Molar Heat Capacity of a General Process

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. You keep giving heat to a gas, and its temperature goes down. Impossible, right? Heat goes in, so the gas should warm up. Well, it really can happen, and by the end of this video you will see when. Our topic is the molar heat capacity of a gas in any process. It comes up often in JEE, so let's begin.

0:26Heat capacity depends on the process

So, what is the heat capacity of a process? Capital C equals d Q over n d T: d Q is a small amount of heat given to the gas, d T the small rise in temperature, and n the number of moles. So C is the heat per mole per kelvin, in that particular process. Okay? Now think. We already know two values for one gas: C v at constant volume, and C p at constant pressure. So does a gas have just one heat capacity? No, it has one for every process! In the course's sign convention, Q is the heat given to the gas, and W the work done by it. So the first law, for a small step, says d Q equals d U plus p d V; small p is the pressure, and d V the small change in the volume, capital V, right? Capital U, the internal energy of an ideal gas, depends on temperature alone. So d U equals n C v d T in every process; C v, the C with a small v below, is the molar heat capacity at constant volume. With me? Now divide each term by n d T, and see what we get. The d U term becomes C v, and the work term becomes p over n, times d V over d T, so C equals C v plus that extra term, for any process. Clear?

1:52Polytropic process: p V to the power x constant

Now, many questions use a special family called polytropic processes. Here p times V to the power x stays constant; x, the small letter up in the power, is a fixed number. Okay? Put this into our general formula, and the extra term becomes R over, one minus x, with capital R the gas constant. So C equals C v plus R over, one minus x: the formula you will use the most. Got it? And C v equals R over, gamma minus one; gamma, the Greek letter, is C p over C v. So C also equals R times the bracket: one over, gamma minus one, minus one over, x minus one. Notice, it is x minus one here, not one minus x, so the sign flips. Right?

2:40Four familiar processes, one formula

Let's test this on four processes you already know. In the figure, small p goes up, capital V goes across, and all four start from one state, p naught, V naught; the small zero means the starting value. Flat line, isobaric; vertical line, isochoric; and the two curves, isothermal, p V constant, and adiabatic, p V to the power gamma constant. Okay? Isobaric first: p times V to the power zero is constant, so x is zero. R over, one minus zero, is just R, so C is C v plus R, which is C p, the C with a small p below, the molar heat capacity at constant pressure. Exactly right! With me? Now isothermal: p V is constant, so x equals one. Put x equals one into R over, one minus x. What happens to C? We are dividing by zero, so C blows up! The arrow means tends to, and the sideways eight is infinity. It makes sense: heat goes in, but the temperature stays fixed, so d T is zero and the heat per kelvin is infinite. Clear? Adiabatic: here x equals gamma. Then R over, gamma minus one, plus R over, one minus gamma: these cancel, giving exactly zero. Of course: no heat flows in an adiabatic process, so d Q is zero, and C is zero too. Got it? And isochoric, the vertical line: the volume is fixed, so d V is zero and C equals C v. One formula, all four processes. Really neat, isn't it?

4:23When x is between 1 and gamma: negative heat capacity

Now, back to our puzzle. Here is the second form of our formula once more, okay? Take x between one and gamma; this pointed sign means less than. Then x minus one is smaller than gamma minus one, so one over, x minus one, is bigger; the sign pointing the other way means greater than. Right? So in our formula, the bigger fraction is the one we subtract. What is the sign of C? It is negative, yes! For x between one and gamma, the molar heat capacity is less than zero. Strange, but true, okay? As the gas expands, it does take in heat, so Q is positive. But it does even more work than the heat it receives: W is greater than Q, clear? So by the first law, delta U, the little triangle meaning change in, equals Q minus W, which is negative. The internal energy drops, so the temperature falls while heat flows in. That is our puzzle solved: heat in, temperature down! Got it?

5:27Example 1: monatomic gas, pV squared and p proportional to V

Alright, time for examples. Find the molar heat capacity of a monatomic gas in the process p V squared equals constant, and in the process p proportional to V; that sign, like a stretched alpha, means proportional to. Pause the video and try it first. Done? Okay, let's check. The gas is monatomic, so C v is three over two R, right? First process: p V squared is constant, so x equals two. C equals three over two R, plus R over, one minus two. That last term is minus R, so C is one half R, okay? Now the second one, where p is proportional to V, so both appear with power one. Then x equals one, right? No, no, careful here! That is the trap. p proportional to V means p over V is constant, that is, p times V to the power minus one. So x is minus one; x equals one is the isothermal case, remember? So C equals three over two R, plus R over, one minus, minus one. One minus minus one is two, so that term is one half R, and three over two R plus one half R gives two R in total. Got it?

6:48Example 2: diatomic gas, find the process with C = minus R

Now a harder one, and it connects to our puzzle. In what polytropic process does a diatomic gas have a molar heat capacity of minus R? Pause the video and try it first. Hint: put C equals minus R into our formula. Done? Okay, let's check. For a diatomic gas, C v is five over two R, so five over two R, plus R over, one minus x, equals minus R, with me? Now cancel R from every term, and move the five over two across. What is one over, one minus x? Yes, exactly: minus one, minus five over two, that is minus seven over two. The double arrow means so: flip both sides, and one minus x equals minus two over seven. Clear? So x equals one plus two over seven, that is nine over seven. The process is p times V to the power nine over seven, equals constant. Okay? Quick check against our puzzle: for a diatomic gas, gamma is seven over five, one point four, and nine over seven is about one point two nine. So x lies between one and gamma, exactly where C is negative. Our answer fits, right?

8:06Quick revision: C formula, special cases, negative C

So, quick revision. For any process, C equals C v plus p over n, times d V over d T, from the first law. For p V to the power x constant, C equals C v plus R over, one minus x. Isobaric gives C p, isothermal infinity, adiabatic zero, and for x between one and gamma, C is negative. Now go and try the practice problems on this page. All the best!