# U(T) for Ideal Gases

#### Ideal Gases

:
Increasing T increases molecular vibrations, rotation and translation
Therefore, U increases as well.
Changes in P do not affect U because of the absence of molecular interactions.
U = U(T)
U as a Function of T at Constant P
U
T
Internal energy is a strong function of temperature.
U as a Function of P at Constant T
U
P
Internal energy is not a function of pressure.
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### Ch 3, Lesson A, Page 3 - U(T) for Ideal Gases

• First we are going to talk about “real” substances.  Later, we’ll discuss a few special cases that are a bit simpler.
• Vibrational, rotational and translational energies are all lumped into a term called the internal energy.
• The symbol for internal energy is U and U-wiggle is the internal energy per mole and U-hat is the internal energy per kg.
• Internal energy is all of the energy associated with the molecules that make up a substance.
• So, U also includes the chemical energies associated with the bonds that make the molecules.
• The good news is that, in this course, we will generally NOT consider systems in which chemical reactions take place.
• Therefore, the chemical component of the internal energy will not change.
• That leaves with ONLY the vibrational, rotational and translational energies !
• That’s cool because as long as we don’t look at chemical reactions, any changes in U must be caused by changes in the energies associated with these 3 modes of molecular motion.
• OK, so then it shouldn’t be a surprise that U increases strongly as the temperature increases, right ?
• The molecules substances at higher temperatures are in a higher energy state and as a result their molecules are vibrating, rotating and translating more energetically.
• I hope the relationship between internal energy and temperature is clear.
• So what does pressure have to do with internal energy ?
• It’s tough to construct a mental model that explains why changes in pressure at constant temperature cause the internal energy to change.
• The good news is that U is a weak function of P.  This means that the pressure must change by a large amount before U changes by a significant amount.
• The graph shown here illustrates this point.
• You can see here that U decreases slightly with increasing pressure.
• But that is NOT always the case.  At certain pressures and temperatures U will actually increase slightly as pressure increases.
• The explanation for this phenomenon is beyond the scope of this course.
• The main thing to remember is that, for real substances, U is a strong, increasing function of T and a weak function of P.