensions of specific heat, molar heat capacity, latent heat and heat capacity. Hint: write each formula, and remember, heat is energy. Pause the video, and try it first. Done? Okay, let's check. Specific heat, small c, comes from Q equals m c delta T, okay? So c is energy divided by mass into temperature. M cancels M, and c is L squared T to the power minus two, K to the power minus one. Molar heat capacity is Q divided by n delta T, with n the number of moles. No mass cancels now, so it is M L squared T to the power minus two, K to the power minus one, mol to the power minus one. Latent heat comes from Q equals m L, okay? Careful: this capital L is latent heat, not length! Latent heat is Q by m, energy per mass, so L squared T to the power minus two. No kelvin here, because there is no delta T in this formula. And heat capacity is simply Q by delta T: M L squared T to the power minus two, K to the power minus one. Notice, this is our bucket of water from the start. Keep this answer in mind.

6:12Example 2: gas constant R and Boltzmann constant

Now a slightly harder one, JEE level, and it solves our puzzle. Find the dimensions of the gas constant, capital R, from p V equals n R T, and of Boltzmann's constant, k B, from E equals three by two k B T, okay? Pause and try. Hint: first find the dimensions of p into V. Done? Okay, let's check. Pressure is M L to the power minus one, T to the power minus two, and volume is L cubed. Multiply: M L squared T to the power minus two. Oh, that is energy! So R equals p V divided by n T, okay? Energy by temperature, so just energy per kelvin, right? No, no, wait! What about n? n is the number of moles, and it carries its own dimension, mol. Don't drop it! So the bottom is mol into kelvin, and R is M L squared T to the power minus two, K to the power minus one, mol to the power minus one. Now Boltzmann's constant. Here E, an energy, equals three by two into k B T, okay? Three by two is a pure number, so drop it. k B is energy by temperature: M L squared T to the power minus two, K to the power minus one. In fact, its unit is joule per kelvin. Now compare with Example one. Boltzmann's constant and heat capacity: exactly the same dimensions! Puzzle solved. Both are simply energy per kelvin. One describes a bucket of water, the other fixes the kelvin itself. Dimensions don't care about size, only about what is divided by what. Beautiful, isn't it?

7:53Quick revision: kelvin, mole and the letter traps

So, quick revision. Heat is energy, temperature brings in K, and moles bring in mol. Every per kelvin gives K to the power minus one, and every per mole gives mol to the power minus one. And watch the letters: the T in sigma T to the four is temperature, and capital K can be thermal conductivity. Now go and try the practice problems on this page. You will find them easy, I promise. All the best!