Starting from the relationship between temperature and kinetic energy for an ideal gas, find the

Starting from the relationship between temperature and...

Key Concepts

Ideal Gas

An ideal gas is a theoretical construct where particles are assumed to have no volume, experience no intermolecular forces, and undergo perfectly elastic collisions. This model is crucial for deriving many thermodynamic properties because it simplifies the behavior of gases compared to real substances, especially when interpreting relationships between temperature, pressure, volume, and energy.

Kinetic Theory of Gases

The kinetic theory of gases explains that the temperature of an ideal gas is directly proportional to the average kinetic energy of its particles. This principle provides a microscopic understanding of thermodynamic properties and serves as a basis for deriving formulas for quantities like pressure and heat capacities.

Molar Heat Capacity at Constant Volume

The molar heat capacity at constant volume (Cv) refers to the amount of heat required to raise the temperature of one mole of a substance by one degree while keeping the volume constant. For an ideal gas, this is directly linked to the internal energy change, which depends solely on temperature due to the absence of work done associated with volume change.

Molar Heat Capacity at Constant Pressure

The molar heat capacity at constant pressure (Cp) is defined as the heat required to raise the temperature of one mole of a substance by one degree while maintaining constant pressure. In this scenario, not only does the internal energy change, but additional work is performed by the gas as it expands. This concept incorporates both the energy required to increase temperature and the work done by expansion, leading to a higher value than Cv.

Transcript

00:01 So recall from chapter 5 that the average kinetic energy of a system is going to be equal to three halves times the number of moles, times the gas constant, times temperature.

00:16 Now, one thing to keep in mind here is that any time you have a change, so let's say we have a change in the energy, i can go ahead and write here that this is the energy of the system, is just the energy of the particle.

00:31 Rather, the kinetic energy and the energy are the same thing.

00:34 So this would then be three halves times the number of moles doesn't change.

00:40 So there's nothing changing there.

00:41 R is a constant times the change in temperature.

00:46 So this tells me that the change in the internal energy of the system is dictated by the change in the temperature.

00:54 Right.

00:55 So one thing to note here, if we have a system where the volume doesn't change, the volume is held constant, another way to write that is that the volume, or rather the change in volume is equal to zero.

01:07 Recall that work is equal to negative pressure times the change in volume.

01:13 In this instance, work is just equal to zero.

01:17 So from there, we have another version of an equation that involves energy, and that's heat plus work.

01:25 Since this is zero, we can just say it's equal to heat.

01:30 So i'm going to then substitute this equation over here.

01:35 And what you find is that heat or q is equal to 3 halves nr delta t for a system where the pressure is changing but the volume is held constant.

01:49 I'm going to rewrite this slightly to say that it is n times three halves are times the change in temperature.

02:00 Now, this might look a little similar to something we've seen before.

02:04 We've seen something that looked like we've seen q is equal to m -c -delta -t, right? where m is mass, c is the specific heat.

02:15 We've also seen this in terms of n molar heat times delta -t.

02:22 And this maybe looks a little bit more familiar where the ends match up.

02:27 But then this, so this quantity here, here is actually serving as the heat capacity.

02:35 So one thing we can then write is that c, so the molar heat capacity at constant volume, excuse me, constant volume is equal to three halves r.

02:51 And that kind of makes sense, because if you look at the units of r, units of r is really equal to, it once you, there's a few different ways to write r.

03:01 But when we're talking about it in terms of kinetic energy of gases, we write the units as joules per mole kelvin, which looks oddly similar to the specific heats or the molar heats that we've been dealing with.

03:18 So it kind of makes sense that this would be our specific or rather our molar heat capacity for a system at constant volume...

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